Universal-cover splitting for compact Kähler manifolds

Proof. Höring’s theorem [19, Theorem 1.3] makes both summands integrable. Since a smooth projective complex manifold is compact Kähler, Theorem 1.1 applies.□

Proof. Holomorphic Frobenius gives a local submersion with kernel E_i for each i. To recall the local construction, choose a coordinate projection that is nonsingular on the distribution and lift its coordinate fields to a frame in the distribution. Their brackets belong to the distribution by integrability and project to zero; the projection is an isomorphism on the distribution, so the brackets vanish. The commuting local holomorphic flows applied to a transverse slice give foliation coordinates and the asserted submersion.

Combine the submersion with kernel E_2 and the submersion with kernel E_1. Their combined differential is an isomorphism by complementarity. The holomorphic inverse function theorem and restriction to a product of polydisks give the desired coordinates. A chart change preserves both distributions, so its wrong-block derivatives vanish. On small product neighborhoods this is exactly (2.2), and invertibility of the chart change makes each block a local biholomorphism.□

Proof. We first obtain a meromorphic extension. We then rule out curves in its graph fibers by comparing two types of level sets: a level isolated at a putative exceptional curve, and nearby levels containing a parameter direction. Finally we show that the remaining finite fibers are singletons.

Meromorphic extension and tangency. The envelope of holomorphy of H is Q. Indeed, expand a scalar holomorphic function on the shell in a Laurent series in v. The coefficients of the negative powers are holomorphic functions of w; they vanish for |w|<e, and hence for |w|<a. Cauchy estimates on any intermediate circle c<|v|<b give normal convergence of the resulting power series on compact subsets of Q. This is the usual Hartogs extension calculation; since the bidisk Q is a domain of holomorphy, it also identifies the envelope.

Ivashkovich’s extension theorem states that a meromorphic map from a domain in a Stein manifold to a compact Kähler manifold extends meromorphically to the envelope of holomorphy [22, Theorem 2.2]; see also [21]. Here H is a domain in the Stein manifold \C^2, and f is holomorphic, so the theorem gives a meromorphic extension to Q.

Let \Gamma\subset Q\times X be its graph, with its reduced analytic structure, and let \rho:\Gamma\to Q be the first projection. The graph is an irreducible analytic surface, \rho is proper, and there is a proper analytic subset A\subset Q, disjoint from H, such that over U=Q\setminus A the graph is the ordinary graph of a holomorphic map F:U\to X. This ordinary graph is dense in \Gamma. The set U is connected: the complement of a proper complex analytic subset of a connected complex manifold is connected. Consequently, the identity theorem applied to the two holomorphic bundle-valued expressions gives

Here P_i:T_X\to E_i is the holomorphic projection determined by the specified splitting.

No positive-dimensional fiber. We use the local irreducible decomposition, dimension theorem, and regular-locus properties of complex analytic sets; see [9, Chapter II, Section 4]. We record first a point that is important when \Gamma is singular or locally reducible. On every local irreducible surface branch of \Gamma, both source coordinates s and t are nonconstant. To see this, choose a relatively open part of that branch away from the other local branches. It meets the dense ordinary graph. Near such a point, \rho is an isomorphism onto an open subset of Q, and s,t are source coordinates. Neither coordinate can therefore be constant on the branch. It follows that, for every s_0, the analytic set

has local dimension at most one; the same holds with t in place of s.

Suppose a fiber over (s_0,t_0) has positive dimension. Since it is contained in (3.2), it has a curve component C. Take a small piece of C in a product chart z=(z_1,z_2) of X. Some scalar component h of z_1 or z_2 is nonconstant on this piece: otherwise the fiber curve, which lies in \{(s_0,t_0)\}\times X, would be a point.

First suppose h belongs to z_1. The curve piece is a component of the reduced set (3.2), since that set has dimension at most one. Choose a general point q of the curve piece such that the reduced set (3.2) is a smooth curve near q and h is nonconstant on its germ. This is possible because the one-dimensional analytic set has locally finitely many curve components, distinct components meet discretely, and the singular points of a reduced curve are discrete. In particular, q is an isolated point of the level set

This assertion concerns the reduced coordinate level, and does not require \Gamma itself to be smooth along C.

Work in ambient complex coordinates on Q\times X around q, and choose a sufficiently small closed Euclidean ball \overline B centered at q. Its closure lies in this coordinate neighborhood, and the indicated level set misses \Gamma\cap\partial B. The image of this boundary intersection under the holomorphic map

is compact and does not contain \Psi(q). Choose points q_j of the ordinary graph tending to q. For all sufficiently large j, the level value \Psi(q_j) also misses that compact image. Therefore

is a compact analytic subset of the open ball B: it has no points on \partial B, so there is no boundary truncation of the analytic level. On the ordinary graph near q_j, however, (3.1) says that h is independent of t. Varying t with s fixed gives a one-dimensional germ in K_j through q_j. This contradicts the fact that a compact analytic subset of an open set in Euclidean complex space has no positive-dimensional component. One way to see the latter fact is to maximize the strictly plurisubharmonic squared coordinate norm on such a component and restrict it to a nonconstant analytic curve germ through a maximum point.

If h belongs to z_2, the identical argument uses \Psi=(t,h) and varies s. Thus every fiber of \rho has dimension zero. Properness then makes each fiber finite.

Singleton fibers and holomorphic removal. Fix p\in Q, and write its fiber as \{(p,x_1),\ldots,(p,x_k)\}. Choose pairwise disjoint target neighborhoods V_1,\ldots,V_k with pairwise disjoint closures and x_i\in V_i. Properness implies that, after shrinking a base ball D about p, every point of \rho^{-1}(D) has its target coordinate in \bigcup_iV_i. Otherwise a sequence of graph points over points tending to p, outside this union, would have a limit in the central fiber. The ordinary locus D\setminus A is connected, so its image under F lies in a single V_i. Density of the ordinary graph then places every x_j in \overline V_i. The disjoint closures imply k=1.

The proper bijection \rho is a homeomorphism. Its inverse followed by the second projection defines a continuous extension \widehat F:Q\to X. Near each source point its image lies in a target coordinate chart. The coordinate functions are continuous and holomorphic off A, hence holomorphic by the removable-singularity theorem for locally bounded holomorphic functions across analytic subsets of a complex manifold. Thus \widehat F is holomorphic. Equation (3.1) extends by the identity theorem, giving the required box tangencies. Uniqueness follows from the identity theorem on the connected bidisk Q.□

Proof. It suffices to work on a disk D with \overline D\subset U. Put d(z)=\operatorname{dist}(z,\partial D) and

If M_j\not\to0, pass to a subsequence with M_j\ge\varepsilon>0. Choose a maximizing point z_j\in D, and set a_j=\norm{\dd h_j(z_j)} and d_j=d(z_j), so a_jd_j=M_j. The maps

are defined for |\zeta|<\varepsilon/4. On this disk, d(z_j+\zeta/a_j)\ge d_j/2, and maximality gives

After taking a subsequence, k_j(0) converges to a point of M. The derivative bound puts the images of a fixed smaller disk in a relatively compact coordinate neighborhood of that point. Montel’s theorem and the Cauchy estimates give a further subsequence converging there together with its derivatives. Its limit has derivative norm 1 at the origin and therefore positive area on a smaller disk. This contradicts the vanishing areas of k_j, which are bounded by the areas of h_j and are unchanged by the affine reparametrization. Thus M_j\to0. On each smaller disk d is bounded below, so the derivatives tend uniformly to zero. A finite covering proves the assertion on any compact subset of U.□

Proof. Use the fixed Kähler form \omega and its Hermitian norm, and recall the bounded projections P_i from (2.1).

Write p=(s_*,t_*). Choose complex-affine coordinates (u,v) with

where the v-direction is complex tangent and the positive real u-direction points outward. All constants below may depend on this boundary point and these coordinates.

Shrinking slices and bounded area. Take a local defining function \rho with B=\{\rho<0\} near p. For real x and small v, Taylor’s theorem gives

because both real derivatives in the complex tangent direction vanish. Choose k>0 sufficiently small. For all x<0 sufficiently close to zero, the closed disks

are compactly contained in B.

Let e_u=\partial_{\operatorname{Re}u} and e_v=\partial_{\operatorname{Re}v}. The box tangencies and complex linearity give the exact identity

Consequently

Holomorphicity makes \norm{\dd f(e_v)} equal to the speed in every unit real direction of the v-plane, including the radial direction.

Set \Omega=f^*\omega and A(x)=\int_{D_x}\Omega. Closedness of \Omega and Stokes’ theorem, applied to the moving disks, yield

Indeed the boundary velocity is e_u+r'(x)\partial_r, and the circles have their counterclockwise orientation. On a boundary circle put q=\norm{\dd f(e_v)}. Since \partial_\theta=rJ\partial_r,

Thus

Here r'(x)=-k/(2\sqrt{-x}). Choose x_0<0 sufficiently close to zero that r'(x)\le-C for x_0\le x<0. We obtain the uniform bound

No product property of the Kähler metric has been used.

A small-image annulus. Let x_j=-\delta_j with \delta_j\downarrow0. Among the scales \lambda=5^\ell\delta_j^{3/4}, with integer \ell\ge0, in [\delta_j^{3/4},\delta_j^{5/8}], the annuli

are pairwise disjoint, and their number tends to infinity. For large j they lie in D_{x_j}. By (4.3), one of them, with scale \lambda_j, has area tending to zero. These choices satisfy

Apply Lemma 4.1 to h_j(\zeta)=f(x_j,\lambda_j\zeta) on 1/2<|\zeta|<2. The derivatives tend uniformly to zero on a neighborhood of 3/4\le|\zeta|\le3/2. This compact annulus has bounded intrinsic diameter, so the image of

has diameter tending to zero. A finite cover of X by product charts has a Lebesgue number. Hence, for large j, this entire image lies in one product chart z:V\to V_1\times V_2. Write its coordinate functions as (g_1(v),g_2(v)).

Figure 1 separates the two scale comparisons.

We have found an annulus whose radius is much larger than the remaining distance \delta_j to the boundary. We now extend it transversely to make a Hartogs shell.

The shell and its agreement with the old box. Put w=u-x_j and c_i=a_i/b_i. Choose \eta>0, independent of j, so that \eta\max(1,|c_1|,|c_2|)<1/8. On

define

Both shifted arguments stay in 3\lambda_j/4<|v|<3\lambda_j/2. The range of z is a product, so combining the two valid factor coordinates in (4.5) needs no further target-chart margin. Moreover,

Thus F_j is a box in the original (s,t) directions.

The two maps agree on an open germ at every point of S_j with w=0. To see this, continuity puts the old map f in V on a neighborhood of that point. There its separated expression is (\varphi_1(s),\varphi_2(t)). Restriction to u=x_j gives the local expressions for g_i, and the displayed identities show that (4.5) reproduces f on a full neighborhood. This is stronger than equality merely on the slice w=0.

The closed slice disk u=x_j, |v|\le5\lambda_j/4 is contained in B. Hence there is 0<e_j<\eta\lambda_j such that the original map is defined on

The overlap C_j\cap S_j is a connected product of a disk and an annulus. The identity theorem for maps into X therefore extends the open-germ agreement to this entire overlap, even if the old map elsewhere leaves V. We have a box on the Hartogs figure C_j\cup S_j.

By Proposition 3.1, it extends to a box on

Both Q_j and B are convex in the underlying real affine space. Their intersection is connected and contains C_j, so the extension agrees with f throughout Q_j\cap B. Finally p has coordinates (w,v)=(\delta_j,0), which belong to Q_j for large j by (4.4). This gives the required extension across p.□

Proof. Product coordinates at the common initial point give a unique box germ: take the first coordinate from \alpha and the second from \beta. Any two boxes extending this germ agree on a connected common domain by the identity theorem.

Fix an integer m\geq1, and consider the convex domains

Let R_* be the supremum of the radii to which the germ extends as a box. The nested extensions agree and give a box on B_m(R_*). Its values on the two axes are the prescribed maps, by the one-variable identity theorem. Suppose that R_*<1.

At a boundary point with st\neq0, the complex tangent line has both components nonzero: both coefficients of \partial(|s|^{2m}+|t|^{2m}) are nonzero there. Proposition 4.2 therefore extends the box across that point. At a boundary point (s_*,0), choose a disk U around s_*, with \overline U\subset\D, such that \alpha(U) lies in a product chart z=(z_1,z_2). Choose s'\in U with |s'|<R_*. For sufficiently small \epsilon>0, the formula

is defined and is a box. Here \D_\epsilon=\{|t|<\epsilon\}. The product range of the chart makes the formula well-defined. Local separation shows that G=F near (s',0). Their common domain (U\times\D_\epsilon)\cap B_m(R_*) is convex, so they agree throughout it. The other axis is treated symmetrically.

These boundary extensions glue. Indeed, restrict each one to an ordinary Euclidean ball centered at its boundary point. If two such balls intersect, their intersection contains a point of the segment joining their centers. That segment lies in \overline{B_m(R_*)} by convexity. Since the intersection is open, it consequently meets B_m(R_*). The two extensions agree there, hence on their connected intersection. A finite subcollection covers the compact boundary, giving an extension to a neighborhood of \overline{B_m(R_*)}. This neighborhood contains B_m(R') for some R_*<R'<1, a contradiction.

Thus the germ extends to B_m(1) for every m. These domains increase to \D^2, and the resulting boxes agree on their connected overlaps. Their union is the required box. The initial uniqueness observation proves uniqueness on \D^2.□

Proof. Choose a Lebesgue number \lambda>0 for a finite product-chart cover of the compact metric space X. Uniform continuity gives one finite rectangular grid such that the image of each closed tile under either map has diameter less than \lambda/3. Whenever the two maps agree at a tile’s lower-left corner, the union of their tile images has diameter less than 2\lambda/3, so both images lie in one product chart. In that chart each box is determined by its bottom first coordinate and left second coordinate. Starting with the prescribed edges and proceeding by increasing sum of the tile indices proves equality on the whole grid.□

Proof. Polydisk automorphisms reduce the problem to p_0=q_0=0. For fixed (p,q), the maps

are holomorphic on disks of radii greater than one. After rescaling, Proposition 5.1 gives a box whose restriction to the closed real square is denoted by R_{p,q}. Set

The constant-axis cases and Lemma 5.2 give the prescribed axis values. It remains to prove holomorphicity and the two tangencies; pointwise existence of R_{p,q} alone does not give parameter continuity.

Fix (p_*,q_*), and choose a finite grid whose closed tile images under R_{p_*,q_*} lie in product charts. We reconstruct this square for nearby (p,q), using these fixed charts. Begin with its two given axes, and process tiles from the bottom and left. The induction maintains the following properties: each constructed tile depends jointly continuously on its real parameters and on (p,q); it is a continuous box in the real parameters; and, at each fixed pair of real parameters, it is holomorphic in (p,q), with the p-directions in E_1 and the q-directions in E_2. These properties hold on the initial axes.

To fill one tile, denote its already constructed bottom and left edges by B(\sigma;p,q) and L(\tau;p,q). At the reference parameter their images lie in the selected product chart z=(z_1,z_2). Joint continuity and compactness of those edges put both images in that same chart after shrinking the parameter neighborhood. Define the tile by

The edges meet at a common corner. Plaque constancy on each edge shows that this formula reproduces both of them. It is jointly continuous and is separated in \sigma,\tau. For each fixed \sigma,\tau, it is holomorphic in (p,q); moreover, the induction hypothesis gives

Thus the two parameter tangencies are preserved as well. At (p_*,q_*), the reconstructed tile is the original tile, so the induction continues. There are only finitely many tiles and therefore only finitely many shrinkings.

The tile maps agree on common edges and form a continuous box across the seams. To check the latter assertion at a seam or a vertex, use a product chart around its common image. On a sufficiently small real neighborhood, the unwanted coordinate along a horizontal or vertical slice is constant on each tile piece and agrees at their endpoints; it is therefore constant across the seam. By Lemma 5.2, the reconstructed square is R_{p,q} for every nearby parameter pair. Its upper-right value is consequently F(p,q), proving that F is a holomorphic box near (p_*,q_*), and hence everywhere. Finally, the axes determine its germ at (0,0), so the identity theorem proves uniqueness.□

Proof. For the first assertion, subdivide the horizontal interval into pieces I_i=[u_{i-1},u_i] and the vertical interval into pieces J_j=[v_{j-1},v_j] on which the respective paths lie in single plaques. Each horizontal piece has a representation

where a_i:I_i\to\D^{r_1} is continuous and A_i:\D^{r_1}\to X is a holomorphic plaque parametrization tangent to E_1. Similarly write \beta(t)=B_j(b_j(t)), with b_j:J_j\to\D^{r_2} continuous and B_j holomorphic and tangent to E_2. Here r_i=\rank E_i.

Fill the tiles I_i\times J_j from the bottom and left. Inductively, the known edges of a tile have the forms A(a_i(s)) and B(b_j(t)), where A and B are holomorphic on the entire polydisks \D^{r_1} and \D^{r_2}, tangent to the respective bundles. They agree at their common corner. Apply Proposition 5.3 with origins a_i(u_{i-1}) and b_j(v_{j-1}), obtaining a holomorphic box G:\D^{r_1}\times\D^{r_2}\to X. Its evaluation

fills the tile. The opposite horizontal and vertical edges have the holomorphic representatives

again on the whole original polydisks. This proves the induction without any shrinking of the edge domains.

For the second assertion, choose a subdivision 0=u_0<\cdots<u_N=1 such that each \gamma(I_i), I_i=[u_{i-1},u_i], lies in a product chart with inverse \theta_i:\D^{r_1}\times\D^{r_2}\to X. Write \gamma(s)=\theta_i(a_i(s),b_i(s)) on that interval. Define the diagonal tile by

These tiles agree at consecutive diagonal vertices, and their edges have the full-polydisk representations used above.

Fill the remaining tiles I_i\times I_j in successive bands away from the diagonal, that is, in increasing order of |i-j|. If i<j, use the already filled bottom and right edges; if i>j, use the top and left edges. The same polydisk box construction works from any corner, since the origins in Proposition 5.3 are arbitrary. The two input edges have matching corner values: for |i-j|=1 this is the prescribed diagonal value, and for |i-j|>1 it follows from the intervening tile closer to the diagonal. More explicitly, when i<j-1, the bottom-right corner is shared with the previously filled tile I_{i+1}\times I_{j-1}. The case i>j+1 is symmetric. Tiles filled at the same stage share no edge, and all their shared vertices already lie on prescribed input edges. Thus every tile is consistent with the previously assigned values, and its output edges retain their full-polydisk representatives.

In either construction, the finite collection of tiles glues continuously. The local plaque condition holds across seams for the same reason as in the proof of Proposition 5.3: in a product chart near a seam point, the unwanted coordinate is constant on consecutive slice pieces and agrees at their common endpoints. The resulting map is a continuous box.□

Proof. The leaf consists of points reachable from o by finite chains of E_i-plaques. Plaque coordinates give its complex atlas: near a point of intersection, two plaques through that point coincide, and their coordinate change is holomorphic. The inclusion into X is an injective continuous immersion. In particular, distinct points of the leaf are separated by inverse images of disjoint ambient open sets, so the leaf is Hausdorff. The definition by plaque chains gives path connectedness and the asserted continuity of plaque paths.

For second countability, take a countable ambient product atlas. Start with a plaque through o. For each plaque already reached and each ambient chart, their intersection has at most countably many connected components: it is an open subset of a second-countable plaque. Each component lies in a single plaque of the new chart, since the transverse coordinate is locally constant. Adjoin these plaques and repeat. There are only countably many plaques after all finite stages, and every plaque path from o is covered by a finite chain of this kind. Their countable coordinate bases form a countable basis for the leaf.

The usual covering-space construction applies because a manifold is locally path connected and semilocally simply connected. Its universal cover is second countable as well. One way to see the latter detail is to refine the countable atlas to simply connected coordinate disks. There are countably many components of their pairwise overlaps. A based loop is described, up to homotopy, by a finite sequence of these charts and overlap components; fixing paths within the charts and components supplies representatives for all such descriptions. Hence the fundamental group is countable. The lifted countable chart basis has only countably many sheets and is therefore a countable basis for the universal cover. Lifting the holomorphic charts gives its complex structure.□

Proof. Transport a first-coordinate germ from the L_1-plaque at \alpha(r) along the vertical path u\mapsto R(r,ru), 0\leq u\leq1. To define this transport, subdivide the path into finitely many subsegments lying in product charts, and choose a local separated transition at each junction. Within a subsegment the first coordinate is constant; at a junction it changes by the first-coordinate local biholomorphism of that transition. Their composition transports a variable first coordinate to the endpoint. Refining the subdivision or changing intermediate charts does not change the germ: separated transitions satisfy the cocycle identity. On a common vertical subsegment, the first-coordinate transition germ at the fixed first coordinate is locally constant, since local separated transitions are independent of the second coordinate. It is therefore constant along the connected subsegment. Common refinements identify any two constructions. Similarly, transport a second-coordinate germ from the L_2-plaque at \beta(r) along u\mapsto R(ru,r). Combining the two transported coordinates in one product chart at \gamma(r) gives a local biholomorphism germ H_r; let K_r be its inverse.

We must show genuine analytic continuation, rather than merely continuous dependence of a collection of germs. Fix r_0. Choose the finite chart chains for the two reference transport paths, a fixed small overlap neighborhood at each junction, and a common final chart z around \gamma(r_0). By joint continuity of R, the same compact subsegments lie in the same charts, and the same junctions lie in the same overlap neighborhoods, for all r sufficiently near r_0. Also \alpha(r) and \beta(r) remain in fixed intrinsic leaf plaques by Lemma 6.2.

Use coordinates a and b on those two starting plaques. Every first-coordinate transition in the vertical chain is now one fixed holomorphic map, independent of the second coordinate. After shrinking the initial coordinate neighborhood, their finite composition is a single local biholomorphism A defined near a(\alpha(r_0)). It describes the transport for every nearby r, not just for r_0. The horizontal chain similarly gives one fixed second-coordinate local biholomorphism B. Consequently one holomorphic map

represents H_r at (\alpha(r),\beta(r)) for every nearby r, and sends that point to \gamma(r). After one further shrinking, its single holomorphic inverse represents all the nearby K_r. At r=0, both transport paths are constant, so these germs are precisely H_o and K_o. This is the required analytic continuation.□

Proof of Theorem 1.1. Every path from (o,o) in L_1\times L_2 consists of two intrinsically continuous leaf paths. Proposition 6.1 provides a box with these as its axes, and Proposition 6.3 continues H_o along the given path. Conversely, every path from o in X is the diagonal of a box, so the same two Propositions continue K_o along every such path.

Let \pi:\widetilde X\to X and q_i:Y_i=\widetilde L_i\to L_i be the based connected simply connected covers, and put q=q_1\times q_2. The product Y_1\times Y_2 is the universal cover of L_1\times L_2. The monodromy theorem, applied on these simply connected source covers, gives holomorphic maps

with the prescribed initial germs. Here the usual monodromy argument applies to manifold-valued holomorphic germs: uniqueness follows from the identity theorem, and finite chains of local representatives are stable under small path variations, so homotopic paths with fixed endpoints produce the same terminal germ.

Since both source manifolds are simply connected, these maps lift to holomorphic maps

Choose the lifts that send the based points to one another. Their germs there are mutually inverse, because H_o and K_o were inverse. Thus \widetilde H\circ\widetilde K and \widetilde K\circ\widetilde H agree with the respective identity maps on neighborhoods of the basepoints. The identity theorem on the connected source manifolds makes both compositions identities everywhere. In particular, \Phi=\widetilde K is a biholomorphism \widetilde X\simeq Y_1\times Y_2.

It remains to retain the specified splitting. Near the basepoint, the initial product chart gives

Equivalently, the two wrong-factor components of \dd\Phi vanish there. They are holomorphic bundle maps, so they vanish on all of the connected manifold \widetilde X. Finally, \dd\Phi is invertible and \dim_\C Y_i=r_i by Lemma 6.2; hence these inclusions are equalities. The factors are connected and simply connected by construction, proving the asserted compatible splitting.□

Proof. The base-point-free theorem applies to D=K_X: D is nef and aD-K_X=(a-1)K_X is nef and big for an integer a>1 [23, Theorem 3.3]. The associated contraction, with its Stein factorization, has normal projective target and connected fibers. For some positive integer m and ample line bundle L on Z,

Bigness makes f generically finite; connected fibers then make it birational. A proper birational morphism onto a normal variety is an isomorphism over an open set whose complement has codimension at least two. On this open set L agrees with \mathcal O_Z(mK_Z). Reflexive extension therefore identifies L with \mathcal O_Z(mK_Z) everywhere. Thus K_Z is \mathbf Q-Cartier and ample.

Choose compatible canonical divisors. The divisor K_X-f^*K_Z is exceptional and \mathbf Q-linearly trivial. A principal exceptional divisor is zero: the corresponding rational function has no divisor on the normal projective variety Z, hence is constant. Consequently K_X=f^*K_Z. The discrepancies on any higher resolution are nonnegative because X is smooth, so Z has canonical, and in particular klt, singularities.

Over Z_{\mathrm{reg}}, the relative Jacobian has zero divisor. It is therefore nowhere zero, and f is a local biholomorphism there. Properness and birationality imply that this restriction is an isomorphism.□

Proof. Proposition 7.1 makes f a smooth projective crepant resolution of a canonical model. The completion statement is [29, Theorem 1.2]. The approximating metrics and their smooth convergence are [29, Equation (4.28) and Lemma 4.1], with \varepsilon=e^{-t}. The cohomology identity follows directly from the Ricci equation.□

Proof. We use the subbundle-curvature method of approximate Kähler–Einstein geometry, developed in the stability results of Enoki [15] and used for parallel splittings by Guenancia [17, Section 3.4, Step 4, Equations (3.12)–(3.13)]. Here the equality of the two summands’ total first Chern class with c_1(T_X) gives a direct cancellation. Give T_X the Hermitian metric induced by g_\varepsilon. Let P_{j,\varepsilon} be orthogonal projection onto E_j, and let B_{j,\varepsilon} be its second fundamental form. These orthogonal projections need not be projections along the specified complementary summands.

Write F_{V} for the Chern curvature of a Hermitian bundle V. The subbundle curvature formula gives

Here and below the second fundamental forms have their complex Hermitian norms. This formula follows by writing the connection in blocks for E_j\oplus E_j^\perp; the off-diagonal blocks are B_{j,\varepsilon} and -B_{j,\varepsilon}^*. In the Kähler case the mean curvature endomorphism \sqrt{-1}\Lambda_{\omega_\varepsilon}F_{T_X} is the Ricci endomorphism

Integrating the trace formula computes first Chern degrees. Since c_1(E_1)+c_1(E_2)=c_1(T_X), and writing dV_\varepsilon=\omega_\varepsilon^n/n!, we obtain

The ranks add to n, so \tr C_\varepsilon=0. Moreover -I\leq C_\varepsilon\leq I, since each P_{j,\varepsilon} is an orthogonal projection. Positivity of A_\varepsilon therefore gives

The last integral is a bounded polynomial in \varepsilon, by Lemma 7.2. No angle bound between the summands is used.

On compact subsets of X^0, the metrics, their connections, the orthogonal projections, and the second fundamental forms converge smoothly. A nonzero limiting second fundamental form would have a fixed positive integral on some compact neighborhood, contradicting (7.1). Both limiting second fundamental forms are therefore zero.

The connection of a Kähler metric is torsion-free. Hence each summand is integrable on X^0. Its Frobenius obstruction

is a holomorphic bundle morphism on X. It vanishes on the dense open set X^0, hence everywhere. Local Frobenius submersions for the two complementary foliations, taken together, give holomorphic coordinates (z,w) in which the summands are the z- and w-directions. Transitions between such charts locally preserve the two factors.□

Proof of Corollary 1.3. A smooth projective complex manifold is compact Kähler. By Proposition 7.3, the supplied summands are integrable. Theorem 1.1 therefore applies and gives precisely the compatible product claimed in the corollary.□

Proof. Let Q_j:T_X\to E_j denote projection along the other supplied summand, and define

Proposition 7.3 implies \nabla^gQ_j=0, hence \nabla^gh=0. Thus \nabla^g is also the Levi-Civita connection of h. It preserves the complex structure, so h is Kähler.

The positive g-self-adjoint endomorphism representing h is parallel. Its eigenvalues are constant on connected X^0. Consequently there are global positive constants a,b such that

In local product coordinates, first- and second-factor real coordinate fields V,W commute. The equality \nabla_VW=\nabla_WV, together with preservation of the complementary summands, makes both sides zero. Since \nabla h=0, the first block of h depends only on first coordinates and the second block only on second coordinates.

A Kähler–Einstein metric is real analytic in holomorphic coordinates and their conjugates [11, Theorem 6.1(c)]. The Q_j are holomorphic, so the displayed formula for h proves its real analyticity.

Uniform equivalence of the metrics makes the identity on X^0 bi-Lipschitz for their intrinsic distances. It therefore extends uniquely to a bi-Lipschitz homeomorphism of their completions. Lemma 7.2 identifies that completion with Z, extending the given identification on X^0. The assertions about d follow from continuity of f and from \rho being a distance.□

Proof. Projectivity and connectedness follow from the construction of f. The rational-chain assertion is the fiber theorem for resolutions of klt varieties [18, Corollary 1.6, arXiv v2].

For simple connectedness we use the resolution-fiber consequence of Takayama’s local argument: the fiber of a proper birational morphism from a smooth complex variety to a klt variety is simply connected in the ordinary topology [30, Theorem 1.1 and the first paragraph on p. 829]. This consequence is also used explicitly in [25, proof of Theorem 3.10, arXiv v4]. Here X is smooth and Z is klt by Proposition 7.1, so it applies to f:X\to Z and gives \pi_1(f^{-1}(z))=1. Replacing the fiber by its reduction does not change its underlying topological space, hence \pi_1(P)=1.□

Proof. Fix s,t and a path \gamma in T from s to t. Set p=u(s), q=u(t) and r=R(s,t), and choose product charts z,\zeta,w at p,q,r, respectively. The path R(s,\gamma) runs from p to r in an \mathcal F_2 leaf. Its holonomy gives a first-coordinate germ h_1:z_1\to w_1. Similarly, R(\gamma^{-1},t) gives a second-coordinate germ h_2:\zeta_2\to w_2. Define

Chart changes cancel in this formula, and (B.2) follows immediately. Both slices are continuous into their leaves. Since T is simply connected, two choices of \gamma give fixed-endpoint leafwise homotopies, so the formula is independent of this choice.

For completeness, coherence concerns the entire formula, not merely its value. Choose finite chains of product charts along R(s,\gamma) and R(\gamma^{-1},t) realizing h_1,h_2. For s' near s and t' near t, continuity on these compact paths ensures that the same chains apply to R(s',\gamma) and R(\gamma^{-1},t'). Join s' to s and t to t' inside small connected parameter neighborhoods. All the resulting endpoint segments lie in the endpoint product charts, so introduce no additional transverse-coordinate change. The same holomorphic representative (B.3) therefore represents c_{s',t'}. At a node one uses a small star neighborhood and connectors through the node; no differentiability of the parameters is involved.

When s=t, choose the constant path to recover (B.1). Finally, a coherent family is analytic continuation along paths in the connected space T\times T of its germ at any chosen diagonal parameter. Uniqueness of continuation of ambient holomorphic germs proves the asserted uniqueness.□

Proof. In either case the divisor has class (1,1) and is a hyperplane section of the Segre embedding of S. Its complement V is therefore a smooth affine, hence Stein, surface. Shrink the given neighborhood to a connected neighborhood W of the divisor. Removing a complex hypersurface does not disconnect W, so W\setminus D is connected, where here D also denotes the diagonal in the first case. The set K=S\setminus W is compact in V, and V\setminus K=W\setminus D. Meromorphic Hartogs extension on Stein surfaces, in the form of [24, Theorem 2.2], extends meromorphic functions from V\setminus K to V.

Embed M in projective space and choose a coordinate hyperplane not containing the image of the given map. Extend its projective coordinate ratios and glue to their original values on W. They define a meromorphic map F:S\dashrightarrow M: the equations of the embedded M hold on W and hence identically. The two slice tangencies also persist on the holomorphic locus by the identity theorem.

The meromorphic graph is a closed analytic subset of the projective manifold S\times M, so it is algebraic by Chow’s theorem [28, Proposition 13]. Thus F is a rational map. Clearing the poles of its finitely many projective coordinate ratios expresses it by a finite-dimensional linear system on S. Remove the fixed divisorial part of this system and resolve its base ideal by point blowups. This is the classical surface elimination-of-indeterminacy argument [1]; the local intersection calculation gives termination in the present setting. Successively blow up the remaining basepoints: the local intersection number of two general members of the resulting linear system drops by their positive multiplicity product at each unresolved basepoint. The finite sum of these intersection numbers therefore forces termination. This gives \mu:\widehat S\to S and a holomorphic map H:\widehat S\to M. We show that every exceptional curve is contracted by H. At a generic point of an exceptional component C over (s_0,t_0), choose coordinates (x,y) with C=\{x=0\}. For local product coordinates s,t downstairs, write

where a,b>0 and U,V are units. The holomorphic vector fields

annihilate \dd(s\circ\mu) and \dd(t\circ\mu), respectively. Off the exceptional locus, slice tangency gives

These inclusions extend holomorphically to C. There both vector fields are nonzero multiples of \partial_y, whence

Thus H is constant on each exceptional component. Fibers of \mu are connected, so each whole fiber has a single image. Around this image choose a target coordinate chart; properness gives a neighborhood downstairs whose inverse image maps into the chart. Its coordinate functions descend holomorphically, since \mu_*\mathcal O_{\widehat S}=\mathcal O_S, equivalently by removal of the isolated singularity downstairs. Hence F is holomorphic. Its slice tangencies give the box condition, and uniqueness follows from the identity theorem.□

Proof. First take T=\Pp^1. Applying local crossing to u(s),u(t) gives coherent holomorphic germs along the diagonal of S. Compactness allows them to be glued on a neighborhood of the diagonal: take a finite family of representatives, shrink their domains, and then shrink around the diagonal so that overlapping representatives agree. Lemma B.4 gives a box on S. In a product chart any box near (s,s) has the form (a(s),b(t)); its diagonal specifies both functions. This proves uniqueness near the diagonal and hence on S.

We also need the corresponding construction from axes. Suppose \alpha:\Pp^1\to M is tangent to E_1, \beta:\Pp^1\to M is tangent to E_2, and \alpha(s_0)=\beta(t_0). Transport the germ of \beta at t_0 along \alpha, using \mathcal F_1 holonomy. There is no path ambiguity: a loop in the parameter sphere contracts, and its image gives a leafwise homotopy. In product charts the resulting map near (s,t_0) has first coordinate supplied by \alpha(s) and second coordinate a holomorphic germ in t. It is jointly holomorphic and the germs are coherent by the finite-chart argument above. Do the symmetric construction along the other axis. Near their intersection both maps are ordinary crossing, so, after shrinking, they glue on a neighborhood of the axes. Lemma B.4 extends the box over S. The two prescribed axes determine its germ at (s_0,t_0), so the extension is unique. This includes constant curves.

For a general tree, first construct the box on the square of each component. Proceed through ordered pairs of distinct components by increasing distance in the dual tree. On C_i\times C_j, prescribe the two axes at the nodes pointing toward each other along the unique component route. These axes have already been defined at smaller distances; they are tangent to E_1,E_2, respectively, and agree at their corner by the previous edge identifications. Apply the axes construction. Every common edge matches: for two adjacent components, their distances to a fixed third component differ by one, and the common edge is an inward axis for the later pair. The finitely many component maps therefore glue continuously, and their separate tangencies imply the box condition also at nodes.

The same induction proves uniqueness, first on component squares and then from the inward axes. Restriction to a subtree is consequently R_u. To compare its ambient germs, compute the two holonomies along paths lying in T. The paths and box values are unchanged in T', so Equation (B.3) gives precisely the same germs.□

Proof. Choose a realizing tree T with markings s,t for the initial germ. Attach two separate copies of D to T, identifying their first markings with s and t, respectively. Map the copies to M by the two components of k. The resulting curve is again a rational tree, and the maps agree at the attachment points. Its box restricts to the original box, including its ambient germs, by the uniqueness and branch compatibility in Proposition B.5 and Lemma B.2.

Move the two parameter points along the copies of a path in D. Lemma B.2 gives exactly the asserted continuation along its image under k. Its terminal markings are smooth and distinct, so the terminal germ is attained. All paths between the markings are homotopic in the simply connected tree; the corresponding leafwise holonomy is unchanged.□

Proof. Choose a finite subdivision and holomorphic representatives on open neighborhoods of the compact subpaths. Consecutive representatives agree on an open neighborhood of each subdivision point. After shrinking the subpath neighborhoods and these overlaps, every sufficiently uniformly close path stays in the same domains and crosses through the same equality overlaps. The same finite chain of representatives works.□

Proof. For each degree d\ge0, take the closed locus S_d in \overline M_{0,3}(\Pp^N,d) of maps into Q with first evaluation q_*. Its second evaluation image F_d is closed algebraic. Every point of Q lies in some F_d: join it to q_* by moving first in one factor and then in the other along rational chains. Represent the chain by a rational tree. Insert contracted components if needed to put the endpoints at distinct smooth markings, add a third marking, and stabilize. This does not change the endpoint evaluations.

Write Q=Q_1\cup\cdots\cup Q_r for its irreducible components. For each i,

If none of the F_d contained Q_i, this would express the compact complex variety Q_i as a countable union of closed nowhere dense sets, contradicting the Baire category theorem. Choose d_i with Q_i\subset F_{d_i}. The disjoint union of the finitely many spaces S_{d_i}, omitting repetitions, is the required S_*.□

Proof. Choose a representative of the limiting stable map in a local finite-quotient chart with a family. A finite quotient map is open, so nearby coarse parameters admit representatives close to this one; no continuous choice is required.

Follow the finite component route between the markings. Away from small segments around its nodes, finitely many relative smooth charts move the chosen arcs to nearby fibers, with the prescribed marked ends. At a node the family is locally xy=a, where a is a function of the parameter. The incoming and outgoing nearby pieces can be joined within an arbitrarily small plumbing neighborhood when a is small; the same assertion holds for a=0. The family map is continuous, so all images in that neighborhood lie close to the limiting nodal value. Parametrize these joining pieces on the corresponding small subintervals. The resulting paths have uniformly close images. Contracted components cause no difficulty, since their limiting images are constant. Restricting the parameter locus to maps into Q keeps all these image paths inside Q.□

Proof. Semialgebraic choice applied to

gives a semialgebraic section. It is continuous on finitely many open subintervals. Compactness of S_* and one-variable semialgebraic monotonicity give one-sided limits at every endpoint. Continuity of e makes those limits lift the correct endpoint values. Extend each interval lift by its one-sided limits. We use the standard semialgebraic choice and one-variable finiteness results in [7, Theorems 2.1, 3.1, and 3.4]; see also [3].□

Proof. Use Lemma C.7 and consider one lifted interval [a,b], with prescribed attained germ g_a at \sigma(a). Continue g_a backwards along a marked route in the tree at a, obtaining an attained germ h at q_* by Lemma C.3. For each t\in[a,b], continue this fixed germ h forwards through the tree over t, and denote the endpoint germ by g_t. The choice of route does not affect it. At t=a, reverse-path uniqueness gives the prescribed g_a.

For t near any t_0, Lemma C.6 permits uniformly close image routes from the common point q_*. Lemma C.4 then supplies one ambient representative giving all g_t near t_0. Thus these germs are analytic continuation along \sigma on the whole interval. Each is attained by Lemma C.3. Repeat from the terminal germ on the next interval. The intermediate germ h at q_* is not assumed to be the original ordinary crossing; only its being fixed on the current interval is used.□

Proof of Theorem C.1. The compact family of routes has supplied continuation along every semialgebraic path, including through singularities and changes of component. It remains to identify the terminal germs and glue them. Triangulate Q semialgebraically [7, Theorem 4.4]. Since P is simply connected, so is Q. Start with the ordinary crossing germ at q_*, and continue it along piecewise linear paths in the triangulation. These paths are semialgebraic in Q, so Lemma C.8 applies. Any two such paths with the same endpoints are homotopic relative endpoints through paths with semialgebraic slices: apply relative piecewise linear approximation to a nullhomotopy in the finite triangulation. For a fixed slice, compact-path stability (Lemma C.4) and uniform continuity of the homotopy show that nearby slices have the same terminal germ. The terminal germ is consequently constant throughout the homotopy. We obtain well-defined ambient germs at every point of Q.

These germs are locally coherent along Q. Indeed, choose a small relative path-connected neighborhood of an endpoint inside the domain of its representative. Appending small paths that are piecewise linear in the fixed semialgebraic triangulation, after subdivision, and lie in that neighborhood shows that the same representative gives the assigned germs there. Such neighborhoods are available from the triangulation.

For completeness, this coherent family has a representative on an actual neighborhood of Q. Choose finitely many representatives c_i:V_i\to M giving the family on Q\cap V_i, and smaller open sets W_i with compact closures in V_i, still covering Q. On V_i\cap V_j, the set where the germs of c_i,c_j agree is a union of connected components, by the identity theorem. Its complement is therefore relatively closed. The compact sets

are disjoint from Q. Removing their finite union and restricting to \bigcup_iW_i gives a neighborhood of Q on which the representatives glue holomorphically. Notice that equality here is equality of ambient germs, not merely of values on Q.

Compactness of P gives an open neighborhood U of P whose square lies in this domain. Since P is connected, take the connected component of U containing it. The glued map restricts to c:U\times U\to M. Its differential identities in (C.1) hold near (p,p) and hence throughout connected U\times U by holomorphicity. The identity on the diagonal extends throughout connected U. In a product chart about any x\in U, these differential identities make the first output coordinate depend only on the first coordinate of the first input, and the second output only on the second coordinate of the second input. The diagonal identity identifies both functions with the corresponding coordinates. Hence c is ordinary crossing near every diagonal point, as required.□

Proof. On the open set

we claim the tensor identities

where v=v_1+v_2 and w=w_1+w_2 denote the supplied splittings of real tangent vectors. The identities hold near any regular diagonal point by the local product expression for h. The complement of V is a proper complex analytic subset of the connected manifold U\times U, so V is connected. Both sides of each identity are real analytic on V. The real analytic identity principle proves (D.3) throughout V.

Properness of f and the topology induced by \rho give r>0 with f^{-1}(B_{\rho}(z,r))\subset U. Consider the first inequality in (D.2). Approximate (a,a',b) in X^3 by triples (a_m,a'_m,b_m) for which all three inputs and both output points are in X^0. Such triples are dense: each excluded condition is a proper analytic subset of U^3, and properness of the output conditions follows from the regular values of c near a regular diagonal point.

For large m all inputs lie over B_{\rho}(z,r/9). Join a_m to a'_m in X^0 by a piecewise smooth path of h-length less than d(a_m,a'_m)+\eta, with \eta>0 arbitrarily small. This is possible because d on X^0 is its intrinsic length distance. Every point of this path lies over B_{\rho}(z,r/3+\eta): the initial point has distance less than r/9 from z, and the path length is less than 2r/9+\eta. In particular the path remains in U, with a margin for perturbation.

For this fixed b_m, the set

is a proper analytic subset, since the output at a_m is regular. It has real codimension at least two. Keeping the endpoints fixed, we may perturb the path to avoid A_m with arbitrarily small length error. Indeed, on a neighborhood of the compact path the metric is smooth; general position relative to an analytic stratification gives such a perturbation within that neighborhood. Along the perturbed path, (D.3) bounds the output speed by the input speed, because the two summands are h-orthogonal. Integration, followed by letting the length errors tend to zero, proves

Continuity of c and d gives the first inequality for the original inputs. Interchanging the two variables proves the second.□

Proof. For the first assertion choose a subdivision 0=t_0<\cdots<t_N=1 such that each \lambda(I_i) has diameter at most \delta, where I_i=[t_{i-1},t_i]. On every diagonal tile I_i^2 put

using (D.4). Its diagonal is \lambda, and each of its partial slices has diameter at most \delta by (D.2). These tiles agree at their common vertices.

Fill the remaining tiles I_i\times I_j by increasing |i-j|. Above the diagonal, the bottom and right edges of a new tile have already been defined; below it, the top and left edges have already been defined. In either case these are one horizontal and one vertical edge, meeting at a corner. Apply (D.5). The induction invariant is that every partial slice of each completed tile has diameter at most \delta. The tile rule proves both agreement on its old edges and preservation of this invariant. Tiles at the same stage share no edge. If they share a vertex, its value is already fixed by an intervening tile from an earlier stage. Thus the finite filling is consistent everywhere.

For the second assertion, subdivide each prescribed axis so that the image of every edge segment has diameter at most \delta. Fill the rectangular array in increasing order of the sum of its two indices. The bottom and left edges of a new tile have already been prescribed, so precisely the same rule and invariant apply.

In both constructions the tile maps glue continuously. The box property also holds across seams: near a seam value, continuity places the adjacent path pieces in one foliation chart, and both follow the plaque through their common endpoint. This proves the assertions.□

Proof. All-path continuation and monodromy give holomorphic maps

With compatible basepoints, the covering lifting theorem gives holomorphic lifts

Their initial germs are inverse. Each composite is therefore the identity near its basepoint, and hence everywhere on its connected source by the identity theorem. The wrong-factor components of their differentials are holomorphic bundle morphisms that vanish in the initial product chart. They vanish identically, giving the asserted compatibility with the supplied splitting.□