Universal-cover splitting for compact Kähler manifolds

1. Integrability gives local products; tangencies remove meromorphic indeterminacy

Combining the two Frobenius submersions gives local coordinates in which each $E_i$ is a coordinate block. Transitions locally separate as $(z_1,z_2)\mapsto(h_1(z_1),h_2(z_2))$. This does not assert a product metric.

Lemma 2.1; Definition 2.2 · pp. 5–6

For a box on a Hartogs figure $H\subset\mathbb C^2$, Ivashkovich's theorem gives a meromorphic extension to its envelope of holomorphy, a bidisk $Q$. The input hypotheses are that $H$ is a domain in a Stein manifold and the target is compact Kähler. The manuscript separately proves holomorphicity. If the projection of the graph $\Gamma\subset Q\times X$ had a curve fibre, some component $h$ of a target product chart would vary along it. If $h$ belongs to the first block, a general point is isolated in a level of $(s,h)$. Near ordinary graph points, however, $h$ is independent of $t$, so the same type of level contains the $t$-direction. A level avoiding the boundary of a small ball would then be a positive-dimensional compact analytic set, a contradiction. Properness, density and connectedness of the ordinary graph make the remaining finite fibres singletons; continuous locally bounded coordinate functions remove the indeterminacy. The affine coordinates defining the Hartogs figure need not be the box coordinates $(s,t)$.

[Iva, Theorem 2.2, p. 3]

Proposition 3.1 · pp. 6–8

2. A closed Kähler form and two scales cross a mixed boundary

At a smooth convex boundary point, assume that the complex tangent has nonzero components in both parameter directions. With $v$ complex tangential and $u$ transverse, the box conditions and the compactness bound $\sup_X\|P_i\|<\infty$ imply $\|df(\partial_{\mathrm{Re}\,u})\|\le C\|df(\partial_{\mathrm{Re}\,v})\|$. On disks $u=x<0$, $|v|\le k\sqrt{-x}$, closedness $d\omega=0$ and Stokes' theorem bound the areas $A(x)$ using the contracting-boundary term.

In slices $x_j=-\delta_j\to0$, many disjoint annuli yield one with area tending to zero. The derivative rescaling in Lemma 4.1 places its image in one product chart. Its radius $\lambda_j$ satisfies both

The first makes the Hartogs shell wider than the remaining distance to the boundary; the second keeps it inside the original tangential disk. Separated target coordinates thicken the shell transversely, and Proposition 3.1 fills it. The Kähler metric is arbitrary: neither orthogonality nor parallelism of the summands is assumed.

Lemma 4.1; Proposition 4.2; (4.2)–(4.5) · pp. 8–11

3. Combine prescribed axes on their entire polydisks

Starting with two holomorphic disks tangent to $E_1,E_2$ through the same point, extend their box germ over $|s|^{2m}+|t|^{2m}<R^{2m}$. The preceding step treats boundary points off the axes; the prescribed disks treat points on the axes. Extension cannot stop at $R<1$, and letting $m\to\infty$ gives the full bidisk. For higher-dimensional axes, first combine radial disks and then reconstruct a finite grid in fixed product charts with parameters. This reconstruction proves holomorphic parameter dependence; pointwise existence of a box would not suffice.

Propositions 5.1, 5.3; Lemma 5.2 · pp. 12–13

4. Two finite grids continue the map and its inverse

Subdivide two leaf paths and express each piece as a holomorphic plaque map evaluated on continuous parameters. After Proposition 5.3 fills a tile, each opposite edge still has a holomorphic representative on the entire original polydisk. The prescribed-axis rectangle can therefore be completed without keeping successive images small. For an arbitrary path $\gamma$, prescribe diagonal tiles first, then fill bands away from the diagonal to obtain $R(s,s)=\gamma(s)$. This second construction is needed to continue the inverse map along all paths.

Proposition 6.1 · pp. 14–15

5. Universal covers of intrinsic leaves give the biholomorphism and compatibility

Give the leaves $L_i$ through the basepoint their intrinsic plaque topologies, rather than the subspace topology from $X$. Lemma 6.2 makes them connected complex manifolds with universal covers $Y_i=\widetilde L_i$. Proposition 6.3 simultaneously continues the inverse local germs $H_o:L_1\times L_2\to X$ and $K_o:X\to L_1\times L_2$ along the rectangles.

Lemma 6.2; Proposition 6.3 · pp. 15–17

Monodromy defines maps on the simply connected covers. Lift them with matching basepoints to $\widetilde H:Y_1\times Y_2\to\widetilde X$ and $\widetilde K:\widetilde X\to Y_1\times Y_2$. Both composites are the identity near their basepoints, hence everywhere by connectedness and the identity theorem. The wrong-factor components of $d\widetilde K$ likewise vanish globally as holomorphic bundle maps. Invertibility of the differential and $\dim Y_i=r_i$ turn the inclusions into equalities. This proves compatibility with the specified embeddings $E_i\subset T_X$, as well as the abstract product isomorphism.

Proof of Theorem 1.1 · p. 17

1. From non-uniruledness to integrability

Höring's Theorem 1.3 makes both factors of a holomorphically split tangent bundle integrable on a non-uniruled projective manifold. The present hypotheses satisfy its assumptions. Its output here is closure under the Lie bracket.

[Hor07, Theorem 1.3, p. 1]

2. From integrability to an ordinary universal-cover product

A smooth projective complex manifold is compact Kähler, so the previous output satisfies Theorem 1.1. The chain is a specified holomorphic splitting, then two integrable distributions, then a compatible universal-cover product. The first step alone is not used to infer the last conclusion in general.

Corollary 1.2 · p. 3

Theorem 1.1 · p. 3

1. The canonical model and approximate Kähler–Einstein metrics

Proposition 7.1 constructs a contraction with $K_X=f^*K_Z$, ample $K_Z$, canonical $Z$, and $f$ an isomorphism over $Z_{\mathrm{reg}}$. Semiampleness uses the cited Kollár–Mori base-point-free theorem: for $D=K_X$ and an integer $a>1$, the divisor $aD-K_X=(a-1)K_X$ is nef and big. Its use in the manuscript was checked; the book's theorem statement and exact page have not been independently checked here.

Proposition 7.1 · pp. 17–18

Song's approximation applies to this smooth crepant resolution. For $0<\varepsilon<1$ it supplies metrics satisfying

The next step uses smooth convergence on compact subsets of $X^\circ$ to a negative Kähler–Einstein metric $g$. Song's additional conclusion identifying its metric completion with $Z$ is not needed for this main-text route.

[Song, Equation (4.28); Lemma 4.1, pp. 27–28]

Lemma 7.2 · p. 18

2. Vanishing second fundamental forms imply global integrability

Let $P_{j,\varepsilon}$ be $g_\varepsilon$-orthogonal projection onto $E_j$ and $B_{j,\varepsilon}$ its second fundamental form. These projections differ from those along the specified direct sum in the main proof. Integrating and adding the two subbundle-curvature identities cancels degrees through $c_1(E_1)+c_1(E_2)=c_1(T_X)$. Since $C_\varepsilon=P_{1,\varepsilon}+P_{2,\varepsilon}-I$ satisfies $\operatorname{tr}C_\varepsilon=0$ and $-I\le C_\varepsilon\le I$,

The bound comes from a polynomial in cohomology classes. Smooth convergence on $X^\circ$ forces both limiting second fundamental forms to vanish, so the connection of $g$ preserves both factors. Its torsion-free property implies integrability on $X^\circ$. The holomorphic Frobenius obstruction

vanishes on a dense open set, hence on all of $X$. This is where the metric argument returns to the holomorphic hypothesis of the main theorem.

Proposition 7.3; (7.1) · pp. 18–19

3. Theorem 1.1 supplies the global product

Integrability and projectivity now permit Theorem 1.1, giving the ordinary universal-cover product compatible with the specified splitting. No completeness of the metric on $X^\circ$ is assumed here, and de Rham's decomposition is not directly applied to that parallelism. Appendices A–D give a different global continuation mechanism through collapsed canonical fibres; it is separate from the route explained here.

Proof of Corollary 1.3 · p. 19