Deforming hypersymplectic four-manifolds to hyperkähler triples

Proof. Use the fixed-class path for the original basis described above and put \alpha_t=\sum_i c_i\omega_i(t). Positivity of the pointwise wedge-product matrix gives \alpha_t\wedge\alpha_t>0. Thus \alpha_t is symplectic, and its class is fixed because each [\omega_i(t)] is fixed.

For the endpoint metric g, set r_c=(c^{\mathsf T}Gc)^{1/2}>0. The endpoint identities show that \alpha_1/r_c is parallel and self-dual with g-norm \sqrt2. It therefore defines a parallel integrable complex structure J^{\mathrm{end}}_c with \alpha_1/r_c=g(J^{\mathrm{end}}_c\,\cdot,\cdot). Hence \alpha_1 is its Kähler form for the hyperkähler metric r_cg.

Since X is closed, Moser’s theorem gives an isotopy \phi_{c,t} with \phi_{c,0}=\mathrm{id} and \phi_{c,t}^*\alpha_t=\alpha_c. Pulling back r_cg and J^{\mathrm{end}}_c by \phi_{c,1} gives the claimed hyperkähler metric and compatible integrable structure on X. Fine and Yao note this Moser implication for a constituent in [10, Section 1].□

Proof. Give X the orientation induced by the triple. The normalization above and Theorem 1.1 give a global hyperkähler triple on X. Choosing one of its complex structures makes X a compact Kähler surface with a nowhere-vanishing holomorphic two-form. The compact hyperkähler surface classification [2, Section 2, p. 162] therefore identifies X itself as a complex two-torus or a K3 surface. A complex two-torus is smoothly T^4, while every complex K3 surface is diffeomorphic to a nonsingular quartic surface in \mathbf P^3, the standard K3 manifold [23, p. 58].□

Proof. Ratios of positive four-forms are smooth functions. Set

Then AT=0 and T^2=A^2>0. Consequently

The form \Omega is complex decomposable and its kernel in the complexified tangent bundle is complementary to its conjugate. Taking this kernel as T^{0,1} defines an almost-complex structure with the given orientation. Replacing y by -y conjugates \Omega and changes J to -J. The real and imaginary parts of \Omega span the anti-invariant forms, and \Omega is a nowhere-zero section of the canonical bundle.

Choose any Hermitian metric for J. The invariant two-forms are the wedge-orthogonal complement of \operatorname{span}(A,C) and have signature (1,3). For the decomposition B=B^{1,1}+B^{2,0+0,2}, the positive-triple condition is exactly (B^{1,1})^2>0. A real (1,1) form in complex dimension two has positive square precisely when its two Hermitian eigenvalues have the same strict sign. The positive component is the cone of forms positive on every complex line. The anti-invariant summand vanishes on such lines, so this is equivalent to taming by B. The other component is positive for -J. Connectedness fixes the choice continuously and globally.

Finally, for independent real vectors u,v, A(u,v)=C(u,v)=0 is equivalent to \Omega(u,v)=0. In a complex two-dimensional vector space with a nondegenerate complex volume form, this says that u,v are complex dependent, equivalently that their real span is a complex line.□

Proof. By Lemma 2.2, one member of the triple is a symplectic form taming an almost-complex structure whose canonical bundle is trivial. The first Chern class of this symplectic form therefore vanishes: the space of its tamed almost-complex structures is contractible and contains compatible structures. Bauer’s bound for a closed symplectic four-manifold with torsion first Chern class gives b^+\le3 [1, Corollary 1.2]. The three independent positive cohomology classes give the reverse inequality.

The characteristic-number identity c_1^2=2\chi+3\sigma, obtained from p_1=c_1^2-2c_2, the Euler-number identity and the signature theorem [33, Corollaries 11.12 and 15.5, Theorem 19.4], together with b^+=3, gives

Moreover w_2\equiv c_1=0\pmod2, so X is spin [33, Problem 14-B]. Rokhlin’s theorem makes \sigma divisible by 16 [42]. Since b^-\ge0, these relations give b_1\in\{0,4\} and the displayed alternatives. Poincaré duality makes the free intersection form unimodular, and the Wu formula makes it even [33, Theorem 11.14]. Classification of indefinite even unimodular lattices now gives the two stated forms [43, Chapter V, Section 2.2, Theorems 5–6]. Only the intersection lattice has been classified at this stage.□

Proof. The first assertion follows from the construction and the self-dual description of Hermitian almost-complex structures in dimension four. At a fixed point, let K be the positive Gram matrix of the triple for g_0. A vector normal to the coefficient plane v^\perp, for this Gram matrix, has coefficients K^{-1}v. Its pairing with B_v is positive. Thus, after normalization, the fundamental forms give the map

in radial coefficient coordinates. Positive definite matrices connect to the identity through positive definite matrices, so this map has degree one. The orientation convention identifying coefficient and twistor spheres can change only its sign.

A closed anti-invariant form is self-dual for g_0, hence harmonic. By (2.2) it is a constant combination of the original triple. It is anti-invariant for J_v exactly when that coefficient vector lies in v^\perp. This proves the final assertion.□

Proof. Set \Omega=A+iD and use its kernel as T^{0,1} as in Lemma 2.2. For vector fields U,V in that kernel, Cartan’s formula and \dd\Omega=0 give

The kernel is involutive. The smooth Newlander–Nirenberg theorem therefore gives integrability [34], and closedness of the (2,0) form \Omega gives holomorphicity.

Under the additional assumptions, C is a positive closed (1,1) form, hence a Kähler form. Its metric volume is C^2/2=A^2/2. The ratio \Omega\overline\Omega/C^2 is constant, so the holomorphic volume form has constant pointwise norm. The Chern connection of the canonical line then makes \Omega parallel. For a Kähler metric this is the connection induced by the Levi-Civita connection. Thus A,D,C are parallel; their normalized self-dual algebra gives the hyperkähler structure. Explicitly, if (\eta_1,\eta_2,\eta_3)=(A,D,C) and \mu=A^2/2, then \eta_i\wedge\eta_j=2\delta_{ij}\mu; thus their normalized matrix \mathcal Q from the introduction is the identity.□

Proof. For (i), apply separation of an open convex set and a disjoint linear subspace in the real locally convex space C^\infty(X,I). The separating continuous functional vanishes on \mathcal Z_J and is nonnegative on its positive cone. Extend it to all two-forms by precomposing with 1-R. Adding a positive multiple of G and taking a limit proves nonnegativity on every semipositive invariant form.

For (ii), choose a smooth closed representative \beta of H and separate the open cone \mathcal P_J from the affine subspace \beta+\dd\Omega^1(X) in \Omega^2(X). If N is the separating functional, the unbounded translation directions in this affine subspace give N(\dd\alpha)=0. Every positive form can be multiplied by any positive scalar, so the separation inequalities imply both N(\mathcal P_J)\ge0 and N(\beta)\le0. Adding an arbitrary real multiple of an anti-invariant form to an element of \mathcal P_J shows that N annihilates anti-invariant forms. The same limiting argument gives positivity on semipositive invariant forms.

For either functional, comparison of an invariant form with sufficiently large positive multiples of G gives

If N(G) were zero, this inequality and anti-invariance would make the functional zero. Normalize N(G)=1. Locally, the functional is therefore represented by a positive Hermitian matrix of measures. Its trace is a positive measure \mu; the Radon–Nikodym matrix relative to \mu is positive semidefinite with trace one. A measurable spectral decomposition of this 2\times2 matrix expresses it as a convex combination of rank-one Hermitian projections. Borel local frames and a Borel partition of X give the global measure \lambda in Equation (3.4). Finally H^3\hookrightarrow C^0 in real dimension four, so every such measure coefficient belongs to H^{-3}.□

Proof. On sections of E consider P=R\dd\dd^*. If a is self-dual, then the self-dual projection of \xi\wedge\iota_\xi a is |\xi|^2a/2: in a frame with first covector \xi/|\xi|, the two self-dual halves of a have equal length. Hence

Thus P is elliptic, self-adjoint, and nonnegative. Its kernel consists of smooth sections with \dd^*a=0. Since a=*a, these sections also satisfy \dd a=0, and hence are harmonic. Conversely a closed anti-invariant form is self-dual and therefore harmonic and in the kernel.

Let H_0 be the orthogonal projection onto this finite-dimensional kernel, and let L be the inverse of P on its orthogonal complement, extended by zero on the kernel. Elliptic Fredholm theory and the parametrix give L as a classical operator of order -2, with H_0 smoothing and PL=\Id-H_0; see [52, Sections 4–5 and 10]. Indeed, if B_0P=\Id+K is a parametrix identity with K smoothing, then B_0(\Id-H_0)=L+KL. Elliptic regularity makes KL smoothing, which gives the asserted order of L. Set

The image of both summands is closed, and RD=PL+H_0=\Id. This proves Equation (3.5).

For closed \xi, the form \xi-DR\xi is closed and invariant, so T(\xi)=N(\xi-DR\xi)=0. The functional Q only depends on R\xi, which proves its anti-invariance and self-duality. Its H^{-3} membership follows because an order-zero operator and its adjoint act boundedly on every Sobolev space [52, Proposition 5.5]. The vanishing on exact forms makes T closed. Its pairings with all closed two-forms vanish, so Poincare duality, or the harmonic projection of *T, gives [*T]=0.

For the family assertion, identify the varying bundles locally in the parameter and use a smooth unitary identification of the L^2 spaces. The elliptic operators then have common domain H^2 in a fixed Hilbert space. Choose a spectral contour enclosing only zero at one parameter value. The resolvent identity and elliptic regularity give a smooth family of resolvents on this contour after shrinking the parameter neighborhood. Its spectral projection has locally constant rank. Since the kernel dimension is constant with that rank, the enclosed spectral subspace consists exactly of the kernel. Thus H_0 depends smoothly on the parameter, and so does

on every Sobolev scale. Equation (3.7) gives the required smooth lifts.□

Proof. We use two precise facts of the order-zero calculus: in local left quantization its symbol is bounded uniformly in the covariable, and its off-diagonal kernel satisfies |K_A(x,y)|\le C\dist(x,y)^{-4} for small positive distance. Smooth remainders have uniformly bounded kernels. These statements hold for bundle-valued classical operators, after finitely many coordinate localizations; see [52, Section 2, Propositions 2.1–2.2].

For a fixed evaluation point x, split the input using a smooth cutoff supported in B_{c\delta}(x) and equal to one on B_{c\delta/2}(x), with a fixed small c. On its support s+\dist(p,y) is comparable to \delta. After translation and dilation by \delta, the localized input has derivatives through order six bounded by C\delta^{-a}. Integration by parts in its Fourier transform therefore bounds its Fourier L^1 norm by C\delta^{-a}, since (1+|\xi|)^{-6} is integrable in dimension four. The bounded symbol gives the same bound for the local contribution to Aa_{s,p}(x).

The remaining part has distance at least c\delta/2 from x. Its portion in B_{C\delta}(p) contributes at most

On a shell of radius r=2^j\delta\ge C\delta about x, distance from p is comparable to r, so its contribution is at most Cr^{-4}r^4r^{-a}=Cr^{-a}. The shells up to the fixed coordinate radius sum to C\delta^{-1} if a=1, and to C(1+|\log\delta|) if a=0. The rest is uniformly bounded. If x stays away from p, the singular input near p has uniformly bounded L^1 norm and the kernel there is bounded; the remaining input is uniformly smooth. This proves the final assertion.□

Proof. Fix first a radius \rho_0>0 for uniform normal coordinates and a smooth radial cutoff which is one on B_{2\rho_0}(p) and supported in B_{3\rho_0}(p). This cutoff will not change when a smaller inner radius is chosen below. The logarithmic potential will detect the mass of B_s(p), while a square-root potential will dominate the inverse-distance errors caused by varying J and by the closed lift. With t=\dist(p,\cdot) define the globally smooth cut-off functions

Smoothness at p follows from the dependence on t^2 and from s>0. For a function f, put

In setup (A), \beta(f) is closed and invariant, and in setup (B) it is exact. Hence N(\beta(f))=0 in both cases.

The principal symbol of \dd\dd_J^c on functions is a multiple of \xi\wedge J^*\xi, which is J-invariant. Thus R\dd\dd_J^c is a first-order operator. Differentiating the radial functions gives, for 0\le j\le6,

On the cutoff annulus these inputs and their derivatives are uniformly bounded. Lemma 3.3 applied to D therefore bounds the two correction terms by C/(s+t) and C(1+|\log(s+t)|) respectively.

For clarity, the positive part of the test can be computed in a fixed Hermitian Euclidean space. If f=f(t) and \ell is a unit complex line, its Hessian trace is

where a is the squared length of the radial unit vector projected onto the line. For the two functions without cutoff this gives

In normal coordinates g-g_p=O(t^2) and J-J_p=O(t), while the first derivatives of J are bounded. Replacing the frozen complex line and operator by the actual ones changes these estimates by at most C/(s+t) for u_s and C for h_s. Indeed the Hessian errors are bounded by Ct|\nabla^2f| and the coefficient-derivative errors by C|\nabla f|. With \delta=s+t, it follows that on B_{\rho_0}(p)

The inequalities include setup (B), where the correction terms are absent.

All constants just obtained use the already fixed cutoff. Now choose 0<\rho<\rho_0 so small that

Next choose a fixed multiplier M\ge2C_1/c_2. If s\le\rho and t\le\rho, Equations (3.12) and (3.13) give \beta(u_s+Mh_s)(\ell)\ge c_1s^2\delta^{-4}. On X\setminus B_\rho(p) all terms are bounded independently of s, including the nonlocal corrections, by the last assertion of Lemma 3.3. Since \delta\le2s on B_s(p), we conclude globally that

Apply N and use N(\beta(u_s+Mh_s))=0 and \mu(X)=1 to obtain Equation (3.9) for s\le\rho. For larger s the same assertion follows from \mu(X)=1. Compactness makes all choices uniform in p.□

Proof. The spectral theorem gives the operator identity

We recall the precise heat-kernel facts being used for the nonnegative Hodge Laplacian on the closed manifold. Its local small-time kernel has a parametrix

with arbitrarily high-order uniform remainder after increasing m; see [30, Theorems 1.1 and 3.1]. Here \chi is supported inside the injectivity radius and equals one near the diagonal. For the Hodge Laplacian a_0(y,z)=j(y,z)^{-1/2}\tau_{z\to y}, with j smooth, strictly positive, and equal to one on the diagonal [29, Section 3, equation (3.4), author version]. After subtracting the leading term, for every prescribed L the kernel is bounded, for 0<t<t_0, by

The local expansion gives the Gaussian part of this bound; away from the diagonal, the global Gaussian estimate makes the cutoff error smaller than every power of t [30, Theorem 3.5]. For t\ge t_0, the heat kernel is uniformly bounded: factor the heat operator through two fixed positive-time smoothing operators and the L^2 contraction e^{-(t-t_0)\Delta}. This last assertion uses the nonnegativity and self-adjointness of the Hodge Laplacian.

Take \chi\ge0 and use its leading coefficient to define

For d<s, integration over 1\le h\le2 gives Equation (3.16). For every fixed k, the elementary inequality e^{-r^2/(ch)}\le C_k(1+r)^{-k}(1+h^{k/2}) shows that the last integral is bounded by C_k(1+d/s)^{-k}. This proves Equation (3.17).

Integrating the t^{-1} heat-kernel error in Equation (3.19) gives

For the uniform remainder, integration over s^2h<t_0 gives C_Ls^{2L}. Since d\le\operatorname{diam}X, this is bounded by the right side of Equation (3.18) when 2L\ge k-2. On s^2h\ge t_0, the factor e^{-h} makes both the heat-kernel contribution and the extended leading term smaller than every power of s. This proves the claimed error for each k.□

Proof. Parallel transport preserves the Hodge star. From Equation (3.2), direct expansion gives

For example, the right side equals -\langle\ell,\tau m\rangle+\tfrac12\langle G_y,\tau G_z\rangle, which is the left side after splitting into self-dual and anti-self-dual parts. The last squared difference is bounded by Cd^2, where d=\dist(y,z).

Express the pairing using the continuous kernel of S_s^2 and Equation (3.4). Its leading scalar is nonnegative, so Equations (3.16) and (3.24) supply the positive term. The Cd^2 term, using Equation (3.17) with exponent at least eight, and the error kernel, using Equation (3.18) with exponent six, are bounded in absolute value by

This is the asserted error. The norm estimate follows by using the absolute kernel bound in the unstarred pairing:

The kernel formulas are legitimate for measure coefficients because the kernel is continuous at fixed s; alternatively they follow by smooth approximation at that fixed scale.□

Proof. Set A_s=S_s^{-1}=(1+s^2\Delta)^3 and U=S_sW. Since \Pi A_sU=\Pi W=0, the distributional identity

holds. The commutator expands as a sum of \binom3j s^{2j}[\Delta^j,\Pi], 1\le j\le3. Because the principal symbol of \Delta is scalar, [\Delta^j,\Pi] is a differential operator of order at most 2j-1. The spectral theorem and elliptic Sobolev equivalence give

If B_j=[\Delta^j,\Pi], its formal adjoint has the same order, so

This proves Equation (3.25), including its meaning for distributional W by the bounded extension of the displayed operator.

Apply this result to R N'=0 and (1-R)Z=0. With u=S_sN' and v=S_sZ, the orthogonal splitting gives

Now use Equation (3.23).□

Proof. Let \mathcal A_s denote the double integral in Equation (3.28) and put x_s=\|S_sQ\|_2. Since Q is self-dual and S_s commutes with *, Equation (3.27) applied to T=N+Q gives

Use Equations (3.22) and (3.21) for the first term and Equation (3.26) for the second. The result is

The right side is independent of s. In setup (A) its intersection term is zero. In setup (B) that term is bounded uniformly for mass-one N: its harmonic coefficients are pairings with fixed smooth forms and are bounded by Equation (3.4). Completing the square in x_s bounds both x_s and s^{-4}\mathcal A_s.

As s\downarrow0, S_sQ converges to Q distributionally. A weakly convergent subsequence of the bounded L^2 family therefore has limit Q, proving Q\in L^2. Its anti-invariance is preserved in this limit, and the spectral theorem now gives S_sQ\to Q strongly in L^2.

Choose smooth anti-invariant Q_j with \|Q_j-Q\|_2\to0; for example apply R to a smooth approximation. By Equation (3.26) and the L^2 contraction property of S_s,

For fixed j, S_s^2Q_j\to Q_j in C^\infty, whereas N'(Q_j)=0. Therefore

The last convergence is uniform over the submeasures, because their trace masses are at most one. First let s tend to zero and then j to infinity to obtain Equation (3.29).□

Proof. Put \nu=\mu+|Q|\,dV_g. Cauchy–Schwarz and the four-dimensional volume bound give

so \nu(B_s(p))\le Cs^2, uniformly in p. Fix p, identify a normal-coordinate ball with a Euclidean ball, and freeze J_p and G_p as constant tensors J_0,G_0. If t=|\zeta|=\dist(p,\exp_p\zeta), define

These evaluations are well-defined because T has measure coefficients. Since G(\ell)=1, G-G_0=O(t), and Q annihilates the invariant form G, we have

Indeed, both error integrals are bounded by Cs^{-2}s\nu(B_s(p)).

Call a radius good if its sphere has zero \nu-measure. All but countably many radii are good. For good 0<r<s, closedness implies

For completeness, replace \log t inside B_s by

The difference \phi_s-\phi_r is compactly supported and C^{1,1}. Inside B_s, the quadratic part has dd^c_{J_0}(t^2/(2s^2))=2G_0/s^2. Expanding T(dd^c_{J_0}(\phi_s-\phi_r))=0 gives Equation (4.5). This use of C^{1,1} tests is justified by smoothing: for fixed positive r,s, their Hessians are uniformly bounded, converge away from the two spheres, and the spheres have zero \nu-measure.

The frozen form dd^c_{J_0}\log t is semipositive on J_0-complex lines and has norm at most Ct^{-2}. A J-complex line at distance t is O(t) from a J_0-complex line. Its evaluation can therefore be negative only by C/t. Likewise the projection of this form onto the actual anti-invariant summand has norm at most C/t, because its frozen anti-invariant projection vanishes. Thus

Quadratic growth, or a sum over the annuli 2^{-j-1}s<t\le2^{-j}s, bounds this integral by Cs. It follows that

The bounded function b_p therefore has a limit along good radii. To see this directly, choose good s_j\downarrow0 realizing its limit inferior and apply Equation (4.6) to all smaller good r; the limit superior is at most the same value. Equation (4.4) gives the same limit for the mass ratio. Squeezing an arbitrary radius between good radii with ratios tending to one gives Equation (4.3).

Measurability follows by taking a fixed sequence of radii tending to zero, and boundedness follows from Equation (4.1). For the last assertion, work in a fixed relatively compact coordinate chart, localizing the measures before extending them to Euclidean space. Apply the differentiation theorem for Radon measures, with smooth four-dimensional volume as denominator [21, Theorem 4.19]. At volume-almost every center, the measure-to-volume ratios of sufficiently small Euclidean coordinate balls are bounded. A geodesic ball of radius s is contained in a coordinate ball of comparable radius, whose smooth volume is O(s^4). Hence \mu(B_s(p))=O_p(s^4) at those centers, and s^{-2}\mu(B_s(p))\to0. A finite chart cover proves the assertion on X.□

Proof. Choose a positive-density point where differentiation of the bounded matrix M with respect to \mu holds in a fixed Euclidean coordinate chart and smooth local trivialization [21, Theorem 4.33 and Corollary 4.34]. This excludes a set of zero \mu-measure. No doubling hypothesis is needed: for fixed R, enclose B_{Rs}(p) in a coordinate ball E_s of radius comparable to Rs. Quadratic growth bounds s^{-2}\mu(E_s), while differentiation makes the mean oscillation of M on E_s tend to zero. Therefore

A smooth change of frame adds a vanishing O(s) error after this normalization. The quadratic mass bound gives subsequential locally weak limits \gamma of the rescaled scalar measures on all of T_pX\simeq\R^4. The existence of the density, followed by approximation of balls from inside and outside, gives

In particular every centered sphere has zero \gamma-measure. The rescaled Q mass on that ball satisfies the sharper estimate

Passing to the limit in closedness of the rescaled T therefore shows that the constant-coefficient current M(p)\gamma is closed.

Identify M(p) with a constant skew-symmetric matrix. The equation \partial(M(p)\gamma)=0 says that all distributional derivatives of \gamma in the column space of that matrix vanish. Equivalently, \gamma is translation invariant along \im M(p). The nonzero positive Hermitian matrix M(p) has real rank two or four. Rank four would make \gamma a nonzero multiple of Lebesgue measure on \R^4, which contradicts Equation (4.7). Thus its rank is two. Trace-one normalization then makes M(p)=\ell_p, the unit bivector of a complex line P_p. The rank-one positive Hermitian matrix is an extreme point of the trace-one positive matrices: if its quadratic form vanishes on a vector, positivity forces almost every matrix in any convex decomposition to vanish there too. Hence \lambda_p is concentrated on \ell_p.

A locally finite measure invariant under translations in P_p has the form \mathcal H^2\!\restriction P_p\otimes\sigma, with \sigma a locally finite measure on P_p^\perp. Equation (4.7) now becomes

Each integrand is nondecreasing in R, and is strictly increasing once R>|z_\perp|>0. Constancy for all R forces \sigma to be supported at zero. Its mass is \vartheta(p)/\pi. Every convergent subsequence has consequently the same limit, proving the asserted convergence.□

Proof. At the positive-density points covered by Lemma 4.4, divide the inner integral by s^2 and pass to the planar tangent measure. The integrand vanishes on that plane, and the limiting sphere has zero measure. At zero-density points the same normalized integral is bounded by s^{-2}\mu(B_s(y)), which tends to zero. These two arguments apply for \lambda-almost every (y,\ell). The quadratic mass bound gives a uniform integrable bound, so dominated convergence proves Equation (4.9).□

Proof. We first smooth the scalar density. Fix a nonnegative function h\in C_c^\infty((0,1)), normalized by \int_0^1h(u)\,du=1. For small s, put

The function f_s is smooth: the support lies inside the injectivity radius, and the cutoff vanishes near its boundary. The mass bound makes f_s uniformly bounded. Integration with respect to the radial distribution of \mu gives

There are no atoms by quadratic growth, so no endpoint term occurs at u=0. Thus f_s\to0 volume-almost-everywhere as well.

Fix a smooth one-form \alpha. To prove that b(\vartheta)N is closed, apply closedness of T=N+Q before passing to the density limit:

The first term has the required limit. Indeed, dominated convergence gives

as currents. The second convergence uses b(0)=0, the volume-almost-everywhere convergence of f_s, and domination by C|Q|.

It remains to show that the derivative term in (4.12) vanishes. The derivative of b is bounded on the common bounded range of the f_s. Write |df_s|_\ell for the norm of the restriction to the plane of \ell. The N contribution is bounded by a constant times \|\alpha\|_{C^0}\int |df_s|_\ell\,d\lambda, and the Q contribution is bounded by a constant times \|\alpha\|_{C^0}\int |Q|\,|df_s|\,dV_g. We therefore need exactly two estimates:

The derivative against Q. Differentiating the radial kernel gives |df_s(z)|\le Cs^{-3}M_s(z). Fubini and the volume bound give

Moreover M_s/s^2 is uniformly bounded and tends to zero volume-almost-everywhere. Therefore

The first factor tends to zero by dominated convergence against |Q|^2dV_g, and the second is bounded. This proves the first estimate in Equation (4.13).

The tangential derivative. Closedness will convert tangential differentiation into normal derivatives. The radial kernel then supplies a transverse displacement factor, multiplied by the difference between nearby complex-line directions. The transverse estimate of Corollary 4.5 and the angular-energy bound will control these two factors together.

Fix a center and line (y,\ell). Choose an orthonormal basis of T_yX whose first two vectors span the oriented plane of \ell, and let \zeta_1,\ldots,\zeta_4 be the corresponding normal coordinates. The estimates below are uniform in this choice, so a measurable choice of such bases suffices when integrating in (y,\ell). Write

First variation of squared distance yields

The sign comes from differentiating the center; the displayed q_i differentiates the radial expression in its coordinate variable.

We replace this scalar integral by T(q_i\omega_{12}). On B_s(y), normal coordinates and parallel transport differ by O(s^2), and hence

The second identity uses that both bivectors have unit length. Since |q_i|\le C/s, the integrated replacement error is at most

Here the double mass is O(s^2), the angular integral is O(s^4), and the last term tends to zero by Equation (4.14). The normal-coordinate term is consequently O(s).

For i=1, closedness of T on d(k_s^y\,d\zeta_2) gives

For i=2, using d(k_s^y\,d\zeta_1) gives the analogous formula, with an irrelevant overall sign. These are legitimate smooth compactly supported tests because the cutoff vanishes near the boundary of the coordinate ball. The mixed forms satisfy, for k=1,2 and a=3,4,

Their value on the transported center line is zero, accounting for the linear angular bound. The normal derivative of the radial kernel has the sharper estimate

Replacing T by N+Q in these mixed terms, the Q contribution again tends to zero by Equation (4.14); the s^2 geometric error contributes O(s). Cauchy–Schwarz for the two remaining angular and transverse factors bounds the rest by

Combining Equations (4.15), (4.17) and (4.20) proves the second estimate in Equation (4.13).

The closed weighted limit. The two derivative estimates now allow passage to the limit in (4.12). Its derivative term tends to zero, and its first term tends to b(\vartheta)N(d\alpha). Hence this last expression vanishes for every smooth one-form \alpha, proving the first assertion.

The remaining weights are obtained in a separate limit, after this argument has been completed for each fixed smooth b. Choose a smooth nondecreasing function \chi\colon\R\to[0,1] which is zero on (-\infty,0] and one on [1,\infty). The smooth bounded functions \chi(nt) converge pointwise to \mathbf1_{\{t>0\}} on [0,\infty), and vanish at zero. For fixed a>0, the functions

are smooth, vanish near zero, and are uniformly bounded by \pi/a. They converge at every t\ge0 to \mathbf1_{\{t>a\}}\pi/t, with value zero at t=a. Dominated convergence and closedness of each smooth-weight current prove Equation (4.10). No derivative bound uniform in n is used: the smoothing limit was taken first for each n. Finally N^d+Q=T-N^+ is closed.□

Proof. For each dyadic shell, quadratic growth gives the summable bound

The same estimate holds once the radius exceeds a fixed small radius, by increasing C and using the finite total mass. This proves uniform boundedness. If \vartheta(z)=0, each fixed shell contribution tends to zero, because s^{-2}\mu(B_{2^{j+1}s}(z))\to0. The summable bound then proves pointwise convergence. A second application of dominated convergence proves Equation (4.21).□

Proof of Theorem 4.1. Suppose that no compatible symplectic form exists. The separator in Lemma 3.1 gives a nonzero positive current N, normalized to have trace mass one, and an anti-invariant Q such that T=N+Q annihilates all smooth closed two-forms. In particular T is closed and \mathfrak c(T)=0. The preceding results apply to this separator.

By Proposition 4.6, T^d=N^d+Q is closed. Equation (4.22) gives

The self-duality of Q accounts for the signs in this expansion. Both mixed terms tend to zero by Proposition 3.8. The positive-current lower bound of Lemma 3.6, applied with second submeasure \lambda^d=\mathbf1_{\{\vartheta=0\}}\lambda, gives

Since S_sQ\to Q in L^2, taking a limit inferior in Equation (4.23) forces \|Q\|_2^2=0. Thus T=N is a nonzero positive current annihilating all closed forms.

Let \eta be a taming symplectic form. Compactness of the bundle of unit complex lines and strict taming give \eta_y(\ell)\ge c_\eta>0. Consequently

contradicting its annihilation property. This proves compatible-form existence. The proof has used neither rectifiability nor a theorem representing integral cycles by curves.□

Proof. Closedness follows from Proposition 4.6. By Equation (4.22) and Lemma 3.6,

The right side tends to zero by Lemma 4.7.□

Proof. Put E_+=\{\vartheta>0\} and \mu^+=\mathbf1_{E_+}\mu. At \mu^+-almost every p, Euclidean coordinate-ball differentiation of \mathbf1_{E_+} gives value one. The coordinate-ball enclosure argument in Lemma 4.4, applied to this bounded function, gives

Thus \mu^+ has the same positive finite two-density and the same planar tangent measure as \mu at almost every such point. Under a smooth coordinate change its limit is the linear image of the same plane measure. Thus, in any fixed coordinate chart, the Euclidean two-density exists and is positive and finite almost everywhere for the pushed-forward measure: every Euclidean sphere has zero mass for the limiting plane measure. Preiss’s rectifiability theorem therefore makes \mu^+ a two-rectifiable measure [40]; see also the theorem’s formulation in [20, p. 771, opening paragraph]. On a countably rectifiable carrier E it has the representation

Here \pi is the area of the Euclidean unit two-disk. The density identification follows from differentiation on a rectifiable carrier. Its approximate tangent plane agrees almost everywhere with the unique plane in Lemma 4.4. Thus N^+ is integration over E with that complex tangent orientation and multiplicity \vartheta/\pi.

Multiplying by the weight defining I_a cancels this multiplicity: I_a is integration with integer multiplicity one over E\cap\{\vartheta>a\}. Its mass is at most \pi\mu(X)/a, and its boundary vanishes by Proposition 4.6. It is consequently an integral current: it is integer rectifiable, has finite mass, and has the zero integral current as its boundary.

Suppose now that J is tamed. By the already proved Theorem 4.1, a compatible symplectic form exists. In particular, the local compatibility hypothesis of the Rivière–Tian regularity theorem holds. That theorem applies to integer rectifiable almost complex cycles and represents them by J-holomorphic curves, with only isolated singularities [41, Theorem I.1]. Integral multiplicity and complex orientation do not depend on the choice of Hermitian metric, so we may use the metric of this compatible symplectic form for that application. Normalize the resulting curve by separating its local branches at the isolated singularities; the local completion is part of the curve representation [41, Section VIII]. Its normalization components are closed. Compactness and local finiteness give only finitely many global components; equivalently, small-ball monotonicity gives a uniform positive lower area for each nonconstant closed component, whereas the total mass is finite. Factoring each component through its somewhere-injective image gives the asserted finite sum of irreducible closed curves with positive integer multiplicities.

Finally, pointwise on \{\vartheta>0\},

For a smooth two-form \alpha, Fubini is justified by

This proves Equation (4.24) with its stated normalization.□

Proof of Theorem 4.2. Assume that the class H has no taming representative. The class separator of Lemma 3.1 gives a nonzero closed positive current N, with Q=0, satisfying

Its positive-density and diffuse parts are separately closed. The anti-invariance of A,C implies

Thus w=\mathfrak c(N^d) lies in V, and w^2\ge0 by Lemma 4.8.

If N^d\ne0, choose a taming form \eta in U. Strict taming on the compact bundle of unit complex lines gives

In particular w\ne0. Since [A],[C] span a positive two-plane and b^+=3, the intersection form on V has signature (1,b^-). The conditions w^2\ge0 and U\cdot w>0 put w in the nonzero closed time cone determined by U. Every timelike H in that component has H\cdot w>0. Explicitly, take a unit timelike vector u in the direction of U, and write H=h_0u+h_-, w=w_0u+w_-, with orthogonal negative components. Then

so

If N^d=0, its pairing is instead zero. This distinguishes the nonzero diffuse case required for strict positivity.

For every a>0, Lemma 4.9 expresses I_a as a positive integral sum of irreducible closed J-holomorphic curves. The hypothesis on H therefore gives

with strict inequality whenever I_a\ne0. If N^+\ne0, the layer identity has positive total mass, so the set of a for which I_a\ne0 has positive Lebesgue measure. Pairing Equation (4.24) with any smooth representative of H then gives H\cdot\mathfrak c(N^+)>0. The pairing is zero when N^+=0. At least one of N^d,N^+ is nonzero, and hence

This contradicts Equation (4.26) and proves the theorem. A smooth closed taming representative is symplectic by pointwise nondegeneracy.□

Proof. Fix a metric g_0 in the conformal class determined by the original triple. All J_v are orthogonal for g_0, and the three closed self-dual forms \omega_i span its self-dual harmonic forms because b^+=3. Denote by R_v the projection to the J_v-anti-invariant two-forms, and consider

where the adjoint is taken with respect to g_0. This is the nonnegative elliptic operator in Lemma 3.2. Its kernel consists exactly of the closed anti-invariant forms: its quadratic form is \|\dd^*\xi\|_2^2, and an anti-invariant form is self-dual, so coclosedness implies closedness. Its kernel is therefore precisely

a smooth vector bundle of rank two over S^2.

Let H_v be the projection onto this explicit kernel. In local parameter coordinates, identify the varying anti-invariant bundles by their orthogonal projections. The kernel projection is then smooth, and L_v^{\mathrm{ell}}+H_v is invertible from H^{s+2} to H^s. The inverse identity and elliptic regularity give smooth dependence on each Sobolev scale, as in the parameter statement of Lemma 3.2. Thus the generalized inverse is

and

is a smooth family of closed lifts.

For any v_0, Theorem 4.1 supplies a compatible form \eta for J_{v_0}. For v near v_0, set \eta_v^{(v_0)}=\eta-D_vR_v\eta. These forms are closed and J_v-invariant, equal \eta at v_0, and remain positive after shrinking the parameter neighborhood. A finite parameter partition of unity patches these local families. Since its coefficients depend only on v, this operation preserves closedness on X; positivity and invariance are preserved by convexity.

The class [\eta_v] is orthogonal to the classes of the two anti-invariant forms, so its projection to P is a multiple of [B_v]. Moreover \int_X\eta_v\wedge B_v>0: the invariant part of B_v is positive, and its anti-invariant part wedges trivially with \eta_v. Since [B_v]^2=1, the multiple is positive.□

Proof. The critical wall-crossing theorem and its section-class version are [27, Theorem 4.10 and Proposition 4.11]. We check the normalization and degree relevant here. A transverse homotopy between two families meets the harmonic wall at isolated parameters in S^2\times[0,1], because its normal rank is b^+=3. Above such a parameter the reducibles form the torus \Pic^0(X). Represent the complex Dirac index on this torus as K-O, with ranks k,o, using a stabilization when necessary. Write x=c_1(\mathcal O(1)) on the projective bundle of lines \pi:\mathbb P(K)\to\Pic^0(X). The obstruction bundle contributes e(\mathcal O(1)\otimes O)=\sum_i x^{o-i}c_i(O). Projective pushforward therefore gives

since Equations (5.1) and (5.3) imply q+o-(k-1)=b_1/2. The normal crossing signs sum to the degree difference, giving Equation (5.6).

The based-gauge normalization makes the universal flat line trivial at p. Thus no Picard-base line is added to x. Writing a determinant connection as a canonical connection plus twice a twisting connection does not alter this conclusion: the remaining circle acts on the spinor with weight one, as used in Equation (5.7).

For a constant regular family, negative ordinary expected dimension gives an empty moduli space. Otherwise the moduli space is the product of an ordinary E^2-dimensional moduli space with S^2, while U_p^q is pulled back from the former and has degree E^2+2. Its pairing is therefore zero.

Every maximal positive harmonic plane projects isomorphically to P: the kernel would lie in the negative definite space P^\perp. This identifies the harmonic-wall bundle with the fixed oriented vector space P. For the perturbation in Equation (5.4), its wall section has leading term (r/4)[\eta_v], with a bounded shift from 2E. By Lemma 5.2, it is nonzero for all sufficiently large r and its normalization has degree one. The same is true after sufficiently small generic family perturbations.□

Proof. Since H^1(S^2)=0, the Kunneth decomposition shows that c_1(\mathcal E_E)=E+eh for some integer e. At p, deform the compatible metrics to g_0 through J_v-Hermitian metrics. The fixed spin-c extension over metric space identifies the positive spinor bundles along this homotopy. For g_0, the first eigenspaces give the tautological line in the projective spinor sphere, pulled back by the coefficient sphere map. The latter has degree of absolute value one by Lemma 2.4, proving |e|=1.

The evaluation transforms with the same gauge weight as the spinor. Its sign can also be checked on a reducible link: evaluation on a kernel line \ell is a map \ell\to\mathcal E_E|_p, hence lies in \operatorname{Hom}(\ell,\mathcal E_E|_p) =\mathcal O(1)\otimes\mathcal E_E|_p. On the quotient it is therefore a section of the tensor product of the based-gauge evaluation line with the pullback of \mathcal E_E|_{\{p\}\times S^2}. Its Chern class is U_p+e\pi^*h. Pairing \pi^*h with the family moduli cycle gives the ordinary dimension-zero invariant \operatorname{SW}_X(E); equivalently, restrict the family to a regular base point. A nowhere-zero evaluation section trivializes its line, giving Equation (5.8).

For the last assertion, suppose the exact family had no evaluation zero. Fixed-r compactness would give the same absence for all sufficiently small perturbations: otherwise a sequence of their zeros would limit to an exact zero. Apply the preceding identity to a regular such perturbation to obtain a contradiction.□

Proof. First take E=0. Taubes’s canonical nonvanishing theorem [46] gives |\operatorname{SW}_X(0)|=1, since b^+>1. By Lemma 5.3, the exact symplectic family has no first-component zeros for all sufficiently large r. Fixed-r compactness transfers this absence to sufficiently small regular perturbations. Applying Equation (5.8) gives |\operatorname{FSW}_0(U_p)|=1. On the other hand, Lemma 5.5 and Equations (5.9) and (5.10) identify this number, up to sign, with

because \operatorname{ch}_1=0 when E=0. It follows that the quadruple cup product is nonzero.

Here is an explicit calculation of the remaining assertion. Choose a basis a_1,\ldots,a_4 of H^1(X;\R), let t_1,\ldots,t_4 be the corresponding Picard-torus basis, and write

The wedge pairing on \bigwedge^2H^1(X;\R) induced by this nonzero volume element is nondegenerate. Thus the cup-product map into H^2(X;\R) is injective. Both spaces have dimension six, by Proposition 2.3, so it is an isomorphism. We may consequently write E=\sum_{i<j}e_{ij}a_ia_j. The graded-product signs give

Thus E^2=0 implies \operatorname{ch}_1^2=0, and the Segre integral equals the same nonzero integral of \operatorname{ch}_2 found for E=0.

Finally, a map S^2\to X pulls back every class in H^1(X;\R) to zero, and therefore pulls back every product of two such classes to zero. These products span H^2(X;\R), so its homology class is zero by Poincare duality.□

Proof of Theorem 5.1. The projection F^+ is nonzero: otherwise F would be a nonzero square-zero vector in the negative definite space P^\perp. Lift F to an integral class and use the fixed spin-c structure \mathfrak s_F.

Its ordinary invariant is zero. Indeed, choose the opposite coefficient direction w=-F^+/|F^+|, so that F[B_w]<0. If \operatorname{SW}_X(F)\ne0, ordinary invariance and fixed-parameter monopole compactness would give solutions of Equation (5.4) for arbitrarily large r at this fixed background. Taubes compactness would give a nonempty positive J_w-holomorphic configuration of class F, contradicting its negative B_w-area.

The spherical U_p pairing is nonzero. For b_1=0, this follows directly from Lemma 5.5; for b_1=4, use Proposition 5.7 as well. Since the ordinary invariant vanishes, Lemma 5.6 forces a first-component zero over p in the exact family for each sufficiently large r. Extracting a convergent parameter subsequence and applying Lemma 5.3 gives a configuration through p in some J_v, with total class F.

The two closed forms whose coefficients are perpendicular to v are J_v-anti-invariant and restrict to zero on every component of this configuration. Thus F^+ is parallel to [B_v]. Its B_v-area is positive, so the proportionality factor is positive, which proves the asserted direction.□

Proof. Lift E to an integral line-bundle class. Its ordinary expected dimension is -2, and its spherical family has dimension zero. By Lemma 5.5, its large-parameter unmarked family invariant is \pm1. For each sufficiently large r, take small regular perturbations and then a fixed-r limit to obtain an actual solution of Equation (5.4) at some parameter v_r. Pass to a sequence with v_r\to v, and apply Lemma 5.3. The limit is a nonempty positive J_v-holomorphic configuration of class E. Every nonconstant component has positive B_v-area, since B_v tames J_v. This contradicts E[B_v]=0, which follows from E\perp P.□

Proof of Lemma 5.3. For a fixed background, this is the compactness and incidence part of Taubes’s existence theorem. For the equations in Equation (5.4), use [48, Theorem 2.2], with its zero-support conclusion as proved in Lemma 7.1 and Equation (7.5) for the first component defined in Equation (3.17) of that paper. The symplectic form has empty zero locus, so the transverse-zero hypothesis is vacuous. The class formula is

The splitting above gives L=K_J^{-1}\mathcal E^2, so this is exactly [\mathcal C]=E over \R. The original fixed-background symplectic statement also includes incidence with prescribed closed sets [47, Theorem 1.3]; the curve-recognition argument is understood in its corrected form [49].

Uniform estimates and support convergence. We explain the uniformity needed here. Smooth convergence provides a common positive normal-coordinate radius, uniform ellipticity, and uniform bounds on all derivatives of the metrics, forms, and induced canonical connections. It also bounds the topological terms c_1(L)[\eta_n]. In the following estimates, constants are independent of n and r after these quantities have been bounded. For readability write w=1-|\alpha|^2 and w_+=\max(w,0), and use the projected spinor covariant derivatives. The estimates used in the compactness proof include

These are the specialization to a nowhere-vanishing symplectic form of [48, Lemma 3.1, Proposition 3.1, Equation (3.27), and Propositions 3.2–3.3]. The global pointwise bound follows directly from the spinor Weitzenbock formula and the maximum principle. The remaining bounds use its two eigenline projections, the curvature equation and its derivative, and the scalar Green kernel. All the geometric coefficients in those arguments are uniformly bounded on the chosen charts. In particular, \int w_+\le\int|1-|\psi|^2|+\int|\beta|^2. Integrating the bound for \beta and absorbing Cr^{-1}\int w_+ for large r therefore bounds r\int w_+. The inequality w\ge-Cr^{-1}, followed by Equation (5.13), then gives

For the nonnegative energy \mathscr E_r(U)=r\int_U|1-|\psi|^2|\,\vol_g, the monotonicity estimate gives uniform constants c,C,s_0>0 such that, whenever Cr^{-1/2}\le s\le s_0,

This is [48, Proposition 4.1], with its universal normalizing factor absorbed into the constants. Consequently the zero sets have covers by at most Cs^{-2} such balls.

Here is also the scale and derivative dependence in the local compactness step. Use normal coordinates for each individual g_n, rescale by r_n^{1/2}, and compare on a ball of fixed radius R. After identifying the center with Euclidean \C^2, the rescaled coefficients satisfy

The first derivatives of g_n' are O_R(r_n^{-1}). The scale in [48, Equation (5.1)] is (r_n|\eta_n|)^{1/2}=2^{1/4}\sqrt{r_n}. A fixed dilation therefore converts these coordinates to that paper’s flat-model normalization; all estimates below absorb this fixed factor, and curvature integrals are unchanged by it. These estimates follow by Taylor expansion in each metric’s own normal coordinates; compare [48, Section 6, Step 4]. For every fixed R,k, the rescaled equations therefore have uniform coefficient bounds through order k. Gauge fixing and interior elliptic estimates give the rescaled local approximation in any prescribed C^k norm by the flat model, as in [48, Proposition 5.2]. Its quantifiers are: for each R,k,\eps>0, a common lower bound on r makes the approximation error smaller than \eps. We retain smooth background convergence, so every derivative order required in this argument is available; no assertion about a minimal finite regularity threshold is needed.

Let \mathcal A_{\mathrm{can},n} be the canonical connection on K_{J_n}^{-1}. The first eigenline connection a_n has curvature F_{a_n}=\tfrac12(F_{\mathcal A_n} -F_{\mathcal A_{\mathrm{can},n}}). The corresponding closed currents

have uniformly bounded mass by Equation (5.15). Their periods on closed test forms are those of E. On the complement of the first-component zero set, its unit section identifies the determinant with the canonical determinant. In that identification the further estimate is

when the distance is at least r^{-1/2} [48, Proposition 6.1].

The ball covers, the lower bound at zeros, and Equation (5.19) now give a subsequence whose zero sets converge to the support Z_* of a weak limit of T_n; the support has finite two-dimensional Hausdorff measure. These are the arguments of [48, Equations (7.1)–(7.5) and Lemma 7.1], using exactly the uniform estimates just listed.

We give the disk argument needed when the backgrounds vary. Write

The sharper estimate [48, Proposition 4.2, Equation (4.9)], specialized to |\eta_n|=\sqrt2, is

Its constants have the same uniform geometric dependence as the estimates above. If \ell is a unit, positively oriented J_n-complex bivector, then \ell^+=\eta_n/2, |\ell^-|=1/\sqrt2, and the curvature equation gives

The leading terms cancel. The canonical curvatures are uniformly bounded, so there is one constant C_0\ge1 such that every J_n-holomorphic disk u satisfies

Here \dd A_u is the induced area density, counted on the source; the inequality also holds at critical points. This derives the disk bound directly for our normalization, including the factor one half in the twisting connection.

The relative integer assignment. First construct the integer assignment independently of positivity. A smooth disk u is admissible when its boundary misses Z_*. For large n, u^*\alpha_n is nonzero on a boundary collar and defines a relative Chern number k_n(u)\in\Z. The unit first component trivializes the eigenline there, and Equation (5.19) gives

The error is uniform for disks with bounded first derivatives whose boundary collars have a common positive distance from Z_*. For each sufficiently large n, relative Chern numbers agree along any fixed admissible homotopy. The same conclusion holds for the varying homotopies below: pointwise short geodesics between uniformly close boundary maps stay in one fixed zero-free neighborhood of the original boundary. Thus their pulled-back unit sections extend over the boundary homotopy, uniformly in large n.

For an immersed disk, take small normal translates, with a smooth averaging function in the two normal parameters. The normal bundle is taken over the source, so self-intersections cause no difficulty. All these disks have the same k_n. Cut off the source near its boundary collar; the removed flux tends to zero by Equation (5.19). The averaged remaining flux is the pairing of T_n with a fixed smooth two-form: the normal coordinate map is a local diffeomorphism, and a finite partition of unity pushes the compactly supported averaging forms to X. Weak convergence of T_n proves convergence of these averages. Thus k_n, being integer valued, is eventually constant. A general admissible smooth disk has an arbitrarily close immersed perturbation, connected to it by an admissible homotopy, and hence has the same conclusion. Denote the eventual integer by I(u). It is zero for disks missing Z_*, invariant under admissible homotopy, additive under subdivision, and multiplied by the degree under proper disk maps; these are the corresponding properties of relative first Chern numbers. This is the Stokes and averaging construction of [47, Proposition 5.6 and Lemma 6.2]. Only positivity for the limiting complex structure remains.

Disk deformations with prescribed incidences. We need the following elementary disk deformation fact. For a holomorphic map on a slightly larger disk, with no boundary condition, the linearized Cauchy–Riemann operator has a bounded right inverse even after prescribing values and complex-linear first derivatives at any finite set of distinct interior source points. Work, for example, in H^k\to H^{k-1}, k\ge4, so these evaluations are continuous. Extend the bundle and operator to a closed surface, and extend the target sections by a bounded Sobolev extension. The closed-surface operator augmented by the finite evaluations is Fredholm. Finitely many smooth forcing terms supported outside the larger disk make it onto. Indeed, an annihilator of all such exterior terms vanishes there. Away from the specified points it is smooth by elliptic regularity and vanishes everywhere by unique continuation for the adjoint real Cauchy–Riemann operator [55, Theorem 2.78 and Proposition 3.12]. It is therefore a sum of distributions supported at the specified points. The invertible principal symbol eliminates derivatives of point masses of positive order: their highest derivative after applying the adjoint has order at least two, whereas the evaluation constraints have order at most one. The remaining term at each point is q\delta. Testing independently the complex-linear and anti-complex-linear first derivatives eliminates both q and the first-derivative constraint covector; testing the value then eliminates its covector. The annihilator is zero, proving the assertion about exterior forcing. A bounded right inverse of this surjective Fredholm operator, restricted to the disk, proves the claimed right inverse.

The implicit function theorem and interior regularity consequently give nearby holomorphic disks for nearby almost-complex structures, with those finite values and complex-linear derivatives prescribed. All domain restrictions are chosen first, and all constraints lie in the smaller disk. The right-inverse bounds persist for nearby operators. Constants may depend on the chosen finite set of points and on a fixed neighborhood of the original map; this is harmless because these choices precede n\to\infty. In particular, while retaining finitely many incident values, one may choose the tangent line at each of them independently in a nonempty open cap. At a critical point one first chooses a sufficiently small nonzero complex-linear derivative. This gives a fixed small J_\infty-holomorphic perturbation of the original disk; its J_n-holomorphic transfers converge smoothly to that perturbation on the smaller closed disk. They need not converge to the unperturbed disk, and admissible boundary homotopy is what identifies their integers.

Positive flux at a retained incidence. Here is the positive contribution at each retained incidence. Choose actual zeros p_{j,n} tending to the finitely many specified values in Z_*. After a joint subsequence, rescaling at these zeros gives centered flat models on all fixed balls. Their first components vanish at the origin. The quadratic energy bound excludes the identically zero first component. Each limiting model has a polynomial divisor by [48, Proposition 5.1]. For any nonempty open cap of complex lines through the origin, choose a direction where the highest homogeneous part of that polynomial is nonzero; only finitely many directions are excluded. Along the chosen line, distance from the divisor grows linearly at infinity. The exponential off-divisor estimate in [48, Proposition 5.1] makes the boundary connection term tend to zero. Stokes’ theorem therefore identifies the total line flux (i/2\pi)\int F_{a_0} with the positive degree m_j\ge1 of the restricted divisor, which contains the origin. The determinant connection in [48, Proposition 5.2] converges to 2a_0, so this is precisely the limiting normalization of f_n, with the bounded canonical curvature disappearing under rescaling. Choose a fixed large radius R_j so that the boundary is zero-free and the flux differs from m_j by less than 1/8. These are the polynomial-model and line-selection steps in [47, Proposition 5.6, Steps 3–4], using the flat limits of [48, Proposition 5.2].

Prescribe those good tangent directions in the disk deformation just constructed. Choose a fixed approximation tolerance making the flux error on each model disk less than 1/8; only then take n large. Because the prescribed derivatives are nonzero and the disk maps have uniform higher derivative bounds, the rescaled maps on microscopic source disks converge to the chosen complex-linear maps. Each resulting source core therefore contributes more than m_j-1/4\ge3/4 to the flux. The finitely many core radii tend to zero, so the cores are source-disjoint. Distinct source incidences may have the same target value: the flux is integrated on the source, and this does not affect the argument. The nonzero derivative sizes, the caps, the models, R_j, and the approximation tolerances are all fixed before the final lower bound on n is imposed.

Positivity for the limiting complex structure. We now prove positivity for any admissible J_\infty-holomorphic disk u meeting Z_*, allowing critical points and self-intersections. Shrink its domain inside a slightly larger disk so that every incidence lies in its interior and its boundary collar has a fixed positive gap from Z_*. First fix a small C^1 neighborhood in which this gap persists and all disk areas are bounded by a number A_*. Put S=u^{-1}(Z_*).

If S is finite, retain all its points. Choose disjoint fixed neighborhoods of them whose total induced area, for all sufficiently close disks, is less than 1/(4C_0). On the compact complement the original disk has a positive distance from Z_*; shrink the allowed disk neighborhood to preserve half that distance. Choose the good tangent caps and the fixed small holomorphic perturbation within this neighborhood, and transfer it to J_n, retaining the chosen zeros. The cores contribute at least 3/4 each. Their complement inside the selected neighborhoods contributes more than -1/4 by Equation (5.21). On the remaining compact domain the flux tends to zero by Equation (5.19). Thus the limiting integer in Equation (5.22) is positive.

If S is infinite, choose a finite number N of distinct source points in it with 3N/4>C_0A_*+1. The area bound was fixed before N. Apply the same finite-constraint construction at these points, keeping the perturbed disks in the chosen area neighborhood. The N source-disjoint cores contribute more than 3N/4, whereas their full complement contributes at least -C_0A_*. The flux is therefore bounded below by a positive number. Its limit is the unchanged integer I(u), so again I(u)>0. Every choice was finite and fixed before the last passage to large n; no rate relating J_n\to J_\infty and r_n\to\infty has been imposed.

Curve recognition and retention of the marked point. In particular the positivity axiom for embedded holomorphic test disks holds. The resulting positive cohomology assignment permits the corrected recognition theorem to be applied to the fixed structure J_\infty, as in [48, Lemmas 7.2–7.3 and Proposition 7.1] and [49]. The recognized normalization map is proper. Since X is compact, its source is compact and has finitely many components; its locally finite exceptional set is finite as well. The multiplicity of a resulting curve at a smooth point is the integer I of a small transverse disk there. The averaging construction identifies this same integer with the transverse coefficient of the weak curvature current: on each smooth branch, a closed current of order zero supported there is a constant multiple of its oriented integration current. The remaining difference is supported on the finite singular set, where a closed two-current of order zero must vanish. Thus the recognized configuration has current equal to the weak limit of T_n, as in [48, Proposition 7.1 and Equation (7.12) in the proof of Lemma 7.4]; in particular its class is E.

The support convergence retains every limit of first-component zeros. If E=0, a retained zero would give a nonempty positive configuration with \int_{\mathcal C}\eta_\infty=E[\eta_\infty]=0, which is impossible. Failure of the final uniform assertion would supply a sequence with r_n\to\infty; compactness of the parameter space would reduce it to the situation just proved.□

Proof. We first construct a rational null subspace with enough integral directions, and then exclude finitely many of those directions.

A positive integral vector and a null lattice. Set

The subspace W_{\R} is rational and negative definite. Hence its orthogonal complement is rational and contains P. Positive square is an open condition, so density of rational vectors in this complement gives a positive rational vector in W_{\R}^\perp. Clearing denominators and dividing out the common integral divisor produces a primitive vector l\in\Lambda satisfying

Write l^2=2n, where n\in\Z_{>0}. We use the following form of Eichler’s criterion: in an even lattice containing two orthogonal hyperbolic planes, the orbit of a primitive vector under the stable orthogonal group is determined by its square and its normalized class in the discriminant group [15, Proposition 3.3(i)]. Here the lattice is unimodular. Thus its discriminant group is zero, and every primitive vector has divisibility one. The criterion sends l to e_1+n f_1, where

are standard bases of the three mutually orthogonal copies of U. In these coordinates the lattice generated by f_1,e_2,e_3 is primitive and totally null. Pulling it back gives a rational three-dimensional totally null subspace L_{\R} and its lattice L=\Lambda\cap L_{\R}, with an element f_0\in L such that

Consequently

is an affine lattice of rank two. Every F\in\mathcal F is primitive in \Lambda, since a nontrivial integral divisor of F would divide l\cdot F=1. It is also nonzero and null. Its projection to P is nonzero by negative definiteness of P^\perp.

Only finitely many cosets can obstruct a direction. The set

is compact. Suppose temporarily that F\in\mathcal F, d\in\Lambda, and c>0 satisfy the three conditions on the left side of Equation (6.1). Then

so r\in K.

Let q:\Lambda_{\R}\to\Lambda_{\R}/L_{\R} be the quotient map. Because L_{\R} is rational and L is saturated, a basis of L extends to an integral basis of \Lambda; the remaining basis vectors project to a lattice basis in the real quotient. Thus q(\Lambda) is discrete and \ker(q|_\Lambda)=L. Since q(d)=q(r)\in q(K), only finitely many cosets d+L can occur in Equation (6.4). This finite set is independent of F.

Avoiding irrational points in a torus. Orthogonal projection restricts to an isomorphism

its kernel is null and lies in the negative-definite space P^\perp, and both spaces have dimension three. Hence \pi(L) is a lattice in P. Each of the finitely many cosets just found determines one point of

because adding an element of L changes the projection by an element of \pi(L).

For a primitive F\in L, let

Two such circles with different directions intersect only in torsion points. Indeed, if t\pi(F)-s\pi(F')\in\pi(L) and F,F' are not collinear, choose an integral linear functional on L which vanishes on F and is nonzero on F'. Applying it to tF-sF'\in L shows that s\in\Q, so the common point is torsion. In particular, any non-torsion point of \mathbb T lies on at most one primitive direction circle.

The elements of \mathcal F have distinct directions: two collinear elements with l-pairing one are equal. Since \mathcal F is infinite, we can choose F\in\mathcal F whose circle contains none of the non-torsion points among the finitely many possible coset points.

For this F, every d,c satisfying the hypotheses of Equation (6.1) gives a torsion point

There is therefore an integer k>0 with kcF\in L. Pairing with l shows that kc\in\Z, so c\in\Q. It follows that r=d-cF\in W_{\Q}, and Equation (6.2) now gives

If m=d\cdot F<0, then m is a negative integer and

Negative definiteness of P^\perp forces r=0, which would imply m=0. This contradiction proves d\cdot F=0, and completes the proof.□

Proof. Represent the irreducible image by a somewhere-injective J-holomorphic map u:\Sigma\to X, with its covering multiplicity factored out. The standard adjunction and singularity theorems for simple curves in real dimension four give

where \delta(u) is the weighted singularity defect and is zero precisely for an embedding; see [56, Theorem 2.8], using the numbering of the cited prepublication version throughout. Positivity of intersections of distinct simple images is [56, Theorem 2.3 and Corollary 2.4]. The critical points are isolated, and a simple curve is locally injective, including at critical points [56, Propositions B.26 and B.41]. These facts prevent accumulation of distinct double-point pairs at the diagonal of the product of the source with itself. Local intersection isolation, unique continuation, and simplicity prevent accumulation off the diagonal. Compactness therefore makes the critical and noninjective sets finite. The definite pair trivializes the canonical bundle by Lemma 2.2, so c_1(TX,J)=0 for either sign of J. In particular, d^2\geq-2.

Both members of the pair vanish on complex lines. Integrating their pullbacks gives [A]\cdot d=[C]\cdot d=0; hence

for a real number c. If c=0, the nonzero integral class d lies in P^\perp. Evenness, negative definiteness, and d^2\geq-2 force d^2=-2. When b_1=0, this contradicts Proposition 5.8. When b_1=4, Equation (6.7) gives g(\Sigma)=\delta(u)=0, so d is the class of an embedded sphere. This contradicts the vanishing of spherical real classes in Proposition 5.7. Thus c\ne0.

Suppose first that c>0. By Proposition 6.1, d\cdot F<0 is impossible. If d\cdot F>0, alternative (i) holds. If d\cdot F=0, the same proposition gives c\in\Z_{>0}, so r=d-cF is integral and lies in P^\perp. Moreover r^2=d^2\geq-2. Either r=0, or negative definiteness and evenness give r^2=d^2=-2. The latter is excluded by Proposition 5.8 when b_1=0; when b_1=4, it would make the original curve u an embedded sphere by Equation (6.7), again contradicting Proposition 5.7. Therefore d=cF.

If c<0, apply the same lattice argument to -d, whose square is still at least -2. This use of Proposition 6.1 is purely arithmetic and does not require a J-holomorphic representative of -d. It gives d\cdot F<0 or d=cF with c\in\Z_{<0}. Indeed, a nonzero residual in P^\perp would again have square d^2=-2. In the b_1=4 case, adjunction is applied to the original curve in class d, making it an embedded sphere and giving the same contradiction to Proposition 5.7. The two alternatives are disjoint because F^2=0.

Finally, the integral of a closed taming form over any nonconstant curve is strictly positive. Its real homology class, and hence its image in \Lambda, cannot vanish.□

Proof. Write a positive configuration as

where the d_i are the classes of its distinct irreducible images. Taming excludes zero real component classes. In the notation of Corollary 6.2,

so c_i>0. The corollary gives m_i=d_i\cdot F\geq0, whereas

Every m_i therefore vanishes. Again by the corollary, d_i=c_iF with c_i\in\Z_{>0}, and comparison of the total classes yields

There is exactly one summand, with n_1=c_1=1. This proves (i), including the exclusion of a nontrivial covering multiplicity.

Apply Theorem 5.1 to the present triple. Its period plane is still P, and Equation (6.6) forces the direction of the configuration to be precisely the one whose anti-invariant plane is \operatorname{span}(\widetilde A,\widetilde C) and whose tamer is \widetilde B. Thus the theorem gives a positive J-holomorphic configuration in F through any prescribed point. Part (i) makes it a single reduced component. Two distinct components of class F cannot meet: their intersection number is F^2=0, while any intersection of distinct simple images has positive local intersection number. This proves (ii).

For the remaining assertion, adjunction becomes

If g(\Sigma)=1, the defect is zero and the curve is embedded. If g(\Sigma)=0, the total defect is one; critical points of the parametrization have not yet been excluded. When the map is immersed, its singular defect is the sum of positive intersection multiplicities between distinct local branches. Total defect one then forces exactly one pair of branches meeting with multiplicity one, which is a transverse positive double point. A tangency has larger multiplicity, and a point with three or more branches contributes at least three. There can be no additional singularity.□

Proof. Let B_\infty be the limiting third form. It tames the limiting structure with a positive lower bound on the unit tangent bundle of a fixed metric. Smooth convergence therefore makes this same fixed form tame every sufficiently late structure. Its area on each sphere is [B_\infty]F=[B]F. The fixed-tamer hypotheses and uniform energy bound in the cited Gromov compactness theorem are consequently satisfied. The nonconstant components in the stable limit give a positive configuration of total class F for the limiting triple. By Proposition 6.3, there is exactly one simple image and its total multiplicity is one. In particular, the surviving map is not a nontrivial cover.

The domain of a genus-zero stable limit is a tree of spheres. With only one nonconstant vertex, a nonempty tree of constant vertices would have a constant leaf with at most one node and the one permitted mark. It would have at most two special points and hence would be unstable. There are therefore no constant components. Gromov convergence is now smooth convergence on the whole sphere after reparametrization, and the marked points converge as well. The same argument gives convergence in the corresponding finite differentiability orders when the backgrounds converge in a sufficiently high finite order.

If the limiting triple had no critical spheres but nearby triples had such spheres, mark a critical point on each. The conclusion just proved would give a critical point on a limiting sphere. This contradiction proves openness. This argument applies uniformly on any compact parameter set of positive triples with the stipulated periods.□

Proof. We prove universal surjectivity in the allowed space of perturbations. We reduce an annihilator to two constants arising from the pullback periods and a distribution supported at the marked point. The elliptic symbol and first-jet tests remove the point-supported part; variations of the coefficient sphere remove the two constants. Fix the complex structure j on S^2. Write v_0=(0,0,1) for the coefficient direction of B. For v close to v_0, a convenient closed anti-invariant pair for the structure J_v is

For any map u in class F, the two pullback periods are

Both pullbacks vanish when u is J_v-holomorphic, so every F-curve forces v=v_0. Exact changes of A,C leave the two periods unchanged, whereas varying v supplies both period directions in the universal linearization. Thus this auxiliary parameter removes the period constraints without adding solutions in other coefficient directions. We retain it in the section

The second component is valued in the real rank-four bundle of complex linear maps T_zS^2\to T_{u(z)}X. On the zero set of the first component, its vanishing is exactly \dd u(z)=0.

Use W^{k,p} maps with p>2, k\geq8, and backgrounds of finite class C^r, with r\geq k+8. The first target is W^{k-1,p}. Sobolev embedding gives W^{k,p}\subset C^{k-1,\alpha} for some \alpha>0, so first-jet evaluation at the moving mark is at least C^{k-2}, hence C^6. The background regularity gives the same C^6 regularity for the composition maps in the first component. Perturbing forms are \dd\beta_A,\dd\beta_C, where the primitives belong to the C^{r+1} closure of smooth one-forms in the indicated support class. This gives separable Banach spaces. We use the ordinary Banach bundle construction for Equation (7.5); compare the first-jet framework in [35, Sections 2–3]. The surjectivity argument for our restricted perturbations is given next.

Suppose (\eta,\lambda) annihilates the derivative of Equation (7.5) at a simple critical sphere. Here \eta is a distribution dual to the Cauchy–Riemann target and \lambda is a covector on the jet target. Testing map variations supported away from z gives D_u^*\eta=0 there. Elliptic regularity makes \eta continuously differentiable away from z, with the further finite regularity supplied by the coefficients. The choice of r gives more derivatives than are needed for all distributional products and integrations by parts below.

Removal of tangential coefficients. Although the index uses a fixed j, we may test every infinitesimal variation \dot j of the domain complex structure. On S^2 the map \xi\mapsto\mathcal L_\xi j is onto: its cokernel is H^{0,1}(TS^2)=0, by the line-bundle surjectivity criterion since \deg TS^2=2>-2 [55, Theorem 3.23]. Naturality under a domain diffeomorphism, with the marked point moved by the inverse diffeomorphism, expresses the \dot j test as a combination of the already allowed map and mark tests. The direct variation of the jet with respect to j vanishes at z, because \dd u(z)=0. Thus

At an immersion point these variations span the tangential Cauchy–Riemann residuals. Hence \eta annihilates that tangential subbundle away from z.

The two pullback coefficients. There are bundle maps L_A,L_C, with the regularity of the coefficients, from Cauchy–Riemann residuals to real two-forms on the domain such that, at a solution,

These identities hold also for variations of the background pair and of v. To see that they extend without loss of regularity at a critical point, choose j\partial_s=\partial_t and put e=(u_s+Ju_t)/2. Anti-invariance gives the exact identity

and the same identity holds for C_v. Differentiation at e=0 proves Equation (7.6). In particular the maps L_A,L_C contain \dd u, not its inverse, and extend continuously across every critical point. At an immersion point they vanish on tangential residuals and identify the real two-dimensional normal residual with the two pullback values: contraction with a nonzero complex tangent vector by a nonzero complex volume form pairs the complex normal line isomorphically with \C.

Consequently, on the injective immersion locus away from z, there are unique continuously differentiable real functions f_A,f_C such that

for tests supported in that locus. Let U be a relatively compact injective disk in the locus. A neighborhood of its image can be chosen to meet no other part of the curve. Restrictions of smooth ambient one-forms supported in this neighborhood are dense in \Omega^1_c(U) in the C^1 topology. Indeed, the available regularity of u gives a compactly supported C^2 ambient extension of any such source one-form in tubular coordinates. Approximate this extension in C^1 by smooth ambient one-forms, keeping their supports in the same isolated neighborhood. Their pullbacks converge in C^1. Testing the variations \dd\beta_A for these smooth primitives with u fixed and passing to the limit gives

since f_A is continuous on the compact source support. The jet contribution of this background variation is zero because \dd u(z)=0. Integration by parts proves \dd f_A=0 on U; independent variations of C prove \dd f_C=0. The complement of the finite exceptional set in S^2 is connected, so these functions are global constants k_A,k_C on that locus.

Define on the whole source the distribution with continuous coefficients

Continuity of the L’s and elliptic regularity away from z show that \eta-\eta_0 is supported at z, including when there are other critical or double points. For every map variation w, closedness of the two forms gives

Elimination of the point-supported remainder. Set R=\eta-\eta_0. The annihilator equation for map variations is

where \mathcal J_z is the linearization of the complex-linear first jet, with the mark fixed. The right side has distributional order at most one and is supported at z. A point-supported distribution is a finite sum of derivatives of the Dirac delta. If its largest nonzero order were m\geq1, its homogeneous coefficient polynomial R_m(\xi) would satisfy

There are enough coefficient derivatives for this comparison: a point-supported functional on W^{k-1,p} in real dimension two has order at most k-2 when p>2, while r\geq k+8. The first-order Cauchy–Riemann symbol is invertible for every real \xi\ne0, so this polynomial must vanish identically, a contradiction. Hence R=q\delta_z for a single covector q. Test Equation (7.8) on variations with w(z)=0. Since \dd u(z)=0, the values of D_uw and \mathcal J_zw at z are respectively the complex-antilinear and complex-linear parts of \nabla w(z). These parts can be specified independently by a smooth variation supported near z. It follows that q=0 and \lambda=0.

Finally, variation of v in Equation (7.4) gives

This map is onto \R^2. Applying the remaining annihilator \eta_0 to these tests yields k_A=k_C=0. Thus the universal derivative is onto. Its range is closed: the map-and-jet operator with background fixed is Fredholm and has finite-codimensional closed range, and adding parameter directions preserves this property.

Index and allowed genericity. The parametrized sphere operator has real index 4, since c_1(u^*TX)=0 [55, Theorem 3.22]. Adding v and the mark and imposing the first-jet constraint gives

For a path, allowing its parameter adds one and gives index 5. The projection from the universal zero set to the space of perturbing primitives is therefore Fredholm of index 4, or 5 for paths. The projection is C^6, and 6>5, so Sard–Smale applies with the chosen regularities [44, Theorem 1.3]. For a regular parameter its fiber would be a manifold of that dimension. But the six-dimensional group \operatorname{PSL}(2,\C) acts freely on simple parametrized spheres, with the marked point transformed contragrediently, and preserves the fiber. Freeness follows because a reparametrization fixing a simple map fixes its injective locus and hence is the identity. A manifold of dimension 4 or 5 cannot contain this six-dimensional orbit. The fiber is empty.

For a path, the local perturbations just used may be multiplied by arbitrary bumps in the path parameter, supported away from its endpoints. If finitely many immersed F-spheres are retained, a critical simple F-sphere cannot have one of those images. It is therefore disjoint from all of them by positivity of intersections, so every injective disk needed in the proof still admits the allowed local variations. Use the closure of smooth primitives supported off the retained images; this preserves them exactly without requiring a fixed forbidden tubular neighborhood. At a fixed model subdisk, every point already lies on a model F-curve. A competing critical sphere is disjoint from the whole subdisk for the same reason. The endpoint assertion follows with perturbations supported in its complement.

There are countably many map-space charts and homotopy components, so the regular parameter choices can be made simultaneously for all of them. The conclusions are open by Lemma 7.2. Approximation in the defining closures by smooth primitives therefore gives smooth perturbations with the same conclusion and the same support restrictions. For paths with immersed endpoints, openness allows fixed small parameter collars at both ends. Smallness ensures positivity with B fixed; the straight segment realizes the initial small perturbation by an exact positive-pair path.□

Proof. Represent nearby immersions by normal graphs over u, removing the infinitesimal reparametrizations. If u is simple, its stabilizer under reparametrization is trivial: an automorphism fixing the map fixes its injective locus and hence is the identity. The normal-graph slice then also parametrizes the local unparametrized moduli. For a multiply covered immersion we make the smoothness assertion only for this slice. The symplectic form A identifies the real normal bundle with T^*L by w\mapsto\alpha=u^*\iota_wA. Write along the source

where the nonzero sign of y is the one giving J tamed by B. Let *_L be the star on one-forms for the induced conformal structure and orientation, with *_L\alpha=-\alpha\circ j. The complex volume identity gives

Cartan’s formula now gives the normal derivative

The nonlinear pullbacks and their derivatives take values in pairs of exact two-forms: their integrals are the two fixed zero periods.

For completeness we solve Equation (7.12). Given exact two-forms \zeta_1,\zeta_2, choose \alpha_0 with \dd\alpha_0=\zeta_1. Adding \dd f changes only the second equation, by

After division by a positive area form its principal part is the nonzero signed multiple y\Delta f of the scalar Laplacian. Its sign is constant on connected L, it has no zeroth-order term, and the strong maximum principle shows that its kernel consists exactly of constants [22, Chapter 1, Theorem 1.6]. Here one first adjusts the overall sign of the operator and then applies the principle at a maximum on the connected closed surface. It has index zero, by homotopy through scalar elliptic operators to the signed Laplacian, with the sign of y held fixed; the Fredholm index is constant along this homotopy [51, Section 7, Propositions 7.3–7.4]. Its image has integral zero by Stokes’ theorem. Its cokernel has dimension one, so its range is exactly the integral-zero two-forms. We can therefore solve

This proves surjectivity onto the asserted target, with bounded elliptic right inverses in the usual Sobolev or Hölder spaces.

If \mathcal D\alpha=0, then \alpha is closed. Every class in H^1(L;\R) has a unique representative of this kind: start with any closed representative and solve Equation (7.13) for an exact correction. Uniqueness follows from the constant kernel of \mathcal P. Thus

is an isomorphism. The implicit function theorem applied to the pullback equations, with exact two-forms as target, proves the first assertion and its parameter version. For L=S^2 the derivative is an isomorphism, giving isolation and unique continuation.

For an embedded torus, the kernel has dimension two. In complex normal notation its elements solve the real-linear normal Cauchy–Riemann equation. Indeed, the normal projection of a linearized map equation is unchanged by variation of the source complex structure, and the two maps in Equation (7.6) identify that normal equation with Equation (7.12). The complex normal line has degree F^2=0. The similarity principle says that a nonzero solution has isolated positive zeros, whose total number with multiplicity is the degree; in the closed case this is the zero formula of [54, Section 2.2, Equation (2.7)]. Therefore every nonzero kernel section is nowhere vanishing. Evaluation at each point of the torus is consequently an isomorphism from the two-dimensional kernel to its real normal plane.

Let u_q, q\in\R^2 small, be the family supplied by the implicit function theorem. The derivative of (x,q)\mapsto u_q(x) is an isomorphism at every (x,0). Compactness of the torus gives a common small parameter disk on which this is a local diffeomorphism. The individual maps remain embeddings; different images cannot meet by F^2=0 and positivity. The immersion slice distinguishes the images, so evaluation is injective after shrinking the disk. It is thus a diffeomorphism onto an open tube, foliated by the u_q(L).□

Proof. The unparametrized sphere space is compact by Lemma 7.2 and discrete by Lemma 7.4, so it is finite. Over a path, the total space of such spheres is compact and, by the parameter version of Lemma 7.4, its projection to the path interval is a local diffeomorphism, also in relative half-interval charts at the endpoints. Hence it is a finite covering of the interval. Equivalently, each isolated sphere has a unique local continuation, and compactness prevents any additional sphere from entering those local descriptions. The number is constant.□

Proof. Denote the node by p and its two preimages on the normalization by p_+,p_-. We first identify the model with a neighborhood of K so that straight interpolation of the two pairs stays positive. We then localize this interpolation by two exact changes. At the node, equality of the forms as tensors makes the radial cutoff correction small. Along the rest of the sphere, the difference of the pairs need not be small; a logarithmic transverse cutoff makes its cutoff error small while retaining the positive interpolation.

Use x,y,\Omega from Equation (7.11) on a neighborhood of K. Choose the qualitative model of Theorem 8.1; write its normalization as u_Y:S^2\to Y with node preimages q_+,q_-. Uniformization supplies a biholomorphism f of the normalized spheres carrying p_\pm to q_\pm.

Matching the derivative and the complex volume. At p, the two complex tangent lines of the branches are transverse. Their prescribed tangent maps, obtained from f, define one complex linear isomorphism

Choose the nonzero complex scale h of the model volume so that T^*\Sigma_Y=\Omega(p). Along each smooth point of the normalization, the tangent map and equality of complex volumes determine the map on the normal quotient. More explicitly, a nonvanishing complex volume identifies the complex normal line with the dual of the tangent line. The prescribed tangent map therefore gives the required normal-line isomorphism globally. The possible lifts to a map of the two ambient complex plane bundles form an affine bundle with fiber the complex-linear maps from the normal line to the target tangent line. A smooth section exists by partitions of unity. Its values at both preimages of the node can be chosen to equal the single map T.

These prescribed first derivatives need only be realized to arbitrary accuracy along K, with exact realization at the node. Here is a construction that addresses the crossing. First straighten the two transverse branches by smooth ambient charts in the source and target. Writing the restrictions of f in these charts as f_+(u) and f_-(v), the product map \Phi_0(u,v)=(f_+(u),f_-(v)) is a genuine local diffeomorphism carrying the branches by f and having derivative T at p. Its derivative along either branch and the prescribed complex-linear derivative have the same tangent restriction and agree at p. They are consequently arbitrarily close on sufficiently short branch collars. On those collars interpolate their normal derivative data, keeping \Phi_0 near p and the prescribed data away from the collars. The interpolated derivatives remain invertible when the collars are sufficiently short.

Away from the crossing the normal tubular construction realizes these jets: in normal coordinates send the zero section by f and use the prescribed normal derivative on the normal coordinate. This can be patched to \Phi_0 where the jets already agree; a partition in normal coordinates preserves the prescribed first jets. Shrinking the tubes gives a diffeomorphism \Phi of neighborhoods of the nodal images. To check injectivity, any hypothetical pair of coincident images in arbitrarily shrinking neighborhoods would limit to two points of K with the same image. The map on the nodal image is a homeomorphism, and the map is a local diffeomorphism at every point, including the crossing. These facts rule out such a pair. We have arranged that

Both volumes vanish when pulled back to the normalization.

Put x_p=x(p), y_p=y(p) and set on this neighborhood

These forms are closed. If the volume matching along K were exact, their pointwise relation to the old pair would be

The ratio y_p/y is positive. Thus every straight interpolation of the two pairs in Equation (7.16) has the same positive anti-invariant plane and remains in the same component relative to B. In particular its triple Gram matrix has a uniform positive lower bound along K, with the interpolation parameter in [0,1]. Choose the approximation in Equation (7.15) small compared with this bound. The actual straight interpolation between (A,C) and (A_*,C_*) is then positive along K, and on a smaller neighborhood by compactness and continuity. The biholomorphism f uses the complex orientation selected by B, so B is positive on the transported model’s complex tangent line at each smooth point of K. Together with positivity of the triple, this selects the model complex structure as the sign tamed by B. That sign persists on a smaller connected model neighborhood, independently of whether y_p is positive or negative.

The first exact correction, at the crossing. For either closed difference \xi=A_*-A or \xi=C_*-C, one has \xi(p)=0 as a tensor, and the pullback of \xi to each branch vanishes. In a crossing chart with p=0, the radial homotopy primitive is

It satisfies |\beta(z)|\leq C|z|^2. Since radial contraction preserves each straightened branch, \beta pulls back to zero on both branches. Choose \chi_r equal to one on the ball of radius r, supported in the ball of radius 2r, and satisfying |\dd\chi_r|\leq C/r. The exact change \dd(\chi_r\beta) has norm O(r): the two terms \chi_r\xi and \dd\chi_r\wedge\beta both have that bound. For sufficiently small r its scalar multiples preserve positivity with B fixed. They preserve the common Lagrangian sphere because the primitive restricts to zero there. Do this for both differences. The resulting pair (A',C') equals (A_*,C_*) on the ball of radius r and differs from (A,C) by O(r) near that ball.

The exact correction along the rest of the sphere. The remaining differences \xi'=A_*-A' and \xi'=C_*-C' are closed, vanish on a neighborhood of the crossing, and pull back to zero on the source. Outside a smaller crossing ball, take normal tubes of the embedded part of the source. Normal contraction h_t(s,n)=(s,tn) gives a relative primitive \beta' satisfying

This is the homotopy identity \dd\beta'=\xi'-\pi^*u^*\xi'. Choose the tubes in the inner collars so that the contraction stays where \xi'=0. Then \beta'=0 on those collars and extends by zero into the crossing ball. The two branch tubes are disjoint outside that ball after shrinking. This gives a single smooth relative primitive on the required tubular region, without a compatibility condition left at the node.

Choose a transverse cutoff equal to one for d\leq r_{\rm in} and zero for d\geq r_{\rm out}, where the outer tube is already small. Using a fixed smooth function of \log(d/r_{\rm in})/\log(r_{\rm out}/r_{\rm in}) gives

The inner radius can be made sufficiently small that the right side is any prescribed \eta>0. For 0\leq t\leq1, the exact change is

and likewise for C'. The last terms have norm at most C\eta by Equations (7.18) and (7.19). The first terms are pointwise straight interpolation with parameter t\chi\in[0,1]. Taking r small in the first correction preserves the positive margin already established along K. Shrinking the outer tubes preserves that margin throughout them. Finally take \eta small compared with the margin. These choices prove positivity along the entire exact path.

The final pair agrees with the model on a neighborhood of the whole nodal image, and its primitives have compact support in the original chosen neighborhood. A sufficiently small complete model subdisk lies in this region: properness of p_Y and compactness of its central fiber imply that p_Y^{-1}(\Delta_\rho) lies in any chosen neighborhood of that fiber when \rho is sufficiently small. Every modification restricts to zero on the normalization, so the same sphere is retained. Disjoint supports prove the last assertion.□

Proof of Theorem 7.1. First apply Lemma 7.3 to make a small exact perturbation with B fixed so that all F-spheres are immersed. Connect it to the starting pair by a small straight segment of positive pairs. There is no assertion about immersion along this initial segment. By Corollary 7.5 the perturbed pair has finitely many nodal spheres K_1,\ldots,K_N, all disjoint. If N=0, the torus argument below already applies.

For N>0, use Lemma 7.6 at all the K_i, keeping B fixed and retaining every K_i along the resulting path. Choose smaller closed model subdisks at its endpoint. Make the endpoint generic by an arbitrarily small exact perturbation outside those subdisks. The relative assertion of Lemma 7.3 applies because any competing F-curve is disjoint from every model fiber. The endpoint now has no critical spheres, and the smaller model neighborhoods remain unchanged. Extend this endpoint perturbation a short distance into the path using a parameter cutoff; all retained K_i are still common Lagrangians.

The two ends of this latter path have only immersed spheres. Apply the path version of Lemma 7.3, keeping its endpoints and every K_i fixed, to remove all critical spheres from its interior. The resulting smooth path stays positive with B fixed. By Corollary 7.5, its endpoints have the same number of nodal spheres. All the original N spheres were retained, so they exhaust the endpoint spheres. This argument is needed to exclude additional immersed spheres outside the inserted model regions.

We now work at the endpoint. By Proposition 6.3, every point of X lies on a unique simple F-curve image. Each is an embedded torus or one of the K_i. Define S as the set of these images, with quotient map p_X. For a torus, Lemma 7.4 supplies an open tube diffeomorphic to T^2\times\Delta. It is saturated: any other F-curve meeting a torus in this tube must equal it by intersection positivity. For a nodal sphere choose an open model subdisk. Its fibers also represent F, since they are homologous to its central fiber. The same argument makes this open subset saturated. Its quotient is its base disk, since the proper holomorphic model projection is open and has connected fibers. Thus the torus tubes and the model subdisks give disk neighborhoods in S and cover it.

These saturated neighborhoods can be chosen arbitrarily small around an entire fiber. For tori this follows by shrinking the parameter disk in the tube; for nodes it follows from properness of the model projection. Two distinct fibers are disjoint compact subsets of X, so they admit disjoint ambient neighborhoods. Choose saturated neighborhoods inside them. Their images are disjoint open neighborhoods of the two points of S. The quotient is therefore Hausdorff.

The disk charts have smooth transition maps. To verify this even at a singular value, choose a smooth point of the corresponding fiber. Near that point both local quotient projections are smooth submersions with the same fibers. A local section of either submersion expresses the other base coordinate as a smooth function of it; interchanging the two gives its smooth inverse. The charts therefore make S a smooth surface without boundary and p_X a smooth map. They also make p_X open. Since X is second countable, images of a countable open basis under this open surjection form a countable basis for S. Compactness and connectedness of X give compactness and connectedness of S.

On regular fibers use their complex orientation and the complex orientation of the normal plane to orient the base. Evaluation along a connected torus preserves this choice, and the holomorphic model gives the same orientation on its regular part. Hence the orientation extends across each singular value. The map is proper because its domain is compact and its base is Hausdorff. Its singularities are precisely the ordinary holomorphic nodes of the models (locally s=z_1z_2 after complex coordinate changes). The pair vanishes on every fiber by construction, and the prescribed model identities hold on the unchanged smaller neighborhoods.

All paths used above consist of exact changes to A,C and keep B fixed. Their finitely many joins can be made smooth by reparametrizations constant to infinite order at the join times. Concatenating them proves the theorem.□

Proof. We compare the families through the degree-three divisors given by three times their sections. Their line bundles embed the fibers as plane cubics. Laurent coefficients at the section, measured in the normalized Abel coordinate, will identify the three-dimensional spaces of sections across the puncture; taking closures will then identify the completed families.

Each p_\rho is a flat proper family of connected reduced curves. For flatness, the local ring of the smooth total surface has no torsion over the local ring of the one-dimensional smooth base, since p_\rho-s_0 is not a zero divisor. Torsion-free modules over this discrete valuation ring are flat. The fibers have arithmetic genus one, and S_\rho is a Cartier divisor in the fiberwise smooth locus.

Set \mathcal L_\rho=\mathcal O_{Y_\rho}(3S_\rho). This line bundle is flat over the base, since it is locally free on the flat family. Its restriction to each fiber is a degree-three very ample line bundle, with h^0=3 and h^1=0. For the complex-linear Dolbeault operator on a line bundle of degree d over a closed connected Riemann surface of genus g, the complex index is d+1-g, and d>2g-2 gives surjectivity [55, Section 3.4, Theorems 3.22–3.23]. On a smooth genus-one fiber this gives the asserted dimensions. After imposing any length-two subscheme, the remaining degree is one and the first cohomology still vanishes. This proves very ampleness on those fibers. For the nodal curve one can check the assertion explicitly. Normalize it by \mathbb P^1, with the two preimages of the node at 0,\infty. A degree-three line bundle pulls back to \mathcal O_{\mathbb P^1}(3), with a nonzero gluing parameter \lambda for its fibers at these points. Its sections are degree-at-most-three polynomials with constant coefficient \lambda times the leading coefficient. The basis w,w^2,w^3+\lambda gives the map

It identifies exactly 0 and \infty; away from these points the ratio of the second coordinate to the first is w. At 0 and \infty the two tangent lines are distinct, as is seen in the affine chart about [0:0:1]. Thus it embeds the nodal curve as a nodal cubic. To compute its cohomology directly, let N denote the nodal fiber, \nu:\mathbb P^1\to N its normalization, and L=\mathcal L_\rho|_N. The gluing condition gives the exact sequence

where the last sheaf is supported at the node and its fiber is \C. In the polynomial trivializations just used, the map on global sections is the constant coefficient minus \lambda times the leading coefficient. It is surjective. The normalization is finite, so the cohomology of its direct image is that of \mathcal O_{\mathbb P^1}(3). The latter has h^0=4 and h^1=0. The cohomology exact sequence therefore gives h^0(N,L)=3 and h^1(N,L)=0.

We use the following proper holomorphic base-change theorem: for a proper holomorphic map to a reduced base and a coherent sheaf flat over that base, constancy of the fiber dimensions h^q implies that the q-th direct image is locally free and its restriction to each base point is the cohomology of that fiber; see [13, Section 7, Satz 5] and the correction [14]. Our map is proper and surjective, the disk is reduced, and \mathcal L_\rho is flat over it. Applying the theorem with q=0 and the constant value h^0=3 gives a holomorphic rank-three bundle

with those spaces as fibers. Evaluation gives a relative closed embedding in \mathbb P(E_\rho^*). Indeed the fiberwise embeddings and base change give the evaluation and separation properties locally over the base. At the nodal point, the fiber’s Zariski tangent space is the whole tangent space of the smooth total surface, so tangent separation there also gives an immersion of the total surface. Properness then gives the closed embedding.

Near the section p_\rho is a submersion. Write \Omega_\rho=f(s,w)\,ds\wedge dw, with w=0 on the section. The coordinate

is holomorphic, has nonzero relative derivative, and is zero on the section. It extends the normalized local Abel coordinate also across s=0. Regard sections of \mathcal L_\rho as meromorphic functions with poles of order at most three on S_\rho. Their Laurent coefficients in degrees -3,-2,-1,0 define

This map is fiberwise injective. A function in its kernel has no pole and hence is constant on the compact connected reduced fiber; its zero constant coefficient makes it zero. Thus its image is a rank-three subbundle, including over zero.

The images agree on the punctured disk by the regular elliptic identifications. Continuity of their Grassmannian-valued maps makes them agree also at zero, identifying the bundles E_\rho. Their relative cubic images agree as well: each is the closure of its image over the punctured disk, and both are reduced analytic subspaces. This gives an isomorphism of the completed families. The holomorphic two-forms agree on the regular part and therefore everywhere.□

Proof. We use the two-cycle comparison from [16, proof of Lemma 4.3]. The annular bundle retracts to the mapping torus of one shear. Its homology exact sequence is

The first term is generated by a fiber, and the last by the invariant a-circle. This circle has a sweep at b=0. These two cycles thus test all real two-periods.

The fiber periods were computed above. The sweep is the connection circle bundle over a base circle in t=\rho/2. The connection circle bundle over the full base disk in this slice is a solid torus bounding it in Y. On the sweep dt=0, and the base has only one tangent direction, so the restriction of dt\wedge\theta+V_\rho dx_1\wedge dx_2 is zero. The invariant comparison form also has zero restriction, because db=0 and the same dimensional observation applies.□

Completion of the proof of Theorem 8.1. The construction and Lemma 8.2 give the fixed holomorphic nodal disk. For general h=|h|e^{i\phi}, rotate \gamma_1+i\gamma_2 by e^{i\phi}, scale the full triple and the metric by |h|, and take

The holomorphic two-form becomes \Sigma=h\,ds\wedge dz, the fiber area becomes \epsilon, and the equal-square identities give Equation (8.3). The scaled annular expression is precisely Equation (8.4), with coefficient |h|^2/\epsilon. The exponential comparison and both period identities follow from Equation (8.23) and Lemma 8.3. The polynomial estimates persist because h is fixed and nonzero. This proves all assertions.□

Proof. In a local translation chart define the average by integrating pullbacks against normalized Haar measure on \R^2/\Z^2. The conjugacy in Equation (9.4) and invariance of Haar measure make the definitions agree on overlaps. Each pullback commutes with \dd, which proves the assertion about differentiation.

For exactness, choose a translation-invariant horizontal splitting of TX^\circ\to p^*TS^\circ and a smoothly varying invariant flat metric on the vertical tori. Such choices exist: make them locally and patch horizontal lifts and vertical inner products by partitions of unity pulled back from the base. The transition rule above preserves invariance, and the relevant sets of choices are affine or convex.

Filter differential forms by horizontal degree. On the associated graded complex the leading differential is the vertical exterior derivative \dd_v. For each horizontal degree, fiberwise Hodge theory [53, Equations (1.11)–(1.22)] provides a homotopy H_v=\dd_v^*G_v on the vertical complex, with the usual sign for its horizontal degree. Here G_v is the inverse of the vertical Laplacian on the complement of its kernel. Its harmonic forms are precisely the translation-invariant forms. On the zero-average subspace, therefore,

These operators act smoothly in the base: in a local torus chart they are the inverses on nonzero Fourier modes for a smoothly varying flat metric. They preserve zero average and use no values at other base points.

Suppose the smallest horizontal degree of the closed zero-average form \xi is r, and denote its component of that degree by \xi_r. Closedness gives \dd_v\xi_r=0. By Equation (9.5), subtracting \dd(H_v\xi_r) removes \xi_r, and the remainder is closed, has zero average, and has horizontal degree at least r+1. Repeat this step. The horizontal degree is at most two, so the procedure terminates. A component of vertical degree zero that is vertically closed and has zero average is already zero. The sum of the primitives gives \beta. Every operation is local in the base, including the differentiations used in the successive remainders. Hence the support assertion follows. Applying the construction to \xi-\operatorname{Av}\xi proves the remaining claims.□

Proof. Choose covectors e_1,e_2 vanishing on L and complementary covectors f_1,f_2 so that

Direct wedge multiplication yields

Writing h=\tr(M)/2 and T=M-h\Id gives

Thus the pair is definite exactly when \det T>0. Set

Then \det T=q^2-r_0^2-u^2. Its two positive components are q>\sqrt{r_0^2+u^2} and q<-\sqrt{r_0^2+u^2}, both convex; h and c are unrestricted. This proves the first assertion. A quotient two-form wedges to zero with both members of a fiber-Lagrangian pair, and its own square vanishes. This proves the assertion about basic additions, also when both members change.

At a fixed point all translated pullbacks of C lie in one of the two components: translations preserve A and can be connected to the identity within the torus. Their compact average remains strictly inside that convex component, as does the segment to the average. The common Lagrangian plane is a complex line by Lemma 2.2. Continuing the choice of sign of the almost complex structure along the path keeps its orientation on this plane, and hence the quotient orientation, fixed.□

Proof. A fixed tamer for J_{t_0} remains a tamer on a parameter neighborhood of t_0, by openness and compactness of X. Choose finitely many such neighborhoods and corresponding fixed representatives. Near 0 and 1 use the prescribed representatives. Choose a subordinate smooth nonnegative partition of unity in the parameter, with the endpoint representative having weight one sufficiently near its endpoint. Their weighted sum is closed on X and represents H, because all coefficients depend only on t and sum to one. It tames J_t by convexity of the taming cone.□

Proof. Let J_t be the structures defined by the pair, with the continued sign. First produce a convenient taming class. On the regular region, J_t preserves the vertical tangent planes and induces an oriented complex structure on each quotient T_xX/\ker\dd p_x. This quotient structure need not be independent of the point in the fiber. Nevertheless the positive base form satisfies

and it vanishes on vertical vectors. The restriction of B_0 to the oriented vertical planes stays positive.

On a compact regular region containing the supports of the pair changes, use a smooth splitting \zeta=v+h into vertical and horizontal vectors and any fixed background norm. Compactness in space and parameter gives positive constants a,b and a finite M such that

Thus B_0+Kp^*\nu_S tames for one sufficiently large K>0. On the unchanged nodal neighborhoods, B_0 already tames and p^*\nu_S is semipositive. Indeed, the model matching in Lemma 7.6 chooses its complex structure with the sign tamed by B_0, and its projection is holomorphic for that sign. The same K therefore works on all of X and at every parameter.

Put H_K=[B_0]+KF and V=[A_0]^\perp\cap[C_0]^\perp. By Equation (9.1), H_K is the class just constructed. It belongs to V, and

Consequently H_K and [B_0] lie in the same timelike component of V. To apply Theorem 4.2, let d be the class, identified with its Poincaré dual, of a nonconstant irreducible J_t-curve. It has H_Kd>0. By Corollary 6.2, either Fd\ne0 and its sign is the sign of the coefficient c in d^+=cF^+, or d=cF. Also [B_0]d=c[B_0]F, with [B_0]F>0. If [B_0]d were nonpositive, the alternatives would give c<0 and Fd\le0, so H_Kd=[B_0]d+KFd<0, a contradiction. Thus every such curve pairs positively with [B_0]. The hypotheses of Theorem 4.2 hold with the already tamed class H_K, and that theorem supplies a tamer in [B_0].

When the initial pair is (A_0,C_0), apply Lemma 9.4 to the family J_t and the fixed class [B_0], using B_0 at the initial endpoint and any prescribed endpoint tamer. This gives the asserted smooth representatives. By Lemma 2.2, their triples with (A_t,C_t) are positive.□

Proof of Proposition 9.1. We first average the pair and construct a closed invariant form of unit fiber area. We then adjust its two total pairings so that exact basic changes of the pair can impose pointwise orthogonality. Throughout, disks may be replaced by smaller nested disks, with closures inside the original model disks.

Averaging the second form. On X^\circ set \overline C=\operatorname{Av}C_0. It is closed and equals C_0 on every model disk, so it extends smoothly over X. The difference \overline C-C_0 is supported over a compact subset of S^\circ and is exact on X by Lemma 9.2. The path

is a definite pair path by Lemma 9.3, with the same periods and the same oriented fiber planes. It admits a completion in [B_0] by Lemma 9.5. For the remaining construction write A=A_0 and C=\overline C. Both are invariant.

A closed form of unit fiber area. The positive number q_0=[B_0]F is the common integral of B_0 over the regular fibers. Start with

This is closed and invariant, and its restriction to every fiber is the invariant two-form of integral one.

We next make it equal to \dd a\wedge\dd b near every puncture. On a punctured model disk, the two real two-homology generators are the fiber and the torus obtained by sweeping the invariant a-cycle around a base circle at b=0. One can see this from the homology sequence for a torus mapping torus: the monodromy is one shear, so the fiber contributes one generator and the invariant part of its first homology contributes one more. The second generator bounds a three-chain in the full nodal neighborhood, as described in Theorem 8.1. The smooth closed form B_0/q_0 consequently has periods 1 and 0 on these generators. Averaging does not change them, by Lemma 9.2. The form \dd a\wedge\dd b has the same periods: its fiber integral is one, and its restriction to the sweep at b=0 is zero. It is a well-defined closed form under the shear transition of the angular coordinates.

It follows that \Psi^{(0)}-\dd a\wedge\dd b=\dd\beta_j on the punctured neighborhood. Replace \beta_j by its average, so it is invariant. Choose a basic cutoff \chi_j that is one on a smaller punctured disk and zero near the outer boundary, and replace \Psi^{(0)} there by

These disjoint modifications give a smooth closed invariant form \Psi^{(1)} on X^\circ, equal to \dd a\wedge\dd b near the punctures. Its fiber restriction is unchanged: the two invariant forms being compared already have the same restriction of integral one, and a differential of a basic cutoff has zero vertical restriction.

The obstruction to an exact orthogonality correction. The form \Psi^{(1)} has the required fiber restriction and nodal behavior. It remains to arrange orthogonality to the pair while preserving the pair’s cohomology classes. Invariance reduces these pointwise equations to a calculation on the base.

Let \Phi be an invariant two-form on X^\circ whose restriction to each fiber has integral one. For any invariant four-form \Xi, we have the pointwise identity

Indeed, only the vertical restriction of \Phi contributes to its wedge with a basic two-form. Invariant top forms have constant vertical density on each fiber and are therefore determined by their fiber integral; both sides of Equation (9.7) have the same integral on every fiber after fixing two base tangent vectors.

In particular, replacing A by A-p^*p_*(\Phi\wedge A) makes it orthogonal to \Phi, and the same holds for C. On the closed oriented surface S, a smooth two-form is exact precisely when its integral is zero. We therefore first adjust \Psi^{(1)} so that both resulting base two-forms have zero integral. Their vanishing near the punctures will allow us to regard them as smooth forms on all of S.

Removing the two total pairings. There are smooth closed invariant forms \widetilde A,\widetilde C on X, representing [A],[C], supported over a compact subset of S^\circ, and vanishing on vertical planes. To construct them, observe that a full one-node neighborhood has the homotopy type of a torus with its primitive vanishing cycle collapsed; its real second homology is generated by the fiber. Hence A,C are exact there, since their fiber integrals vanish. Subtract differentials of local primitives multiplied by basic cutoffs equal to one near the nodal fiber. This gives global cohomologous forms that vanish on smaller model neighborhoods. Their restrictions to vertical planes remain zero, because both the original forms and the differentials of the cutoffs have that property in the required restriction. Averaging these forms, and extending by zero near the nodes, gives \widetilde A,\widetilde C. Their cohomology classes are unchanged by Lemma 9.2.

On a punctured model disk, \Sigma_j=h_j\dd s\wedge(\dd a+\tau\dd b): the term b\tau'(s)\dd s in \dd z disappears after wedging with \dd s. Thus A,C have only mixed base–fiber terms there, and

Consequently the integrals

are well-defined: their integrands vanish over smaller punctured disks. The matrix of pairings of \widetilde A,\widetilde C with A,C is their cohomology Gram matrix, hence the identity under our normalization. Set

The form \Psi remains closed, invariant, of unit fiber area, and equal to \dd a\wedge\dd b near the punctures. By construction,

Without the normalization one would solve the same two equations using the positive invertible period Gram matrix.

Removing the pointwise pairings. Fiber integration of the invariant four-forms gives base two-forms

They vanish near the punctures and so extend by zero to smooth forms on S. Their integrals are zero by Equation (9.9). Since S is a closed connected oriented surface, there are smooth one-forms \kappa_A,\kappa_C on S with \zeta_A=\dd\kappa_A and \zeta_C=\dd\kappa_C.

Now make the simultaneous basic deformation

Both changes are globally exact, since the primitives are the smooth pullbacks of \kappa_A,\kappa_C. The changed forms are invariant, have zero vertical restriction, and agree with the original pair on smaller model neighborhoods. Their wedge-product matrix is unchanged by Lemma 9.3, so the path remains definite. Equation (9.7) shows that its endpoint satisfies

Moreover, contraction of a basic two-form with a vertical vector is zero. Hence the vertical Hamiltonian fields of base functions for A_t' are the same as those for A, and their period lattice and torus translations are unchanged.

Finally concatenate the two pair paths, making their parameters flat at the join. They preserve the original pair periods, oriented fibers, and smaller nodal models. Lemma 9.5 gives a smooth completion by forms in [B_0], with initial value B_0. This proves all the assertions. We henceforth denote the endpoint pair again by A,C and its chosen third form by B.□

Proof. We first check the three types of region in the construction. On a model core, A and C are fixed constant linear combinations of the two parallel real components of \Sigma, and E_\eps is the local parallel Kähler form. The model metric has volume A^2/2. Consequently it is exactly g_\eps there. The intrinsic charts and bounds of Theorem 8.1 therefore apply. This argument transports the metric, forms, and charts together through the holomorphic identification of the model disks. It requires no bound on derivatives of that identification in a separate fixed smooth atlas.

On the complement of smaller nodal disks, use finitely many fixed regular bundle charts. The coefficients in Equation (10.5) and all their fixed-order derivatives grow at most polynomially in \eps^{-1}. Its products with A,C vanish and its square is A^2. The remaining two-by-two wedge block is that of the fixed definite pair. It is uniformly positive on this compact region, and the same assertion holds on the cores because there its coefficients are fixed constants. The exponential annular gluing errors preserve these bounds on all of X.

For completeness, the passage from these form bounds to metric bounds is algebraic. Normalize the frame by the positive square root of its wedge Gram matrix, obtaining \Theta_1,\Theta_2,\Theta_3 with \Theta_i\wedge\Theta_j=2\delta_{ij}\nu and with the frame orientation chosen for a positive metric. In oriented coordinates, the identity

follows by evaluating on a standard self-dual orthonormal frame. In the fixed regular charts this gives polynomial coefficient and derivative bounds for g_\eps. Its determinant in these charts is fixed by \nu, so the adjugate formula gives polynomial bounds for the inverse metric as well. The core metric and inverse bounds were already supplied by the transported model charts. Matrix square roots and inverses have bounded derivatives on the uniformly positive Gram matrices in question. This also proves the claimed frame bounds. On the fixed annuli all comparisons remain polynomial after the exponential changes. Curvature and any fixed number of its covariant derivatives are consequently polynomially bounded.

We spell out why local chart control gives a polynomial global cover. Shrink the charts uniformly by powers of \eps so that every point has a chart containing an intrinsic ball of radius \rho_\eps\ge c_1\eps^{a_1}. In that chart both the coordinate domain radius and the upper and lower metric eigenvalue bounds are controlled by powers of \eps. A coordinate ball of polynomial radius lies in a fixed smaller intrinsic ball, and its volume density has a polynomial lower bound. It follows that every intrinsic ball of radius, say, \rho_\eps/8 has volume at least c_2\eps^{a_2}, after a further common polynomial shrinkage if needed. A maximal disjoint collection of these balls has at most (2c_2\eps^{a_2})^{-1} members because the total volume is 1/2. Their enlarged balls cover X and still fit in controlled charts. Coordinate cutoff functions and a normalized partition of unity have polynomial derivative bounds. Even the crude overlap bound by the number of charts is polynomial. This proves (i) with the asserted global control.

Euclidean Sobolev estimates on this cover, followed by the polynomial coordinate and covariant-derivative comparisons, give polynomial intrinsic Sobolev constants. Since the dimension is four and k\ge6, derivatives of order at most \lfloor k/2\rfloor are bounded in L^\infty by the H^k norm with such a constant. Apply the intrinsic product rule to each derivative of h\wedge\widetilde h, placing the factor with fewer derivatives in L^\infty and the other in L^2. This proves Equation (10.12).

Before normalization the residuals vanish on the cores and outside the gluing annuli. On the annuli their fixed-coordinate derivatives are exponentially small; all conversions just described cost only powers of \eps^{-1}. The constant corrections in Equation (10.9) introduce exponentially small coefficients multiplying forms with polynomial intrinsic bounds. The fixed total volume then gives Equation (10.14) and proves (iv).□

Proof. Put E_1=A, E_2=C, and E_3=D^0_\eps, and let M be their pointwise inner-product Gram matrix for g_\eps. The equations uniquely specify the self-dual part \phi=h^+: if f_i=t_i\nu, then

Moreover

Each E_i is closed and self-dual, hence harmonic. Proposition 2.3 gives b^+(X)=3, so these three linearly independent forms are a basis of the harmonic self-dual forms. Thus Equation (10.19) is exactly the solvability condition for \dd^+a=\phi.

One may see existence directly from the Hodge Green operator G for g_\eps [53, Equations (2.1)–(2.2)]. Since \phi is orthogonal to the harmonic self-dual forms and the Hodge Laplacian commutes with the star, G\phi is self-dual and \Delta G\phi=\phi. For a self-dual two-form \psi, 2(\dd\dd^*\psi)^+=\Delta\psi. Therefore

is exact and has self-dual part \phi. This formula proves existence for each fixed metric; we will not estimate the Green operator itself.

If h is exact, Stokes’ theorem gives

In particular an exact anti-self-dual two-form is zero, proving uniqueness. This identity is also the required L^2 estimate, independent of any positive spectral gap.

For higher derivatives use

The integrated Weitzenböck identity for two-forms gives

To iterate, apply the same identity to the tensor-valued form \nabla^j h. Commuting covariant differentiation with alternating derivative and divergence produces only curvature derivatives multiplied by lower derivatives of h. Using Equation (10.22), this yields, for fixed j,

where K_{j,\eps} is a polynomial in finitely many suprema of curvature and its derivatives. Those suprema are polynomially bounded by Lemma 10.3. Induction starting with Equation (10.21) thus bounds \|h\|_{H^k} by a polynomial multiple of \|\phi\|_{H^k}, without losing derivatives. Finally Equation (10.18) and the frame and inverse-Gram bounds give the same estimate in terms of f. Smooth approximation, or the fixed-metric Green formula, gives the Sobolev version. Uniqueness is always asserted for h, rather than for its one-form primitive.□

Proof of Theorem 10.1. Fix once and for all an integer k\ge6. Every constant and exponent in the following argument uses only finitely many derivatives at this order. Let \mathcal E^k_\eps be the closed subspace of H^k two-forms consisting of exact forms. Equivalently, its elements are distributionally closed and have zero harmonic projection; they possess H^{k+1} primitives for this fixed metric.

Writing D_\eps=D^0_\eps+h, the desired equations become the fixed-point equation

This map is defined on all of \mathcal E^k_\eps. Indeed, the first two data have integral zero by Equation (10.10), as does the free term in the third slot. For each exact iterate, including Sobolev iterates, \int_Xh\wedge h=0 by approximation and Stokes’ theorem. Thus every iteration satisfies precisely the three mean conditions of Lemma 10.4.

Let \delta_\eps=\|\mathcal R_\eps\|_{H^k}, let P_\eps be an inverse bound from Equation (10.17), and let Q_\eps be a multiplication bound from Equation (10.12). Increasing them if necessary, write

On the ball of radius R_\eps=2P_\eps\delta_\eps in \mathcal E^k_\eps,

Choose \eps so small that

This is possible because its left side is at most C\eps^{-(2p+q+s)}e^{-c/\eps}, which tends to zero. The ball is invariant and the Lipschitz constant is less than 1/2. The contraction theorem gives an exact solution with

If the residual is zero, the same conclusion follows directly by taking h=0. Polynomial losses can always be absorbed by replacing c with any fixed smaller positive exponent. Thus the correction is exponentially small also in C^0, by the polynomial Sobolev embedding bound. The uniformly positive Gram matrix of the approximate triple stays positive, with the same choice of sign. Equation (10.23) gives Equation (10.2) exactly.

We finish by proving spatial smoothness; no simultaneous estimate over infinitely many derivative orders is needed. For the now fixed \eps, choose a Coulomb primitive a\in H^{k+1} of h, so \dd a=h and \dd^*a=0. The three equations for D^0_\eps+\dd a, together with this gauge, form a first-order nonlinear system for the four components of a. At the solution the principal symbol of its linearization sends a covector-valued unknown b at a nonzero covector \xi to the three wedge slots of \xi\wedge b against A,C,D_\eps, together with \iota_{\xi^\sharp}b. If the three wedge slots vanish, \xi\wedge b belongs to the negative-definite wedge-orthogonal complement of the positive three-plane. But (\xi\wedge b)^2=0, so \xi\wedge b=0. It follows that b is a multiple of \xi, and the gauge slot then forces b=0. The symbol is square, hence invertible.

Since k\geq6 in real dimension four, the initial Sobolev order gives a\in C^{4,\alpha} and at least C^{3,\alpha} coefficients for this elliptic linearization, for some 0<\alpha<1. Differentiate the system in a smooth coordinate chart. The derivative of a satisfies the linearized elliptic system, with right-hand side involving only smooth background derivatives and already controlled first derivatives of a. These coefficients and the right-hand side are C^{3,\alpha}, so local first-order elliptic estimates with finitely differentiable coefficients [50, Section 4, Theorem 4.3] give \partial a\in C^{4,\alpha} and hence a\in C^{5,\alpha}. Repeating this argument gives smoothness. All constants in this bootstrap may depend on the fixed \eps. The resulting D_\eps is therefore smooth and proves the theorem.□

Proof of Proposition 10.2. We identify integral curve classes with their Poincaré duals. Choose disjoint closed nodal neighborhoods N_i whose boundaries lie outside all gluing cutoffs, and put U=X\setminus\bigcup_i\operatorname{int}N_i. The restriction of E_\eps to U is D^{\mathrm{sf}}_\eps. If there is at least one node, U fibers over a compact surface with nonempty boundary, whose second real cohomology is zero. Thus the basic term in Equation (10.5) is exact on U, and

In particular its restriction divided by \eps is bounded.

When there is a node, the kernel of the restriction map H^2(X;\R)\longrightarrow H^2(U;\R) is exactly \R F. To verify this assertion, excision and Poincaré–Lefschetz duality identify

A one-node neighborhood has the homotopy type of a torus with its primitive vanishing cycle killed, and its second homology is generated by the fiber. Under the map to H^2(X;\R) these generators give the Poincaré dual fiber classes, all equal to the nonzero class F. The long exact sequence of the pair gives the assertion. Hence restriction induces an injective map from H^2(X;\R)/\R F onto its image. Its inverse on that finite-dimensional image is bounded. Equation (10.29) proves that the class [D^0_\eps] modulo \R F is O(\eps).

The nonlinear correction is exact, so [D_\eps]=[D^0_\eps]\in V. Choose a fixed linear complement of \R F in V and use it to lift the quotient class divided by \eps to a bounded vector u_\eps. This gives [D_\eps]=\lambda_\eps F+\eps u_\eps. Choose any regular fiber outside the gluing region. The original glued form has integral \eps there; the subtracted forms A,C have zero fiber integrals, and the correction is exact. Therefore

The square normalization and F^2=0 now give

Boundedness of u_\eps and Equation (10.30) prove Equation (10.3).

If there are no nodal fibers, \Psi is defined on all of X and the basic term in Equation (10.5) is a multiple of F in cohomology. The form D^{\mathrm{sf}}_\eps already solves the equations, so one takes it for D_\eps; the same conclusions follow with u_\eps=[\Psi], r_\eps=1, and the coefficient computed by Equation (10.31).

It remains to establish the uniform implication. By Corollary 6.2, every curve class under consideration with [D_\eps]d>0 satisfies d^2\ge-2 and the following exhaustive alternatives: either m=Fd\ne0 has the same sign as [B]d, or d=cF for a nonzero real number c. The latter alternative has no sign difficulty, since both [D_\eps]F and [B]F are positive.

Choose a null vector G_1\in V with FG_1=1. Such a choice is obtained from any vector pairing to one with F by subtracting half its square times F. The orthogonal complement of \operatorname{span}(F,G_1) in the Lorentzian space V is negative definite. Decompose

where z,w_\eps lie in that complement, and use the Euclidean norm |z|^2=-z^2 there. We have \alpha_\eps=u_\eps F\ge a_0 and uniform bounds for \alpha_\eps,\beta_\eps,w_\eps.

Suppose that m<0. Integrality of F and d means m=-a for an integer a\ge1. The square bound gives

Complete the square in the negative-definite component:

The constant K is independent of a,z,d, and of the future almost-complex structure. It follows that

whenever \eps^2<c_0/K (with no restriction from this inequality if K=0). This contradicts [D_\eps]d>0 and excludes every negative integer m simultaneously. The remaining alternative m>0 gives [B]d>0, proving Equation (10.4).

Finally, both [D_\eps] and [B] have positive square and pair positively with the same nonzero null vector F. In a Lorentzian space the two components of the positive cone are distinguished by the sign of pairing with a fixed nonzero null vector: no positive vector is orthogonal to it. Thus the two classes are in the same component. All restrictions on \eps in this proof depend only on fixed construction data and the original periods. They can therefore be imposed before choosing any later pair deformation.□

Proof. The compact-surface Kähler criterion gives a Kähler form \kappa because b_1 is even [3, Theorem 11]; see also [24, Corollaire 5.7]. Taming gives

For the second inequality, pointwise \beta\kappa=\beta^{1,1}\kappa>0, since the invariant part of a tamer is positive definite. Together with H^2>0, these inequalities are the numerical Kähler-class criterion on a surface [3, Corollary 15]. Equivalently, the segment from [\kappa] to H has positive square and remains positive on every curve, so the component condition in the numerical characterization of the Kähler cone is satisfied [4, Theorem 0.1].

Choose a Kähler representative \kappa_H\in H and set f=\log(\nu/\kappa_H^2). The volume hypothesis says \int_X e^f\kappa_H^2=\int_X\kappa_H^2. Yau’s theorem produces a smooth cohomologous Kähler form \beta_1 with \beta_1^2=e^f\kappa_H^2=\nu [58, Section 4, Theorem 1].□

Proof. Every holomorphic two-form is f\Omega for a global holomorphic function f, hence is a constant multiple of \Omega. Thus H^{2,0}(X)=\C[\Omega]. The real plane generated by its real and imaginary parts is nondegenerate and positive for the intersection form: their squares are equal and positive and their mixed product vanishes. Hodge decomposition and type considerations make its orthogonal complement exactly H^{1,1}(X;\R).□

Proof of Theorem 1.1. We trace the actual form paths and then construct the endpoint; Figure 1 summarizes the final stages. All coefficient changes below are constant orthogonal changes, and will be undone at the end.

The prepared pair.. Choose the primitive null class F of Proposition 6.1 and rotate the coefficient basis so that [B] points in the direction of F^+ and the pair [A],[C] spans its orthogonal plane in P. The rotated initial triple is denoted (A,C,B); its cohomology Gram matrix is still the identity.

The families result and Proposition 6.3 and Theorem 7.1 give an exact pair deformation to a torus fibration with the stated standard nodal neighborhoods, keeping B fixed. Proposition 9.1 then gives a further path of positive closed triples to invariant regular data with the auxiliary form \Psi. Both stages preserve the three individual classes and start at the actual representatives supplied by the preceding stage. Retain the notation (A,C,B) for the resulting triple. In particular

The auxiliary solution.. Fix a sufficiently small \epsilon>0 for Theorem 10.1 and Proposition 10.2, and set D=D_\epsilon. The form D is closed and smooth and satisfies

The triple with forms A,C,D is positive. The same choice of \epsilon gives the numerical conclusion of Proposition 10.2 for every later definite pair with the classes [A],[C]. These choices precede the Kähler constructions that follow.

The first Kähler step.. By Lemma 2.5, A\pm iD is a nowhere-zero holomorphic two-form for an integrable complex structure I; choose the sign so that C tames I. This is possible by Lemma 2.2 and positivity of (A,D,C). Proposition 2.3 gives even b_1, so the surface is Kähler. The period relations (11.1) and (11.2) give [C][A]=[C][D]=0. Therefore Lemma 11.2 puts [C] in real type (1,1).

Apply Lemma 11.1 with H=[C] and \nu=A^2. Its integral condition holds by (11.1). We obtain a Kähler representative C_1\in[C] satisfying

Let C_t=(1-t)C+tC_1. Both endpoints tame I, so the entire segment does. Thus (A,D,C_t) is positive, and (A,C_t) is a definite pair in the original pair classes. Moreover D tames its associated almost-complex structure with the continuously chosen sign. Indeed D is wedge-orthogonal to both pair forms by (11.2) and (11.3), and has positive square.

Keeping the third class.. Let J_t now denote the structure associated with the moving pair (A,C_t) and tamed by D. For every irreducible J_t curve, its class d has [D]d>0. Proposition 10.2 implies [B]d>0, and places [B] and [D] in the same open timelike cone of the fixed space V from (2.3). The hypotheses of Theorem 4.2 therefore hold with U=[D] and H=[B]. Thus [B] contains a tamer for every J_t.

At t=0 this sign of J_t agrees with the sign previously tamed by the actual third form B. To check this pointwise, take a regular fiber with its prepared orientation. Both B and D restrict positively to its tangent planes, which are complex lines for the prepared pair. They therefore select the same sign there. On the connected manifold X this choice of sign cannot jump, so B tames J_0 everywhere. We will choose the family of tamers after fixing its final representative in the next step.

The second Kähler step.. The final pair (A,C_1) is orthogonal with equal square by (11.3). Thus the structure J_1 already selected by D is integrable, with holomorphic volume form A\pm iC_1 for the corresponding sign. The preceding step supplies a representative of [B] taming this same J_1. Even b_1 gives Kählerness again, and (11.1) together with Lemma 11.2 puts [B] in type (1,1). Apply Lemma 11.1 with H=[B] and \nu=A^2. This yields a Kähler representative B_1\in[B] such that

Now apply Lemma 9.4 to the family J_t, prescribing the actual prepared form B at t=0 and the newly constructed B_1 at t=1. It gives closed tamers B_t\in[B] with these two endpoint values. Thus (A,C_t,B_t) is a path of positive closed triples in the original classes, joining the prepared triple to (A,C_1,B_1) itself.

Concatenate the finitely many triple paths, reparametrizing each to be constant to all orders at its endpoints. This produces a smooth path on [0,1]. Each component is closed and remains in its original class, and positivity holds at every parameter. By (11.3) and (11.4), if (\eta_1,\eta_2,\eta_3)=(A,C_1,B_1) and \mu_1=A^2/2, then

By Lemma 2.5, these forms are parallel and self-dual for a hyperkähler metric with volume \mu_1. Finally undo the initial orthogonal change of coefficients. Orthogonality preserves the displayed identity and restores the original ordered classes, proving the theorem.□