Prescribed large-volume charges on threefolds with trivial canonical bundle

Proof. The rational Chern character identifies the Grothendieck group with the rational Chow group [28, Tags 0FDI, 0FEW, 0FB2 and 0FB5]. In Riemann–Roch the Euler pairing is the intersection pairing after applying the invertible operations of dualizing a class and multiplying it by \td(X). Its radical therefore gives precisely the numerical quotient of the Chow group. In dimensions zero and three this quotient is \QQ; in the other two dimensions it is the divisor–curve pairing in (2.1).

For a rational ample class A, the Hodge index theorem makes (D,D')\mapsto ADD' nondegenerate on N^1(X)_\QQ; see [15, Theorem 6.4 and Corollary 6.5] for its real-ample formulation. The divisor–curve pairing is nondegenerate by numerical equivalence, so multiplication by A is an isomorphism N^1(X)_\QQ\to N_1(X)_\QQ. Divisors, their products with A, and A^3 consequently span every summand of (2.1).

Choose an affine slice transverse to the ray of H_* and a rational simplex in its ample cone whose interior contains that ray. Its q=\rho(X) vertices are linearly independent. Taking sufficiently large integral multiples makes them very ample without changing their positive cone. This also explains the rank-one case, where a single very ample class suffices.

Pushforward is defined on numerical groups by adjunction with perfect complexes. If i_*v=0 numerically, the projection formula gives

Chern characters of line bundles on Y span its polynomial Chow ring, and their restrictions span (2.1). Nondegeneracy of the Euler pairing gives v=0. The spanning argument here uses dimension three; it makes no analogous claim in arbitrary dimension.□

Proof. We use the product-with-a-curve construction of [24, Lemma 5.7 and Remark 5.8], the positive-genus product construction of [22, Theorem 4.5], and the support induction of [21, Lemma 6.6]. For n=1, the usual slope stability on C has charge -x_{\varnothing}+iw_1x_{\{1\}} and the assertion is immediate.

Suppose that the scaled support bound has been proved in dimension n-1. Omitting factor r splits the coordinates into x',x'', where x' consists of those containing r, with that index removed. The product construction has charge

It initially gives support with respect to its own quotient lattice. Every choice of omitted factor gives the same exponential charge and the same phase 1 for point sheaves. The uniqueness theorem [22, Theorem 3.11] therefore identifies the slicings: that theorem permits the two support lattices to differ. We may consequently test all the product support forms on the same semistable objects, before proving support on the full coordinate lattice.

Write M_{\widehat r} for scaling the remaining coordinates by their products of weights. The scaled coordinates split as

The lower-dimensional charge in scaled coordinates is a fixed linear map \widehat Z. Increase its support constant C_{n-1} if needed and choose the support form

It is nonnegative on lower-dimensional semistable classes. Crucially,

not merely for semistable classes. This is the norm form used in the product support estimate.

For clarity, that estimate has the following coefficient condition. Write Z_{n-1}(x')=\alpha+i\beta and Z_{n-1}(x'')=\gamma+i\delta. With equal positive product parameters s=t=w_r, the form

is nonnegative on product-semistable classes when 0\le\eta_r\le w_r/C_{n-1}^2 [24, Lemma 5.7]; this is the substitution in [21, Lemma 6.6]. Choose 0<\lambda_r<C_{n-1}^{-2}, independently of R, and put \eta_r=\lambda_rw_r. If

then multiplication of the product form by w_r gives

All powers of R have disappeared. This form is nonnegative on the common semistable objects just constructed.

The total charge in these coordinates is (\widehat\gamma+\widehat\beta) +i(\widehat\delta-\widehat\alpha). On its kernel,

It is nonpositive there, and vanishes only when y'=0; the kernel condition then also gives \widehat Z(y'')=0. Sum (2.4) over all omitted factors. If the sum vanishes on the total charge kernel, then y'=0 for every factor. This kills every coordinate with I\ne\varnothing, and the charge kills the remaining top-degree coordinate. The summed form is thus negative definite on the charge kernel and nonnegative on semistables. Compactness on the unit sphere of its nonnegative cone gives \|y(u)\|\le C_n|Z_{\mathbf w}(u)| for every semistable class u. The forms and all choices were made in the scaled coordinates. In particular they are independent of R and can be kept uniform on compact sets of positive ratios. This completes the support induction. The geometricity assertion is part of the positive-genus product construction; see also [21, Theorem 4.1(1)].□

Proof. Cheng proves the two-block construction and phase estimate in [8, Proposition 4.3 and Remark 4.4]. We apply the same quotient construction block by block, using Lemma 2.2 to control all coordinates uniformly in the volume.

First take R and the positive weights s_h rational. Put n=\sum_hd_h and give each of the d_h elliptic factors in block h the weight w_r=2Rs_h. The lemma supplies the starting stability condition and its scaled support bound. Only the coordinate lattice of the elliptic power is used here; full numerical support will follow on Y from the explicit pullback calculation below.

First apply [21, Proposition 5.1] to the stability condition just constructed on the coordinate lattice. It supplies the descent by (\ZZ/2)^n to (\mathbb P^1)^n and the Bayer property with k=0 for every effective line bundle \OO(a_1,\ldots,a_n), with all a_r\ge0. We then take the symmetric quotients one block at a time.

Here is the filtration used at a symmetric step. For Q=(\mathbb P^1)^d\to\mathbb P^d, there is a finite filtration of \OO_{Q\times_{\mathbb P^d}Q} whose successive quotients are

where p_2 is the second projection and \Gamma(g_j) is the graph of the permutation g_j [21, Definition 3.19 and Example 3.20(3)]. At a block stage the map is f:Q\times M\to\mathbb P^d\times M, where M is the product of the other blocks, and its fiber square is (Q\times_{\mathbb P^d}Q)\times M. Flat pullback to this product preserves the filtration and its exact sequences. Its graph automorphisms are g_j\times\operatorname{id}_M, and its line bundles remain pulled back from Q.

The finite-flat descent criterion [21, Proposition 3.21] therefore requires precisely invariance under these permutations and the Bayer properties with k=0 for these effective line bundles. Indeed, for a phase-\phi object F, the factors of f^*f_*F are \mathcal L_j^{-1}\otimes g_{j*}F; the two properties put their largest phases at most \phi. On the elliptic product, signs and permutations within each equal-weight block preserve the charge and point phases, so the uniqueness theorem used above gives their invariance. Block permutations normalize the full sign group; their invariance descends through the completed involution quotient. For subsequent block quotients, the required properties pass to the remaining blocks as follows [21, Lemma 3.9 and Remark 3.14(3)]. At each quotient f:V\to V', pullback computes extremal phases for the induced slicing \mathcal S'=f_\#\mathcal S. If a line bundle L comes from an unquotiented block, then

The Bayer property for f^*L therefore descends to L. An automorphism of another block commutes with f, so its invariance descends by the same pullback description. Thus every remaining quotient has the required Bayer properties and invariance, for any finite number of blocks.

For completeness, let g be the resulting map from the elliptic power to Y. Pullback satisfies

If |I\cap\text{block }h|=d_h-j_h, then the scaled I-coordinate of g^*u is

The fixed multiplicities are \prod_h\binom{d_h}{j_h}. Pullback is thus injective on the complete Chern-character lattice of Y, with norm comparison constants independent of R. Semistability pulls back, and Z_{\mathbf w}(g^*u)=\deg(g)U_R(u). This proves (2.6).

For the phase estimate, vary the real tensor parameter on each elliptic factor successively. The one-factor argument in the proof of [9, Theorem 2.4], used in [8, Remark 4.4], gives its absolute variation divided by \pi w_r. A twist e_h\xi_h pulls back with coefficient 2e_h on the d_h factors in block h; its factor 2 cancels the factor 2 in w_r. Adding the variations gives (2.7). Finally the deformation and multiplication-isogeny argument in the proof of [8, Proposition 4.3] extends the rational-weight construction continuously to positive real weights. It applies factor by factor: the scaled support forms just constructed remain fixed on compact positive weight sets, the exponential charges vary continuously, and uniqueness on overlapping deformation neighborhoods identifies the lifts. The closed support inequalities and the phase bounds persist. This also extends R to all real R\ge1. Point stability in this extension follows separately from [21, Theorem 4.1(1)] on the elliptic power and preservation of geometricity under finite pushforward [21, Remark 3.11]. Concretely, for the finite faithfully flat quotient g, the sheaf g^*\OO_y has a finite filtration by point sheaves, all stable of phase 1. A phase-one subobject and quotient of \OO_y pull back to objects whose Jordan–Hölder factors are among these points, so both pullbacks are coherent finite-length sheaves. Faithful flatness descends this cohomological concentration. The original subobject and quotient are therefore sheaves, and simplicity of \OO_y rules out a proper nonzero subobject. Continuity and the charge value -1 normalize the point phase to 1.□

Proof. This is the point-object argument of [7, Lemma 10.1]; we include it to specify the dimension and boundary statements being used. Let E be stable with phase 0<\phi\le1 and not a point of phase 1. Phase vanishing gives \Hom(E,\OO_x[j])=0 for j<0. For j\ge n, Serre duality identifies this space with \Hom(\OO_x,E[n-j])^*, which is zero by phases. At j=n,\phi=1, use nonisomorphism of the two stable objects. The canonical bundle causes no change, since its restriction to a point is trivial. A minimal free complex at each closed point consequently has nonzero terms only in degrees [-n+1,0]. Its bottom cohomology is a subsheaf of a locally free sheaf and hence is torsion-free. At phase 1 we also have \Hom(E,\OO_x)=0 for every x, so \mathcal H^0(E)=0. Point sheaves themselves satisfy the stated cohomological bounds. Finite Jordan–Hölder filtrations and extensions give these assertions for the whole heart; in phase 1 its zeroth cohomology is an extension of sheaves supported at finitely many points.

A coherent sheaf cannot receive a nonzero morphism from an object whose phase is greater than 1, by the cohomological bound just proved and a shift. It cannot map nontrivially to an object of phase at most 1-n: such an object is in D^{\ge1}, including the endpoint by the phase-one observation. Applied to its extreme Harder–Narasimhan factors, these vanishings prove \Coh(V)\subset\mathcal R(1-n,1]. Shifting this inclusion and the heart bound proves (2.10).□

Proof. If E is strictly semistable, take a first stable subobject G in a finite Jordan–Hölder filtration. Its class cannot be real proportional to [E]. Indeed their aligned charges would make [G]=c[E] with 0<c<1, whereas primitivity makes an integral class on this line an integral multiple of [E]. Choose a phase-window heart of length one centered at their common phase. The triangle G\to E\to E/G remains a short exact sequence in this heart for small deformations: all three objects stay strictly inside its phase window. An independent small perturbation of the two charge values makes the phase of G greater than that of E. This contradicts the assumed semistability of E in the entire neighborhood.□

Proof. Take V bounded in the b_h coordinates, and restrict W by |\ell_h/s_h-1|<\eta. In the scaled coordinates (2.5), expansion of the two exponentials gives

To see the uniformity, the coefficient of u_{\mathbf j} in the uncorrected exponential has degree n-|\mathbf j| in t. Varying the positive coefficients s_h to \ell_h changes its ratio to the weight of M_t by O(\eta). A codimension-d>0 term of e^{-\sum b_h\xi_h}\gamma-1 lowers that degree by d and hence contributes O(t^{-d}). There are finitely many monomials, and their coefficients are bounded on the chosen neighborhoods. This proves (2.17) independently of the object.

Combine it with (2.6). Choose \eta first and then T so that

To check the support hypothesis of the deformation theorem explicitly, put

It is nonnegative on the original semistable classes. If U'(u)=0 and \|(U'-U_t)M_t^{-1}\|\le\delta<1/D, then

Thus Q_t is negative definite on every charge kernel along the straight deformation, and throughout the indicated open charge region. The support-property deformation theorem [2, Theorem 1.2] lifts the straight path while preserving Q_t. On its nonnegative cone, \|M_tu\|\le D|U_t(u)|, so the relative charge change is less than \sin(\pi\epsilon). The quantitative phase estimate [2, Lemma 2.9] then makes the slicing distance less than \epsilon as long as it is less than 1/4. It is zero at the start; continuity along the path and \epsilon<1/8 prevent a first exit from that 1/4 neighborhood. This proves the asserted strict distance bound. The charge bound is strict, so for each fixed t it also holds on an open neighborhood in all ambient numerical charges. This is the space on which we deform, before applying restriction.□

Proof. Set A=\OO_Y(1,\ldots,1). Choose a locally free resolution P^\bullet\to i_*\OO_X with P^0=\OO_Y and with each P^{-k} a finite sum of line bundles A^{-b} for k\ge1. Such a resolution is obtained by repeatedly generating the successive coherent kernels after sufficiently positive twists. Fix N>\dim Y+1 and truncate after the terms in degrees 0,\ldots,-(N-1). The remaining kernel is shifted by [N], so the cofiltration remainder lies in D^{\le-N}(Y).

For a defining ambient linear form of a smooth section S_h, first form the cone of the untruncated map P^\bullet(-\xi_h)\to P^\bullet. It resolves (i\circ j_h)_*\OO_{S_h}, and its term in degree -k is

Truncate each cone resolution after its terms in degrees 0,\ldots,-(N-1). Its remaining kernel is again shifted by [N], so its cofiltration remainder also lies in D^{\le-N}(Y). The line bundles in all these terms are independent of the chosen smooth section. Include all their summands, in particular the mixed twists A^{-b}\otimes\OO_Y(-\xi_h), together with A and \OO_Y(\xi_h), in one finite list. This list is fixed before the parameters B,H,t and the sections are chosen.

Let \mathcal R be a lift from Lemma 2.6. Tensoring is an isometry for slicing distance, so (2.7) gives, after increasing the single threshold T,

for every line bundle L in the finite list, uniformly in the claimed sector. The triangle inequality bounds the left side by 2\epsilon+O(t^{-1}). The same strict inequalities hold in a full open neighborhood of each lifted charge.

In particular tensoring by A^{-1}[1] has the Bayer property. [21, Proposition 3.27] makes every point sheaf on Y semistable throughout that open neighborhood. Its numerical class is primitive, because \chi(\OO_Y,\OO_y)=1. Lemma 2.5 upgrades semistability to stability. At the charge (2.15) its value is -1, and closeness to \mathcal R_t fixes its phase to be 1.

We can now apply Lemma 2.4 on Y. It supplies (2.11) with n_1=-\dim Y and n_2=0. Each resolution term L[k], k\ge1, has the Bayer property: (2.18) says that tensoring by L lowers no extremal phase by as much as one, and the shift [k] restores the required lower bound. The initial \OO_Y term is the identity. The chosen N exceeds the truncation bound. Thus the finite-map criterion applies to the closed immersion i and, when needed, to every i\circ j_h. It supplies slicings with the indicated charges and Harder–Narasimhan filtrations. Definition (2.12) and uniqueness of those filtrations show that pushforward computes their extremal phases.

Pushforward is exact for these hearts and detects zero objects. A proper destabilizing subobject of a point would therefore give one for the stable point on Y. The induced points are stable of phase 1. If E is semistable on X, the ambient support inequality and the injectivity in Lemma 2.1 give

This is support on the full Euler numerical group of X. For each auxiliary surface the ambient lattice through pushforward suffices; no injectivity assertion for its full numerical group is needed. Finally the support bound on a finite-rank lattice gives local finiteness: in a sufficiently short phase interval, the projection of a nonzero semistable charge onto its middle ray has a fixed positive lower bound. Additivity of this projection bounds the length of any chain of strict subobjects or quotients of a fixed object in that interval. The induced slicings are thus locally finite, as asserted.□

Proof. Lemma 2.4 gives \mathcal R(0,1]\subset D^{[-1,0]}(S) and \Coh(S)\subset\mathcal R(-1,1]. The comparison of these two bounded t-structures makes \mathcal R(0,1] a tilt of \Coh(S), with torsion pair

For instance their torsion sequence is obtained by truncating a sheaf at phase 0; the cohomological range ensures that both pieces are sheaves.

If G\in\mathcal T_0, every quotient sheaf also belongs to \mathcal T_0 and has nonnegative imaginary charge. A nonzero positive-rank quotient cannot have imaginary charge zero: it would lie in \mathcal R(1), whereas Lemma 2.4 makes its zeroth cohomology zero-dimensional. Thus all positive-rank quotients of G have strictly positive slope. Applying this to its last slope Harder–Narasimhan quotient proves \mathcal T_0\subset\mathcal T_{\omega,\beta}. If F\in\mathcal F_0, then F[1]\in\mathcal R(0,1], so F is torsion-free by the same lemma. Every subsheaf remains in \mathcal F_0 and has nonpositive imaginary charge. Its maximal-slope factor therefore has slope at most zero, giving \mathcal F_0\subset\mathcal F_{\omega,\beta}.

These inclusions identify the torsion pairs. Indeed the torsion sequence for either pair, applied to an object in the corresponding part of the other, leaves a quotient or subobject in the intersection of its two orthogonal parts, hence zero. This proof includes slope zero; it does not use a generic-parameter argument.□

Proof. The range and torsion-freeness follow from Lemma 2.4. Write F=\mathcal H^{-2}(E) and G=\mathcal H^0(E). For each h, choose a general smooth S_h which is a nonzerodivisor on the finitely many cohomology sheaves and the subobjects and quotients used below. Those sheaves and sequences then restrict without Tor. Such choices are available inside the complete very ample linear system.

By (3.5), E(L_h)|_{S_h} has largest phase at most 2. The surface tilt (3.3), applied to its lowest ordinary cohomology, implies F(L_h)|_{S_h}\in\mathcal F_h. Similarly (3.4) places E|_{S_h} in \mathcal P_h(>0), and the highest ordinary cohomology of the surface tilt gives G|_{S_h}\in\mathcal T_h. These conclusions can also be read directly from the aisles: \mathcal P_h(\le2) has its degree -2 cohomology in \mathcal F_h, while \mathcal P_h(>0) has its degree 0 cohomology in \mathcal T_h.

If F\ne0, restrict its maximal-slope subsheaf F'\subset F. The class \mathcal F_h is closed under subsheaves, so the shifted twist in the definition of the surface slope gives

If G has a positive-rank last slope quotient G'', quotient-closure of \mathcal T_h gives

For a torsion sheaf G its minimum slope is +\infty instead. Multiply the displayed inequalities by the positive \ell_h and sum. Since \sum_h\ell_hL_h=H, they are exactly (3.7). Positivity of the weights is what permits this argument for a real ample H.□

Proof. Let D\in\mathcal T_{H,B} be a sheaf with m^-_{H,B}(D)>0. Lemma 2.4 places all its phases in (-2,1]. If a last Harder–Narasimhan factor G had phase in (-2,-1], then G[2]\in\mathcal P(0,1]. It follows that G\in D^{[0,2]} and m^+_{H,B}(\mathcal H^0(G))\le-c. The canonical nonzero map D\to G is tested on its degree-zero cohomology:

Here the equality follows from G\in D^{\ge0} and standard truncation; the vanishing is the slope comparison, including the case of torsion D since the target is torsion-free. This contradiction excludes that phase interval and gives D\in\mathcal P(-1,1].

Now let J\in\mathcal F_{H,B}, so J is torsion-free with m^+_{H,B}(J)\le0. The coherent-sheaf range gives J[1]\in\mathcal P(-1,2]. If a first Harder–Narasimhan factor G had phase in (1,2], then G[-1]\in\mathcal P(0,1], G\in D^{[-3,-1]}, and m^-_{H,B}(\mathcal H^{-1}(G))>c. The canonical nonzero map G\to J[1] is tested on its top cohomology:

again by truncation and slope. Thus J[1]\in\mathcal P(-1,1]. Extensions prove (3.8).

The final assertion is the comparison theorem for two bounded hearts one of which lies in two adjacent degrees of the other. Explicitly, truncate an object of \mathcal B at phase 0 in \mathcal P. The inclusion (3.8) and its corresponding inclusions of aisles make both truncations objects of \mathcal B; they give its torsion sequence with the two stated intersections. Tilting that sequence recovers \mathcal P(0,1].□

Proof. Let E belong to the stated intersection. Its ordinary cohomology is J=\mathcal H^{-1}(E)\in\mathcal F_{H,B} and U=\mathcal H^0(E)\in\mathcal T_{H,B}. Lemma 2.4 makes U zero-dimensional. Choose the smooth sections S_h to avoid its finite support and to restrict J without Tor. If J\ne0, it is torsion-free of positive rank, and E|_{S_h}=J|_{S_h}[1]. Inequality (3.4) gives \phi^-_{\mathcal P_h}(J|_{S_h})\ge0. Its phases as a coherent sheaf are also at most 1, by Lemma 2.4. Hence \Im Z_h(J|_{S_h})\ge0 and (3.2) gives

The positive weighted sum says m_{H,B}(J)\ge c>0, contrary to m^+_{H,B}(J)\le0. Therefore J=0 and E=U.□

Proof. Let J=\mathcal H^{-1}(E) and U=\mathcal H^0(E). The first torsion pair makes each contribution to H^2\ch_1^B(E)=H^2\ch_1^B(U)-H^2\ch_1^B(J) nonnegative. If the sum is zero, J, when nonzero, is slope-semistable of slope zero, and U has dimension at most one. Indeed positive-rank quotients in the positive torsion part have strictly positive degree, and a nonzero effective divisorial cycle has positive intersection with H^2. For real H, the Kähler Bogomolov–Gieseker inequality follows from [23, Theorem 1.1, Equation (1.6)], applied with zero Higgs field to the reflexive hull J^{**} when J\ne0. This hull is semistable for the same H: intersecting a subsheaf of J^{**} with J preserves its rank and first Chern class. The quotient Q=J^{**}/J has codimension at least two, and, writing r=\rk J, additivity gives

where \ch_2(Q) is an effective curve cycle, so the inequality descends to J. The same argument on a surface uses an effective zero-cycle correction; neither application changes the real polarization. Classical Bogomolov–Gieseker and the real-ample Hodge index theorem [15, Theorem 6.4], applied to H^2\ch_1^B(J)=0, give H\ch_2^B(J)\le0. The curve cycle of U is effective. Consequently

If J\ne0, the last term is strictly positive. If J=0 and U has a nonzero one-dimensional part, its curve degree is strictly positive. Thus equality is possible exactly when E=U is zero-dimensional.□

Proof. Compare (\mathcal T',\mathcal F') with the torsion pair (\mathcal T_0,\mathcal F_0) of Lemma 3.3. For D\in\mathcal T', take its latter torsion sequence 0\to D_1\to D\to D_2\to0. A nonzero D_2\in\mathcal F_0 has \Im Z(D_2)\le0. Its denominator cannot vanish: Lemma 3.5 would make it zero-dimensional, whereas zero-dimensional sheaves are in \mathcal P(1) and \mathcal F_0\subset\mathcal P(-1,0]. Thus D_2 has positive denominator and \nu^{\mathrm{phys}}_{B,tH}(D_2)\le0, contradicting the strictly positive slopes of every nonzero quotient of D. Hence \mathcal T'\subset\mathcal T_0.

For D\in\mathcal F', consider D_1 in the same torsion sequence. A nonzero subobject of D has second slope at most zero, so D_1 has positive denominator and \Im Z(D_1)\le0. But D_1\in\mathcal T_0\subset\mathcal P(0,1] makes its imaginary charge nonnegative. It must therefore vanish, and D_1\in\mathcal P(1). Lemma 3.4 makes it zero-dimensional, whose second slope is +\infty, a contradiction. Thus \mathcal F'\subset\mathcal F_0. The two inclusions identify the torsion pairs, and their tilts coincide.□

Proof. Write V=\mathcal H^{-1}(E) and U=\mathcal H^0(E) for ordinary cohomology. The defining torsion pair gives d_B(V)\le0 and d_B(U)\ge0, whence d_B(E)=d_B(U)-d_B(V)\ge0. If it is zero, a nonzero V is torsion-free and slope-semistable of slope zero, whereas U has dimension at most one. Bogomolov–Gieseker and Hodge index give w_B(V)\le0; effectiveness of the curve cycle gives w_B(U)\ge0. Thus

Equality forces V=0 and the curve cycle of U to vanish, so U is zero-dimensional. The converse is immediate.

In a short exact sequence 0\to F\to E\to G\to0 with d(E)>0, a subobject with d(F)=0 has slope +\infty while d(G)>0. It is forbidden by semistability. If both denominators are positive, additivity makes \nu(E) their weighted average, proving the desired equivalences. If d(G)=0, then n(G)\ge0, so \nu(F)\le\nu(E) and comparison with \nu(G)=+\infty is automatic.□

Proof. The first inequality of (4.10) gives a^2r^2+2rn\le d^2, and hence |r|\le 2|n|/a^2+d/a. The equality w=n+a^2r/2 bounds |w|. Finally the second inequality gives

which proves the norm bound. These expressions also prove uniformity on compact a-intervals.□

Proof. Let \nu(E)=u\in\RR. If E is not stable, there is a proper nonzero subobject F with \nu(F)=\nu(E/F)=u and positive denominators. The object F is semistable by Lemma 4.1. To obtain a splitting into two semistable objects, we may have to absorb a zero-dimensional subobject of the quotient back into F. Put G=E/F and take its Harder–Narasimhan filtration. Its last factor is a quotient of E, so, if its slope is finite, that slope is at least u. The other finite slopes are no smaller. There must be a finite factor since d(G)>0. Additivity of n-ud, together with the nonnegativity of n on infinite-slope factors, now forces all finite slopes to equal u and all infinite-slope factors to have d=n=0. Thus G is either semistable or has a zero-dimensional initial subobject T followed by a semistable quotient of slope u. Replace F by the preimage of T. It is still a proper equal-slope subobject of E, hence semistable, and its quotient is now semistable as well.

Repeated splitting cannot produce arbitrarily many nonzero factors. For a semistable factor of slope u, Lemma 4.2 gives |r|\le C_a(1+|u|)d. If r\ne0, integrality of rank therefore bounds d below by 1/(C_a(1+|u|)). If r=0, then d=H^2\ch_1/h is positive and belongs to h^{-1}\ZZ, so d\ge1/h. Each factor has denominator at least the smaller of these two positive constants, and their denominators sum to d(E). A finite number of splits consequently gives stable factors.

For the extension assertion, intersect a subobject with the first term and take its image in the last term. A nonzero intersection or image has positive denominator and slope at most u by semistability. Their sum has the same properties, proving the subobject criterion.□

Proof. Since w_b=ud_b+a^2r/2, the twist formulas give w_c=R^2r/2 and d_c=d_b-ur. Thus \Delta=d_c^2-R^2r^2\ge0. If r\ne0, the negative alternative d_c\le-R|r| would give d_b=d_c+ur\le-(R-|u|)|r|<0. It follows that d_c\ge R|r|; for r=0 we have d_c=d_b>0. On the whole open arc, therefore,

Consider the set of points on the arc where E is semistable in the corresponding heart. It contains the original point. It is closed within the arc: the formal charge has positive real part, and continuity of the largest and smallest phases makes both converge to its finite phase. In particular, this argument also ensures membership in the limiting heart. It is open within the arc as well. At a semistable point take the finite stable filtration from Lemma 4.3. Every factor has the same slope there and thus the same c,R in (4.12). Openness of stability keeps all these finitely many factors stable in nearby hearts, with their common slope -x. Their fixed triangles are short exact sequences in those hearts, and their extensions are semistable by the preceding lemma. Connectedness of the arc proves the assertion.□

Proof. For rational \omega=\sqrt3aH and rational B, [4, Proposition 5.1.3(b)] gives the triangle and semistability directly: its hypothesis is finite maximal tilt slope, which holds for a finite-slope semistable object. In that triangle \widetilde E has ordinary cohomology in degrees -1,0, whereas T_0[-1] is in degree 1. Hence \widetilde E is the stated ordinary truncation; it depends only on E.

First suppose E is stable at real parameters. By openness it is stable at a sequence of nearby parameters with rational \omega,B. The rational triangles all have the same truncation \tau^{\le0}\mathbb D(E) and the same sheaf \mathcal H^1(\mathbb D(E)). Their mirror finite phases converge to the phase of slope -u, with the denominator still positive. Continuity of extremal phases therefore gives semistability of this fixed truncation in \mathcal B_{H,-B}.

For a semistable E, use Lemma 4.3 and induct on its number of stable factors. For an extension 0\to E_1\to E\to E_2\to0, duality reverses the triangle. We must show that its degree-zero cohomology in the mirror tilt heart is semistable and that its degree-one cohomology is zero-dimensional. The cohomology sequence is

and all other tilt cohomology vanishes. Subobjects and quotients of a zero-dimensional sheaf in this heart are zero-dimensional: additivity first forces both denominators to vanish, then both nonnegative numerators to vanish, and Lemma 4.1 applies. Consequently the kernel of \widetilde E_1\to T_{0,2} has the same positive denominator and the same slope as \widetilde E_1; it is semistable by the subobject criterion. The degree-zero term is an extension of it by \widetilde E_2, and is semistable of slope -u. The degree-one term is zero-dimensional. This proves (4.13), including its truncation description, for the extension.

Finally the shifted dual has characters (-r,c_B,-\ch_2^B,\ch_3^B) at twist -B. The term T_0[-1] has third character -\operatorname{length}(T_0), so taking characters of the triangle gives (4.14).□

Proof. Write

The characteristic-zero semistable case of Langer’s restriction inequality [18, Theorem 3.1] has zero extremal-slope term. On X it gives, for raw restricted slopes s_i=h\mu_i,

Since \sum r_i\mu_i=r\mu(G), the left side is h^2r\sum_i r_i(\mu_i-\mu(G))^2. Division gives (5.2). If the restriction is semistable, its left side is zero, and the inequality follows directly from \xi_G\ge0. Refinement at a fixed slope leaves the sum unchanged. For a surface restriction, apply the same theorem with its very ample divisor mH. The raw slopes are s_i=mh\mu_i and the right side is m^2h\operatorname{Disc}(G)=m^2h^2r^2\xi_G; the factors m^2h^2r cancel in the same way.

We use the corrected very-general formulation of the theorem [17, p. 1211]. In the ample polarizations used here, that addendum also permits general divisors: the stated difficulty concerns the remaining nef polarizations when they are not ample.

For each original factor, the weighted mean of its restricted slopes is unchanged. Subtracting a fixed scalar \mu(T) from these slopes therefore does not change their variance. The elementary identity

proves the second-moment assertion. From X to S, restriction preserves the total w_T; hence \sum r_i\delta_{G_i}^T=\sum r_i\mu_T(G_i)^2-2w_T(G) changes by exactly the same amount. Sum these identities over the original factors. Their restricted and refined filtrations are exact for a suitable choice of the divisor, as justified next.□

Proof. Choose m so large that (m-1)H-K_X is ample, H^1(X,\OO_X((m-1)H))=0, and m\ge\mu(K_S)+1. The last slope is independent of S, by adjunction K_S=(K_X+H)|_S. Kodaira vanishing on S [10, Chapter VII, Theorem 3.3] gives the first equality in (5.3), and the restriction sequence on X gives surjectivity on sections.

Bertini’s theorem supplies a smooth pair meeting transversely. Shrink an analytic neighborhood of that pair so that its closure is compact and all intersections remain smooth. The universal surfaces and curves over this closure are smooth proper families. Local trivializations on a finite cover give uniform bounds for the metrics, their derivatives, coordinate comparisons, and the partitions of unity used below. Surjectivity on sections allows the curve on each chosen surface to vary in an analytic open subset of its complete linear system.

For a fixed coherent sheaf, a general defining section avoids all its associated points, and is thus a nonzerodivisor. Applying this to the quotients in a finite collection of exact sequences preserves their exactness under restriction. To ensure torsion-freeness, embed a torsion-free sheaf into a vector bundle and require the defining section to be a nonzerodivisor on its cokernel. Restriction then embeds the restricted sheaf into the restricted bundle. Such an embedding exists because a sufficiently positive twist of the dual is globally generated and the original sheaf injects into its double dual. Repeating the argument on the surface gives the curve assertion. Applied to the ordinary cohomology sheaves of a bounded complex, the nonzerodivisor condition kills the first Tor terms in the restriction spectral sequence, giving the asserted cohomology identification.

These are finitely many nonempty Zariski-open conditions at each stage. A very-general condition removes at most countably many proper algebraic closed subsets. Each such subset has empty analytic interior, so the Baire theorem ensures a choice in every one of our nonempty analytic neighborhoods. The geometric constants were fixed before this choice and do not depend on the sheaves being restricted.□

Proof. The uniform charts and partitions of unity give the Nash inequality

For completeness, in a Euclidean chart split the Fourier integral at frequency R: its low-frequency part is bounded by CR^n\|f\|_1^2, and its high-frequency part by R^{-2}\|df\|_2^2. Optimize in R, and use a fixed finite partition of unity. Derivatives of the partition produce the additional \|f\|_2^2 term. All comparison constants are uniform over the chosen families.

Set L_{\mathrm{sc}}=\Delta_{\mathrm{sc}}+1 and f_s=e^{-sL_{\mathrm{sc}}}f. The scalar semigroup is positivity preserving and contracts L^1. For y(s)=\|f_s\|_2^2, (5.5) and y'=-2\langle L_{\mathrm{sc}}f_s,f_s\rangle imply

Integration gives \|e^{-sL_{\mathrm{sc}}}\|_{1\to2}\le Cs^{-n/4}. Selfadjointness and composition give \|e^{-sL_{\mathrm{sc}}}\|_{1\to\infty}\le Cs^{-n/2} for every s>0. Multiplication by e^s gives the stated estimate for k_s when s\le1.

For a section-valued heat solution, Kato’s inequality and the maximum principle give

One obtains the differential inequality by differentiating (|v|^2+\varepsilon^2)^{1/2} and then letting \varepsilon\downarrow0; compatibility of the connection with the metric is the only bundle input. Approximate a vector-valued point mass at y in this inequality. This bounds the operator norm of K_s(x,y) by the scalar kernel of e^{-sL_{\mathrm{sc}}}, and proves (5.4).□

Proof. On a curve G is locally free. On a surface its reflexive hull W=G^{**} is locally free and slope-stable: intersecting a subsheaf of W with G preserves rank and first Chern class, since W/G is zero-dimensional. The Hermitian–Einstein theorem [11, 29] therefore gives a metric on W for which i\Lambda F_W is the scalar determined by \mu(G). Its proportionality constant is fixed by the dimension and our volume normalization.

For a holomorphic section s of W, the Bochner identity gives

Scalar heat comparison consequently yields |s(x)|^2\le e^{C u\mu(G)_+}(e^{-u\Delta_{\mathrm{sc}}}|s|^2)(x). Take u=(1+\mu(G)_+)^{-1} and apply Lemma 5.3. The evaluation operator, with the L^2 norm on H^0(Y,W), satisfies

Give H^0(Y,J)\subset H^0(Y,W) the induced L^2 norm and choose an orthonormal basis s_1,\ldots,s_N. Outside a proper analytic subset, their evaluations lie in a subspace of dimension at most q. Thus

Integration proves (5.6); the volumes are uniformly bounded. The case q=0 means J=0. Dualization and tensoring with a line preserve slope stability, so the same proof gives the last assertion.□

Proof. The compact positive operator L^{-1/2}PL^{-1/2} has at least \dim U eigenvalues greater than or equal to 1: apply min–max to the subspace L^{1/2}U and (5.8). Its nonzero eigenvalues, with multiplicity, are those of P^{1/2}L^{-1}P^{1/2}. Define an operator with an additional time variable by

Integration of the heat semigroup gives \mathcal B^*\mathcal B=P^{1/2}L^{-1}P^{1/2}.

Fix 0<\eta<1 and split \mathcal B=\mathcal B_{\le}+\mathcal B_> by inserting the fiberwise spectral projections \mathbf1_{sP\le\eta} and \mathbf1_{sP>\eta} immediately after P^{1/2}. Heat contraction gives

Indeed, each positive eigenvalue p contributes \int_0^{\eta/p}p\,ds=\eta, and a zero eigenvalue contributes zero.

Let K_s be the kernel of e^{-sL}. The semigroup identity and Lemma 5.3, in real dimension four, imply

The last identity follows eigenvalue by eigenvalue from \int_{\eta/p}^\infty s^{-2}p\,ds=p^2/\eta. All integrands are nonnegative, so Tonelli’s theorem applies.

On the spectral subspace where \mathcal B^*\mathcal B\ge1, the triangle inequality gives \|\mathcal B_>v\|\ge(1-\sqrt\eta)\|v\|. Apply this to an orthonormal basis of that subspace and sum. Taking \eta=1/4 proves (5.9).□

Proof. We prove the bound for one factor G, writing r=\rk G, C_1^{\mathrm{num}}=c_1(G)/r, and \rho=\|C_1^{\mathrm{num}}-T\|. We separate \rho\ge1, where the numerical error already controls rank, from \rho<1, where the curvature argument is needed.

Suppose first that \rho\ge1. The class T is uniformly bounded once A is fixed, so 1+\|C_1^{\mathrm{num}}\|^2\le C_A\rho^2. Lemma 5.4, applied to G(tH), gives h^0(G(tH))\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2). Put W=G^{**}. Serre duality identifies H^2(G(tH))^* with \Hom(G,K_S(-tH))=H^0(W^*\otimes K_S(-tH)); the equality follows by extending across the finite support of W/G. The dual version of Lemma 5.4 gives the same bound for h^2.

Surface Riemann–Roch reads

The Hodge norm controls both the intersection form and pairing with K_S. Consequently |\chi(G(tH))|\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2+\xi_G). Since I is finite, h^1=h^0+h^2-\chi is at most Cr(\rho^2+\xi_G), as required.

Now assume \rho<1. The hull sequence 0\to G\to W\to Q\to0 has Q zero-dimensional and

Here W is stable, as in the proof of Lemma 5.4, and \xi_W\ge0 by the classical Bogomolov–Gieseker inequality. The long exact cohomology sequence gives h^1(G(tH))\le h^1(W(tH))+\operatorname{length}Q. It remains to bound h^1(W(tH)) by Cr\xi_W.

Equip W with its Hermitian–Einstein metric. Write its Chern curvature as

The real (1,1)-form \vartheta is closed and has constant contraction, hence is harmonic. It represents C_1^{\mathrm{num}}. The Hodge star formula on primitive (1,1)-forms identifies its L^2 norm with a fixed multiple of the divisor norm. Uniform elliptic estimates on our compact smooth families therefore give a uniform bound for the supremum norm of a harmonic representative in terms of this divisor norm. In particular, since \rho<1 and |b|\le1,

The fixed constant C_1 bounds the harmonic representatives of B_0|_S+bH and the unit ball of errors. Here \omega_S is the induced Kähler form, and the line bundle \OO_S(H) has curvature 2\pi\omega_S. Choose A_0 so that the right side is strictly positive with a fixed lower bound. This choice uses only the upper endpoint M, not the lower endpoint of I.

The trace-free curvature F_0 is primitive. Chern–Weil and the primitive Hodge star formula give

Indeed \operatorname{Disc}(W)=-r\int_S \operatorname{tr}((iF_0/2\pi)\wedge(iF_0/2\pi)), and the last integrand is the negative squared norm because F_0 is primitive. The fixed conversion constant depends on the curvature convention, not on r.

By Serre duality and the Dolbeault theorem, h^1(W(tH)) is the dimension of the space U of harmonic (2,1)-forms with values in W^*(-tH). On this bundle the Bochner–Kodaira–Nakano identity [10, Chapter VII, Corollary 1.3] is

For the central curvature, (5.12) makes the commutator on (2,1)-forms at least \lambda I for a fixed \lambda>0. To see the sign directly, diagonalize the central (1,1)-form in a unitary frame. On top holomorphic degree, its commutator is the sum of the positive eigenvalues whose antiholomorphic indices occur; there is one such index here. Denote the remaining selfadjoint curvature endomorphism by K_0. Wedge and contraction act on fixed-dimensional form spaces, so

We explain explicitly how to retain an elliptic operator in (5.14). Let \nabla be the total metric connection on (2,1)-forms with these bundle coefficients. For a pure-type form the terms in \partial u+\bar\partial u have different types, as do their adjoints. Hence the quadratic form of the connection de Rham Laplacian is the sum of the forms of \Delta' and \Delta''. On U, the latter vanishes, so the de Rham form equals the \Delta' form. The Weitzenböck formula, restricted to this type, consequently gives

where R_0 is selfadjoint and |R_0|_{\mathrm{HS}}\le C|F_0|_{\mathrm{HS}}. The bounded term contains the tangent curvature and the central bundle curvature; its bound C_2 is uniform for fixed A,I. The formula follows by anticommuting wedge and contraction in d_\nabla and d_\nabla^*: the second-order part is \nabla^*\nabla, and the remaining terms contract the tangent and bundle curvature. Only the fixed form dimension enters the contraction constants.

Choose 0<\varepsilon\le\min\{1,\lambda/(C_2+1)\}. On U, retain \varepsilon\Delta' in (5.14), discard the nonnegative remainder, and use (5.16). This gives

Our choice ensures \lambda-\varepsilon C_2\ge\varepsilon. Thus, with spectral absolute values of selfadjoint matrices,

Moreover,

This uses the Hilbert–Schmidt norm of the matrices, without replacing them by their operator norms times the identity.

Lemma 5.5, followed by (5.13), now proves h^1(W(tH))=\dim U\le Cr\xi_W. Together with (5.11), this gives the required bound for the factors with \rho<1 as well. Sum over the factors. For an extension 0\to G'\to G\to G''\to0, the cohomology sequence gives h^1(G(tH))\le h^1(G'(tH))+h^1(G''(tH)); induction proves the final filtration assertion.□

Proof. For each line bundle in the fixed list \OO_X(\pm D),\OO_X(\pm H),K_X^{\pm1}, choose three successive surjections from sums of negative powers of \OO_X(H), with locally free kernels. They exist by global generation after twisting; a surjection onto a locally free sheaf has locally free kernel. Spread these finitely many sequences and a regularity bound for \OO_X to a model, and shrink so that they remain exact on the fibers.

There is consequently an integer v_0, independent of the fiber, such that tensoring by any one line bundle in the list increases by at most v_0 a threshold above which all higher cohomology vanishes. Indeed, tensor its three-step presentation by the original line bundle and use the cohomology sequences. The three dimension shifts end in cohomological degree greater than three. If the finitely many twists in the presentation are bounded by v_0, every intervening sum of lines has vanishing higher cohomology above the shifted threshold. Starting with the fixed bound for \OO_X proves the claim by induction on the number of tensor factors.

The line K_X^{1-p}\otimes L^{-1} is a product of at most c_0p+c_1 members of this fixed list, uniformly for |b|\leq1 and for both choices of L. It therefore has no higher cohomology after twisting by sH whenever s\geq c_2p+c_3. Projection formula gives

for a fixed sufficiently large j. Castelnuovo–Mumford regularity then gives the evaluation surjection in (6.3), using a basis of H^0(G_{p,L}(j)), and gives surjectivity of multiplication of these sections at every higher twist.

The cohomology sequence for its kernel gives a uniform regularity bound j_2. For degrees at least two use regularity of G_{p,L} and of \OO_X(-j); in degree one use the just-proved multiplication surjectivity. Increasing j_2 makes the kernel generated at j_2. Finally, the higher cohomology of K_X^{1-p}\otimes L^{-1}(pj) vanishes, so

This Euler characteristic is constant under specialization for each of the indicated line bundles. On the complex fiber, Riemann–Roch is a polynomial of degree at most three in exponents bounded by a fixed multiple of p. This proves N_{p,L}\leq C_0p^3. All the choices preceded any choice of a tilt object.□

Proof. Finite duality and projection formula give

Indeed F^!P_1=F^*P_1\otimes K_X^{1-p}, so adjunction applied to K_X^{1-p}\otimes L^{-1} proves the identity. The semilinearity of absolute Frobenius does not change these dimensions. Apply (6.3). The obstruction to injectivity of

is the image of

The equality uses the ordinary cohomological range of P_1. Generation of K_{p,L}(j_2) and the assumed vanishing make this group zero. The Hom map is therefore injective, and its target has dimension N_{p,L}h^0(P_1(j))\leq C_0p^3h^0(P_1(j)).□

Proof. Put N=\dim W and take the tilt image I and kernel K' of the evaluation map \OO_X(-tH)^N\to P at the side point. We first control the rank, first character, and H-degree of the second character of I by e, independently of N. We then use the evaluation kernel, whose rank is close to N, to cancel all but O(e) of the sections in the evaluation source. Ordinary cohomology gives a sheaf K' and an exact sequence

where V_I=\mathcal H^{-1}(I), U_I=\mathcal H^0(I) and V_P=\mathcal H^{-1}(P). Since I\subset P, 0\leq d(I)\leq d(P)=O(e). The sheaf U_I(t) is globally generated. Its first Chern class has nonnegative H-degree, including its divisorial torsion, and H^2B_0=0; hence

Thus r(U_I)=O(e). The inclusion into V_P gives r(V_I)=O(e) as well.

In the positive decomposition I\subset P\subset E, so n(I)\leq c d(I). In the negative decomposition P/I is a quotient of E and satisfies n(P/I)\geq c d(P/I), also when its denominator is zero. Additivity and the numerical bounds on P therefore give n(I)\leq O(e) in both cases.

Every finite Harder–Narasimhan slope of I is at least -C_1. Indeed, its last factor receives a nonzero map from one of the lines in the evaluation map. The tilt image of this map is a quotient of that line and a subobject of the last semistable factor; the line comparison forces the claimed lower bound. If the last factor has infinite slope there are no finite factors. Denominator-zero factors have nonnegative numerator. Writing d_i,n_i for the factors, we obtain

Lemma 4.2, applied at the fixed side volume and summed over the filtration, now gives

The twist change to I(t) is bounded. Its rank is O(e) by (6.19), so

The sheaf K'(t) is torsion-free and has rank N+O(e). Before twisting it belongs to the positive part of the torsion pair, while (6.19) bounds its maximum slope by -t. Its slopes after twisting therefore lie in (-u,0]. Its first and second characters are the negatives of those in (6.20). For its refined ordinary slope factors this proves

In detail, the bounded nonpositive slope interval and the total degree O(e) bound the first sum; subtracting the total 2w_0 bounds the second.

The next step is to obtain almost as many sections of this kernel on a surface as the trivial evaluation source has. Its small first character and projected second character control the Riemann–Roch error; the same numerical bounds will control the cohomology lost from the negative term V_I.

Choose a very general S in the fixed neighborhood and a very general curve C_m\in|mH|_S in its fixed neighborhood. Impose also the finitely many open conditions that all the sheaf sequences and filtrations used here restrict exactly, with torsion-free restrictions where required. Such choices are possible by Lemma 5.2. Intersect the refined restriction filtrations of K'(t) with V_I(t). The successive intersections are subsheaves of the stable restriction factors and have total rank O(e). By Lemma 5.1, the second slope moments on both S and C_m are O(e). Applying Lemma 5.4 to the intersections, after the additional fixed twist m, yields

The reason this has no N term is that each bound is a constant times the rank of the intersection plus its rank times \mu_{i,+}^2; its rank is at most r_i, and the second moments in (6.21) control the sum.

We also have h^2(K'|_S(t+m))=O(e). A stable factor of K'(t)|_S with slope \mu_i>\mu(K_S)-m has no homomorphism to K_S(-m). For all the other factors, (6.6) gives \mu_i\leq-1, so their total rank, as well as their second slope moment, is O(e). Serre duality and Lemma 5.4 applied to their dual hulls, twisted by K_S(-m), bound their h^2 by O(e). Surface Riemann–Roch and (6.20) give

Indeed the error consists of w_0(K'(t)) and the intersection of c_0(K'(t)) with the fixed class mH-K_S/2; here K_S=(K_X+H)|_S. Consequently

Restrict the triangle K'\to\OO_X(-tH)^N\to I and twist by (t+m)H. The kernel on H^0 from K'|_S(t+m) has dimension at most h^{-1}(I|_S(t+m))=h^0(V_I|_S(t+m)). Equations (6.22) and (6.23), with r(K')=N+O(e), therefore give

The source of the next map has dimension Nh^0(\OO_S(m)). Subtracting the displayed lower bound shows that the image of

has dimension O(e). This is the cancellation which removes the unrestricted dimension N from the estimate.

It remains to remove the auxiliary twist m. Multiplication by the section defining C_m gives a map from \HH^0(I|_S(t)) to this latter group. Its kernel has dimension at most h^{-1}(I|_{C_m}(t+m)), which is again O(e) by (6.22). The image of H^0(\OO_S)^N in \HH^0(I|_S(t)) consequently has dimension O(e). Composing with I\to P proves the assertion for W. All equalities involving h^{-1} use derived restriction with nonzerodivisors on the ordinary cohomology sheaves; no exactness of a tilt-image operation under restriction was assumed.□

Proof. For the positive decomposition, apply Lemma 6.4 to all of H^0(P(t)) and the triangle P(t-1)\to P(t)\to P|_S(t). The increase in h^0 is at most the dimension of the restriction image, at most C_2e. Telescope over the fixed set of integers t_0<t\leq j, starting with (6.18).

For the negative decomposition, apply the lemma to the image of H^0(E(t)) in H^0(P(t)). The kernel of

has dimension at most h^1(V_0|_S(t))=O(e) by (6.13) and (6.15). Thus the restriction image of H^0(E(t)) itself has dimension O(e). Telescope the restriction triangle for E from the same vanishing at t_0. Finally every slope of the negative cohomology sheaves in (6.25), after twisting by j_2, is at most b-A+j_2<0, by (6.7). They have no sections.□

Proof. Embed G in \OO_Y(u)^k by choosing generators for a twist of its dual and using the injection G\to G^{**}. Spread this injection and make its cokernel flat, so it remains an injection on the fibers. We describe a finite-type set containing every destabilizing saturated subsheaf on every geometric fiber.

For a saturated subsheaf J of rank s, 0<s<r(G), its generic Plücker coordinates extend across codimension two to sections of \OO_Y(su)\otimes(\det J)^{-1}. If J destabilizes, the degree of the zero divisor of a nonzero coordinate is bounded above by a number depending only on u,s and the degree of G. An effective divisor of degree at most d_0 is contained, with its multiplicities, in the zero divisor of a section of \OO_Y(v) with v\leq\max(1,d_0). To see this, choose a finite linear projection to projective space of dimension \dim Y. Each prime component has hypersurface image of degree at most its own degree. Pull back an equation for each such image and multiply with the required multiplicities. The projection is finite because its pullback of \OO(1) is ample.

Divide the Plücker tuple by one nonzero coordinate, then multiply by this section to clear its pole divisor. Normality extends the resulting tuple to global sections of \OO_Y(v). The saturated subsheaf J is the kernel, inside G, of wedging with this tuple, viewed as a map to a fixed sum of \OO_Y(u+v): the kernels agree at the generic point and are saturated. The finitely many possible values of s,v give finite-type spaces of tuples after shrinking the base so that their section spaces commute with base change. On each parameter space, flatten the cokernel of the universal wedge map. The kernel then commutes with taking fibers, and its rank and degree are constructible, as read from its Hilbert polynomial. Hence the condition that some tuple gives a kernel of smaller positive rank and larger slope than G is constructible on the base. Allowing tuples not satisfying the Plücker equations does not cause a problem: a kernel with those numerical properties is itself a destabilizing subsheaf.

This constructible subset excludes the generic point. Otherwise a destabilizer over its algebraic closure would extend to one over \CC. A constructible subset of an integral noetherian scheme whose closure is the whole scheme contains a nonempty open set and hence the generic point. Its closure is therefore proper; removing that closure proves the assertion.□

Proof. We use the Cartier-descent argument of [19, Theorem 4.1 and Proposition 4.2], keeping the chosen cotangent twist in the slope normalization. Generation of a fixed positive twist of the tangent bundle gives an embedding \Omega_Y^1\hookrightarrow\OO_Y(c_Y)^k with locally free cokernel. It can be chosen on the model and retained on its fibers. Frobenius is finite flat because the fiber is smooth over a perfect field.

Suppose two consecutive Harder–Narasimhan slopes of F^*G differ by more than c_Y. For the truncation J above that gap, the second fundamental map for the canonical connection is

It is \OO_Y-linear. Its source has minimum slope larger than the maximum slope of its target, the latter bounded using the displayed embedding, so the map is zero. The connection thus preserves J. The canonical connection has zero p-curvature, as does its restriction to J. Cartier descent therefore descends this inclusion to a subsheaf of the Frobenius twist of G. Its slope is \mu(J)/p>\mu(G), contradicting semistability. Here the Frobenius twist of the algebraically closed base field is an automorphism and does not change the slope assertion. Thus no adjacent gap is greater than c_Y. There are at most r factors, so the maximum slope is at most the mean plus c_Y(r-1), as claimed.□

Proof. We first compute along the slope-zero curve of a fixed class. Its positive-denominator branch is

For rank zero, d and b remain constant. If the branch starts at a semistable object, then \Delta\ge0, and its denominator stays positive at every positive a. Indeed d(a)\ge a|r| when r\ne0, whereas d is a positive constant when r=0. The identities dz_b/db=-w_b and w_b=a^2r/2 give

For example, when r\ne0 one has d'=ar^2/d, b'=-ar/d, and z'=a^3r^2/(2d); these give both equalities directly. The rank-zero case follows from the definitions. For a differentiable nonnegative function k, therefore,

There are three comparisons. For the corrected bound below A, take k(a)=C+(A^2-a^2)/6 and move upward toward A. The right side of (7.6) is k(a)ar^2S_a/d(a)^2\ge0, so a strict violation persists upward. To rule out positive defect at a starting value R>R_0, take k(a)=a(R-a)/3 on [A,R] and move downward. Since ar^2/d(a)^2\le1/a,

A strict violation again persists in the direction of movement. Here k(R)=0 and k(A)=A(R-A)/3>C. For the corrected bound in the remaining interval, on any interval [A,R] use k(a)=Ca/A and move downward. The same upper bound for the last term gives a derivative at most -aS_a/3\le0, and k(A)=C.

Each comparison preserves a strict violation while the class remains stable. We next show how to continue through a change of stability, with a strict decrease of a discrete discriminant at each replacement. For an equal-slope-zero filtration with stable factors E_i, all d_i are positive and w_i=a^2r_i/2. Additivity and weighted Cauchy–Schwarz give

Thus a strict inequality D_a(E)>k(a)S_a(E), with k(a)\ge0, passes to at least one stable factor. Such a factorization is finite by Lemma 4.3.

Start with a violating stable factor and follow its branch (7.4) toward A as long as it remains stable. Stability is open. At an endpoint in the positive a-interval, continuity of extremal phases gives semistability, since the formal denominator remains positive. The strict violation persists at the endpoint: along the movement D-kS is at least its initial positive value. If the endpoint is A, this contradicts (7.2). Otherwise the endpoint is nonstable, since stability there would extend the interval, and we split again and select a violating stable factor.

At every such genuine wall, the discriminant of each factor is strictly smaller than that of its parent. Indeed,

Every term is nonnegative because d_i\ge a|r_i|. If all cross terms vanished, all r_i would be nonzero with the same sign and d_i=a|r_i|; hence all factors would have \Delta(E_i)=0 and proportional triples (r_i,d_i,w_i). Twisting preserves this proportionality. Openness keeps the finitely many stable factors stable near the wall, so their fixed extension triangles make E an extension of objects with equal finite slope immediately before the wall as well. The filtration has at least two factors, so this contradicts the preceding stability of E. Some cross term in (7.8) is therefore positive, and every factor has smaller discriminant.

Because B_0 is rational, formula (4.9) puts all these discriminants in N^{-1}\ZZ for one positive integer N depending only on X,H,B_0. They are nonnegative on the semistable factors. Strict descent consequently permits only finitely many wall splits. This also excludes an accumulating sequence of walls with changing factors: every change consumes at least 1/N of discriminant, and every final stable interval either reaches A or has an interior semistable endpoint at which another such split is necessary. There is no denominator-zero endpoint inside the interval, by (7.4). The proposed violation must therefore reach A, giving a contradiction in each of the three comparisons.

The first comparison proves D_a\le(C+(A^2-a^2)/6)S_a for 0<a\le A. The second proves D_a\le0 for a>R_0. The third proves D_a\le(Ca/A)S_a for A\le a\le R_0. Finally use S_a\le d_b and (7.3).□

Proof of Theorem 1.2. First use the very-ample and transverse-twist reductions of Section 4. Proposition 6.1 supplies A>0 and C\ge0, chosen independently of E and b, such that

At slope zero put e=d_B-A|r|. Since d_B\ge A|r|,

The hypothesis of Proposition 7.1 follows. Multiplying its conclusion by h gives

and gives the same inequality without the correction for a>R_0. Undo the polarization rescaling as described in Section 4. This gives constants k\ge0 and R_*>0 for the original X,H,B_0, uniform in both E and b.

Enlarge k to a rational nonnegative number. The curve class \Gamma=kH^2\in N_1(X)_{\QQ} then satisfies \Gamma\cdot H=kh\ge0, and the correction is exactly \Gamma\cdot\ch_1^B(E). This proves the corrected statement with its required rational class, as well as the independent uncorrected large-volume threshold. No canonical-bundle hypothesis has entered either argument.□

Proof of Corollary 1.3. Take the threshold of Theorem 1.2 with B_0=0. Set b=\beta and a=\sqrt{2\alpha-b^2}. The numerator in (7.9) is h(w_b-a^2r/2) and its denominator is hd_b, so its slope is exactly \nu_{a,b}. Expanding (7.10) with (4.2)–(4.3) gives

If d_b(E)=0, this is nonnegative by (4.10); it includes zero-dimensional objects.

Otherwise put u=\nu_{a,b}(E), c=b+u, and R=\sqrt{a^2+u^2}. Lemma 4.4 makes E semistable of slope zero at (R,c), with d_c>0. Using w_b=ud_b+a^2r/2,

in (7.11) gives the exact identity

Since \alpha>f(b) implies R\ge a>R_*, the expression in parentheses is nonpositive by the uncorrected tail. This proves the claim. The passage uses the entire semistability arc and the inequality R\ge a; an all-parameter equivalence alone would not justify a conclusion on this truncated region.□

Proof. Write \ell=H|_D, s=\ell^2=D^3>0, \rho=\rk(F), and C=B_0|_D. Put

Every exact sequence of sheaves on D pushes forward to an exact sequence of torsion sheaves in the first-tilt heart. Their tilt slopes differ from their C-twisted normalized surface slopes by the common constant 1/2-b. Twisting shifts all surface slopes equally, so tilt semistability of i_*F implies slope semistability of F for \ell. The classical surface Bogomolov–Gieseker inequality [4, Theorem 3.1.4] and the Hodge index theorem give

The first expression is unchanged by twisting the Chern character by C, so these conclusions hold for the specified B_0.

The normal line bundle has first Chern class -\ell, and hence \td(N_{D/X})^{-1}=1+\ell/2+\ell^2/6. Grothendieck–Riemann–Roch now gives

Since the threefold rank of i_*F is zero, its zero-slope condition is b=\mu+1/2. The degree-three term of the same formula yields

This proves (8.4); the two strong bounds follow by subtracting a^2/6.□