A product X = Y1 x Y2 of complex manifolds has holomorphic tangent bundle
T_X = pr1^* T_Y1 (+) pr2^* T_Y2,
pr_i being the projection onto the i-th factor. A holomorphic splitting of a compact complex manifold is a fixed decomposition T_X = E1 (+) E2 into holomorphic subbundles of positive rank. A subbundle E of T_X is integrable when the Lie bracket of any two local holomorphic sections of E is again a section of E.
./kobayashi-nomizu-foundations-of-differential-geometry-vol2.pdf
Holomorphic Frobenius gives integrability a concrete local meaning: every point of X has coordinates (z1, z2) in a product of polydiscs D^r1 x D^r2 in which E1 is the z1-direction and E2 the z2-direction. The question is whether those local pieces fit together globally, not whether the local pieces exist.
./kobayashi-nomizu-foundations-of-differential-geometry-vol2.pdf
Two obstructions keep an integrable splitting from producing a global product. The holonomy of one foliation along leaves of the other may fail to close; and even when X itself is a product, a given splitting may not come from any product decomposition of X. The second obstruction is why integrability is assumed rather than derived.
Beauville's example makes the second obstruction concrete. Take X = A x P^1 with A a complex abelian surface. On A choose independent translation-invariant vector fields U, V with [U, V] = 0; on P^1 choose vector fields S, T with [S, T] != 0; view both families on the product. Set E1 to be the rank-two span of U + S and V + T, and E2 = T_(P^1). Transversality gives T_X = E1 (+) E2, but
[U + S, V + T] = [U, V] + [S, T] = [S, T] in T_(P^1)
is not in E1. The splitting exists, X is a product, and yet no product decomposition of X realises E1 and E2 as its coordinate tangent bundles. Integrability of the specific summands is a real constraint.
./beauville-2000-complex-manifolds-with-split-tangent-bundle.pdf
Beauville stated the compatible universal-cover conjecture for compact Kaehler manifolds with integrable partial sums of the tangent summands (Beauville 2000, Section 2.3) and settled the Kaehler-Einstein and compact-Kaehler-surface cases there (Theorems A and C). For two summands the partial-sum hypothesis amounts to simultaneous integrability of E1 and E2. Hoering later proved automatic integrability on non-uniruled projective manifolds (Hoering 2007, Theorem 1.3), which turns the conjecture into a theorem in that class.
./hoering-2007-uniruled-varieties-with-split-tangent-bundle.pdf
Prior art. de Rham's splitting theorem is the Riemannian prototype: a complete simply connected Riemannian manifold whose tangent bundle splits into parallel complementary distributions is a metric product of integral manifolds. The compact Kaehler-Einstein analog for a holomorphic split of the tangent bundle is due to Yau, refined by Beauville. Under additional flatness or partial-sum hypotheses, Druel, Brunella-Pereira-Touzet, Pereira-Touzet, and Druel-Pereira-Pym-Touzet prove compatible universal-cover splitting. Campana-Peternell and Demailly supply the tools that let Hoering prove automatic integrability of split tangent bundles on non-uniruled projective manifolds and, in follow-up work, on normal Q-factorial Fano-type varieties.
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Main theorem. Let X be a compact connected Kaehler manifold of complex dimension n >= 2, and T_X = E1 (+) E2 a holomorphic splitting into integrable subbundles of positive ranks r1, r2. Let pi : Xtilde -> X be the ordinary universal covering. There are connected simply connected complex manifolds Y1, Y2 and a biholomorphism Phi : Xtilde -> Y1 x Y2 with
dPhi(pi^* E_i) = pr_i^* T_Yi, i = 1, 2.
In particular dim_C Yi = ri.
The equation dPhi(pi^* E_i) = pr_i^* T_Yi asks more than existence of some product structure on Xtilde: it asks that the specified subbundles become the two coordinate directions after lifting. The factors are the universal covers of the two foliation leaves through a chosen basepoint and can be noncompact. The conclusion concerns the ordinary universal cover and says nothing about finite covers of X.
Two consequences. On a smooth connected non-uniruled projective complex manifold every holomorphic splitting into positive-rank subbundles is integrable by Hoering's theorem (Hoering 2007, Theorem 1.3), and the main theorem applies. On a smooth connected projective X of dimension >= 2 with K_X nef and big, every holomorphic splitting into positive-rank subbundles is integrable by the second-fundamental-form calculation for the canonical Kaehler-Einstein metric, and the same compatible product follows.
For the analytic part let X be a compact connected Kaehler manifold of complex dimension n >= 2 and T_X = E1 (+) E2 a splitting into integrable subbundles of ranks r1, r2. Write D for the open unit disk.
./kobayashi-nomizu-foundations-of-differential-geometry-vol2.pdf
Product charts. Every point of X has a holomorphic coordinate neighbourhood (D^r1 x D^r2, z = (z1, z2)) in which E_i is tangent to the z_i-direction. Changes between two such charts are locally of the separated form
z1' = h1(z1), z2' = h2(z2),
with h1, h2 local biholomorphisms.
Proof. Fix a point x. Choose a local holomorphic coordinate projection pi: V -> C^r1 whose differential is nonsingular on E1 near x. Lift the ambient coordinate vector fields to sections of E1. Integrability keeps the brackets of the lifts inside E1, so their projections to C^r1 vanish; the projection is an isomorphism from E1 to the base tangent, so those brackets vanish. The commuting local flows of the lifts give holomorphic coordinates (z1, z2) with E1 = ker(d z2). Repeating for E2 and combining both coordinate systems on a smaller polydisc gives the required pair. A chart change preserves E1 and E2, so its off-diagonal blocks vanish on small product neighbourhoods; invertibility of the chart change then makes the diagonal blocks invertible.
./kobayashi-nomizu-foundations-of-differential-geometry-vol2.pdf
A first plaque is a slice of a product chart with z2 fixed; a second plaque is a slice with z1 fixed. A continuous path is a first (respectively second) plaque path if every point of it has a neighbourhood on which the path lies in a first (respectively second) plaque. Transitions have to be separated only locally; disconnected overlaps need not admit a single formula.
Holomorphic box. Let U be open in C^a x C^b with parameter blocks (s, t). A holomorphic map f: U -> X is a box if at every point of U
df(T C^a (+) 0) is contained in E1, df(0 (+) T C^b) is contained in E2.
The conditions ask for tangency only; f need not be an immersion. In a product chart a box reads f(s, t) = (f1(s), f2(t)) locally, and this local separation conversely implies the two tangencies. The scalar case a = b = 1 appears first.
Fix a Kaehler form omega on X with Hermitian metric g(u, v) = omega(u, J v), J denoting the complex structure. Real and holomorphic tangent bundles are identified as complex vector bundles, so multiplication by i on real tangent vectors is J. The projection P_i: T_X -> E_i determined by the splitting is complex-linear. Compactness gives
sup_{x in X} ||P_i(x)|| < infinity, i = 1, 2,
and no orthogonality of E1 and E2 is assumed.
Hartogs extension for boxes. Let (s, t) be coordinates on C^2 and (w, v) any invertible complex-affine coordinate system. For real numbers 0 < e < a and 0 < c < b, put
H = {|w| < e, |v| < b} union {|w| < a, c < |v| < b},
Q = {|w| < a, |v| < b}.
Every holomorphic box f: H -> X with df(partial_s) in E1 and df(partial_t) in E2 extends uniquely to a holomorphic box Q -> X with the same tangencies.
./ivashkovich-2008-extra-extension-properties.pdf
Proof. Step 1: meromorphic extension. Scalar holomorphic functions on H extend to Q: expand a function on the shell {c < |v| < b} in Laurent series in v; the coefficients of the negative powers are holomorphic in w, vanish on {|w| < e}, hence by continuation on {|w| < a}; Cauchy estimates on an intermediate circle give normal convergence on compacta of Q. So Q is the envelope of holomorphy of H. Ivashkovich's meromorphic extension theorem (Ivashkovich 2008, Theorem 2.2; originally Ivashkovich 1992) extends any meromorphic map from a domain in a Stein manifold into a compact Kaehler manifold meromorphically to the envelope of holomorphy. Since H is a domain in C^2 and f is holomorphic, f extends meromorphically to Q.
./ivashkovich-1992-hartogs-type-extension-for-meromorphic-maps.pdf
Step 2: graph. Let Gamma in Q x X be the meromorphic graph with reduced analytic structure and rho: Gamma -> Q the first projection. Gamma is a proper analytic subset of Q x X, so rho is proper. There is a proper analytic A in Q, disjoint from H, with the graph over U = Q minus A being the ordinary graph of a holomorphic map F: U -> X. The ordinary graph is dense in Gamma and U is connected. Applying the identity theorem to P2 dF(partial_s) and P1 dF(partial_t) gives
P2 dF(partial_s) = 0, P1 dF(partial_t) = 0 on U,
with P_i: T_X -> E_i the projections of the splitting.
Step 3: no positive-dimensional fiber of rho. On every local irreducible surface branch of Gamma both s and t are nonconstant: choose a relatively open part of the branch away from the other branches; that part meets the dense ordinary graph, where rho is a local isomorphism and s, t are coordinate functions of Q. Hence for every s0 the analytic set Gamma cap {s = s0} has local dimension at most one; sym for t.
Suppose a fiber over (s0, t0) had positive dimension. It is contained in Gamma cap {s = s0}, so it has a curve component C. Take a small piece of C inside a product chart z = (z1, z2) of X. Some scalar component h of z1 or z2 is nonconstant on this piece, else the curve piece, contained in {(s0, t0)} x X, would be a point.
Suppose h is a component of z1. The curve piece is a component of the reduced set Gamma cap {s = s0}; its smooth points carry dimension one. Choose a general q where the reduced set is smooth and h nonconstant. Then (s, h) has an isolated zero at q: locally q is the only point of Gamma with (s, h) = (s0, h(q)). Work in ambient coordinates on Q x X near q and choose a closed Euclidean ball Bbar around q small enough that Gamma meets the (s, h)-level of Psi(q) = (s0, h(q)) only at q. The image of Gamma cap (partial B) under Psi is compact and misses Psi(q). Choose q_j on the ordinary graph tending to q. For large j, Psi(q_j) misses that compact image, so
K_j = Gamma cap Bbar cap Psi^{-1}(Psi(q_j))
is a compact analytic subset of the open ball B. Near q_j, on the ordinary graph, the identity P2 dF(partial_s) = 0 says h is independent of t; varying t with s fixed produces a one-dimensional germ in K_j through q_j. This contradicts the standard fact that a compact analytic subset of an open subset of C^N has no positive-dimensional component (maximize the strictly plurisubharmonic squared coordinate norm on a component and restrict to a curve germ through the maximum).
If h is a component of z2, the same argument uses Psi = (t, h) and varies s. Hence every fiber of rho has dimension zero; properness makes it finite.
Step 4: singleton fibers. Fix p in Q and write its fiber as {(p, x1), ..., (p, xk)}. Pick pairwise disjoint neighbourhoods V1, ..., Vk of the x_i with pairwise disjoint closures. Properness gives a base ball D around p such that every point of rho^{-1}(D) has its target in the union of the V_i: otherwise a sequence of graph points over points tending to p, outside that union, would have a limit in the central fiber, contradicting the choice of V_i. The ordinary locus D minus A is connected, so its image under F lies in a single V_i. Density then places every x_j in the closure of V_i, and the disjoint closures force k = 1.
The proper bijection rho is a homeomorphism; its inverse followed by the second projection defines a continuous Fhat: Q -> X. Near each source point the image lies in a target coordinate chart, and its coordinates are continuous and holomorphic off A. The removable-singularity theorem for locally bounded holomorphic functions across analytic subsets of a complex manifold makes them holomorphic; Fhat is holomorphic. The tangencies extend to all of Q by the identity theorem. Uniqueness follows from the identity theorem on the connected bidisk Q.
Two features of the extension are used by the mixed-boundary argument: no rank condition on the box is needed, and the Hartogs figure may be aligned with any invertible complex-affine change of the box coordinates (s, t).
Small-area lemma. Let M be a compact Hermitian manifold with positive (1,1)-form omegaM, and U a domain in C. If h_j: U -> M are holomorphic with int_U h_j^* omegaM -> 0, then sup_{z in K} ||d h_j(z)|| -> 0 on every compact K in U. Closedness of omegaM is not required.
./brody-1978-compact-manifolds-and-hyperbolicity.pdf
Proof. It suffices to treat a disk D with Dbar in U. Put d(z) = dist(z, partial D) and Mj = max over Dbar of d(z) ||d h_j(z)||. If Mj does not tend to 0, pass to a subsequence with Mj >= epsilon. Let z_j be a maximizer and set a_j = ||d h_j(z_j)||, d_j = d(z_j). For |zeta| < epsilon/4 put k_j(zeta) = h_j(z_j + zeta/a_j). On that disk d(z_j + zeta/a_j) >= d_j/2 and maximality gives ||d k_j(zeta)|| <= 2, ||d k_j(0)|| = 1. After subsequence, k_j(0) converges to a point of M. The derivative bound puts images of a fixed smaller disk in a relatively compact coordinate neighbourhood of that point; Montel's theorem and the Cauchy estimates give a further subsequence converging there with its derivatives. The 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 those of h_j and unchanged by the affine reparametrization. Hence Mj -> 0; on each smaller disk d is bounded below, so derivatives tend uniformly to zero. A finite cover proves the assertion on any compact subset.
Mixed-boundary extension. Let B in C^2_(s,t) be convex and f: B -> X a holomorphic box. Suppose p in partial B has a C^2 defining function with nonzero differential. If the complex tangent line to partial B at p is generated by a vector whose components in the s and t directions are both nonzero, then f extends as a holomorphic box to a neighbourhood of p, agreeing with f on its intersection with B.
Proof. Use the fixed omega, its Hermitian norm, and the bound on the projections P_i. Write p = (s*, t*).
Complex-affine coordinates at p. Choose (u, v) with
(s, t) = (s*, t*) + (a1, a2) u + (b1, b2) v, b1 b2 != 0,
where the v-direction is complex tangent and the positive u-direction points outward from B. Constants may depend on p and these coordinates.
Let rho be a local defining function with B = {rho < 0} near p. Taylor's theorem, using that both real derivatives of rho in the complex tangent direction vanish, gives rho(x, v) <= c x + M(|x|^2 + |v|^2) with c > 0. Choose k > 0 small enough that for all x < 0 close to zero the closed disks
D_x = {u = x, |v| <= r(x)}, r(x) = k sqrt(-x),
are compactly contained in B.
Let eu = partial_{Re u} and ev = partial_{Re v}. Box tangencies and complex linearity give
df(eu) = sum_{i = 1, 2} (a_i / b_i) P_i df(ev).
Hence
||df(eu)|| <= C ||df(ev)||, C = sum_{i = 1, 2} |a_i / b_i| sup_X ||P_i||.
Holomorphicity makes ||df(ev)|| the speed in every unit real direction of the v-plane, including radial.
Set Omega = f^* omega and A(x) = int_{D_x} Omega. Closedness of Omega and smoothness of D_x in x let Stokes be applied to the moving disks:
A'(x) = int_0^{2 pi} Omega(eu + r'(x) partial_r, partial_theta) |_{u = x, v = r(x) e^{i theta}} d theta.
On a boundary circle put q = ||df(ev)||. Since partial_theta = r J partial_r,
Omega(partial_r, partial_theta) = r q^2, |Omega(eu, partial_theta)| <= C r q^2,
hence
A'(x) <= int_0^{2 pi} r(x)(C + r'(x)) q^2 d theta.
With r'(x) = -k / (2 sqrt(-x)). Choose x0 < 0 close enough to zero that r'(x) <= -C for x0 <= x < 0. Then
0 <= A(x) <= A(x0) < infinity, x0 <= x < 0.
No product property of the Kaehler metric appears.
Let x_j = -delta_j with delta_j -> 0. Among the scales lambda = 5^l delta_j^{3/4}, integer l >= 0, contained in [delta_j^{3/4}, delta_j^{5/8}], the annuli lambda/2 < |v| < 2 lambda are pairwise disjoint and their number tends to infinity. For large j the annuli lie in D_{x_j}. The area bound gives one of them, at a scale lambda_j, area tending to zero. The scales satisfy
delta_j / lambda_j <= delta_j^{1/4} -> 0, lambda_j / sqrt(delta_j) <= delta_j^{1/8} -> 0.
Apply the small-area lemma to h_j(zeta) = f(x_j, lambda_j zeta) on 1/2 < |zeta| < 2. Derivatives tend uniformly to zero on a neighbourhood of 3/4 <= |zeta| <= 3/2. This compact annulus has bounded intrinsic Euclidean diameter, so the image of
u = x_j, 3 lambda_j / 4 < |v| < 3 lambda_j / 2
has diameter tending to zero. A finite cover of X by product charts has a positive Lebesgue number, so for large j the whole image lies in a single product chart z: V -> V1 x V2. Restrict the box to the slice u = x_j in this chart, writing it as (g1(v), g2(v)).
Put w = u - x_j and c_i = a_i / b_i. Choose eta > 0 independent of j with eta max(1, |c1|, |c2|) < 1/8. On
S_j = {|w| < eta lambda_j, lambda_j < |v| < 5 lambda_j / 4}
define
F_j(w, v) = z^{-1}(g1(v + c1 w), g2(v + c2 w)).
Both shifted arguments stay in 3 lambda_j / 4 < |v| < 3 lambda_j / 2, so F_j is well defined. The range of z is a product, so combining the two factor coordinates needs no extra target-chart margin. Moreover
v + c1 w = (s - s* - a1 x_j) / b1, v + c2 w = (t - t* - a2 x_j) / b2,
so F_j is a box in the original (s, t).
F_j agrees with f on an open germ at every point of S_j with w = 0. Continuity puts f into the same chart V on a neighbourhood, where its separated expression is (phi1(s), phi2(t)). Restriction to u = x_j gives the local expressions for g1, g2, and the displayed identities show that F_j reproduces f on a full neighbourhood, not only on the slice.
The closed slice disk {u = x_j, |v| <= 5 lambda_j / 4} lies in B, so there is 0 < e_j < eta lambda_j with f defined on
C_j = {|w| < e_j, |v| < 5 lambda_j / 4}.
The overlap C_j cap S_j is a connected product of a disk and an annulus. The identity theorem for maps into X extends the open-germ agreement to this overlap, so f and F_j together give a box on the Hartogs figure C_j union S_j.
Hartogs extension gives a box on Q_j = {|w| < eta lambda_j, |v| < 5 lambda_j / 4}. 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 (w, v) = (delta_j, 0), which belongs to Q_j for large j. This extends the box across p.
Two things matter for the disk-axes argument: no product property of omega is used, and the annulus scale lambda_j sits strictly between delta_j and sqrt(delta_j). The upper inequality lambda_j / sqrt(delta_j) -> 0 keeps lambda_j well inside D_{x_j}; the lower inequality delta_j / lambda_j -> 0 makes the shell wide enough in the u-direction to reach past delta_j. The exponent 3/4 is the geometric mean of the two constraints.
Disk axes. Let alpha, beta: D -> X be holomorphic with d alpha(T D) in E1, d beta(T D) in E2 and alpha(0) = beta(0). There is a unique holomorphic box F: D^2 -> X with
F(s, 0) = alpha(s), F(0, t) = beta(t)
for all s, t in D.
Proof. Product coordinates at alpha(0) = beta(0) give a unique box germ at (0, 0): first coordinate from alpha, second from beta. Any two boxes extending this germ agree on a connected common domain by the identity theorem.
Fix an integer m >= 1 and let B_m(R) = {(s, t) in C^2 : |s|^{2m} + |t|^{2m} < R^{2m}} for 0 < R <= 1; these domains are convex. Let R* be the supremum of radii to which the germ extends as a box. Nested extensions agree and combine to a box on B_m(R*), with values alpha, beta on the axes by the one-variable identity theorem. Suppose R* < 1.
Boundary points with s t != 0. At such a point of partial B_m(R*) the complex tangent has both components nonzero: the real defining function of the boundary is |s|^{2m} + |t|^{2m} - R^{2m} and its complex s- and t-derivatives are both nonzero when s t != 0. The mixed-boundary extension extends the box across the point.
Boundary points (s*, 0). The complex tangent degenerates and the mixed-boundary extension does not apply. Reconstruct on a small product using one coordinate from alpha and one from a known interior point. Choose a disk U around s* with closure in D and alpha(U) in a single product chart z = (z1, z2). Fix s' in U with |s'| < R*, so F is defined at (s', t) for small t. For epsilon > 0 small enough,
G(s, t) = z^{-1}(z1(alpha(s)), z2(F(s', t))), (s, t) in U x D_epsilon,
is a holomorphic box: first coordinate from alpha, second from F, product range of z needs no target-chart margin. Near (s', 0), G agrees with F by local separation. Since (U x D_epsilon) cap B_m(R*) is convex, the two extensions agree on the whole intersection. This extends across (s*, 0). The points (0, t*) are treated symmetrically.
Gluing. Each boundary extension is defined on some neighbourhood of its point. Restrict each to an ordinary Euclidean ball centred at its boundary point. If two such balls intersect, their intersection contains a point of the segment joining the centres; the segment lies in the closure of B_m(R*) by convexity. Since the intersection is open, it also meets B_m(R*), where the two extensions agree; hence the two extensions agree on their whole connected intersection. A finite subcollection covers the compact boundary partial B_m(R*), giving an extension to a neighbourhood of the closure of B_m(R*). This neighbourhood contains B_m(R') for some R* < R' < 1, a contradiction.
Hence the germ extends to B_m(1) for every m. The domains B_m(1) increase with m to D^2, and the resulting boxes agree on connected overlaps; their union is a holomorphic box F: D^2 -> X. Uniqueness follows from the initial uniqueness observation.
Continuous box. A continuous map R: [0, 1]^2 -> X is a continuous box if it is locally separated in product coordinates: every horizontal slice locally follows an E1-plaque and every vertical slice locally follows an E2-plaque. If a whole subrectangle maps into one product chart its first coordinate depends only on the horizontal parameter and its second only on the vertical parameter; the unwanted coordinate is locally constant along connected slices.
Uniqueness. Two continuous boxes on [0, 1]^2 with the same bottom and left edges are equal.
Proof. Choose a Lebesgue number lambda > 0 for a finite product-chart cover of X. Uniform continuity gives a finite rectangular grid such that each closed tile, under either candidate, has diameter < lambda/3. Whenever the two maps agree at a tile's lower-left corner, the union of their tile images has diameter < 2 lambda/3, so both lie in one product chart. In that chart a box is determined by its bottom first coordinate and left second coordinate. Starting with the prescribed edges and proceeding by increasing sum of tile indices proves equality on the whole grid.
Polydisk axes. Let S, T be unit polydisks of positive dimensions. Suppose alpha: S -> X, beta: T -> X are holomorphic with alpha tangent to E1, beta tangent to E2, and alpha(p0) = beta(q0) for some (p0, q0) in S x T. There is a unique holomorphic box F: S x T -> X with
F(p, q0) = alpha(p), F(p0, q) = beta(q).
Proof. Polydisk automorphisms reduce to p0 = q0 = 0. For fixed (p, q) in S x T the maps sigma -> alpha(sigma p), tau -> beta(tau q) are holomorphic on disks of radius greater than one. After rescaling, disk axes give a holomorphic box; restrict it to the closed real square and call the result R_(p, q): [0, 1]^2 -> X. Set F(p, q) = R_(p, q)(1, 1). Constant-axis cases and uniqueness of continuous boxes give the prescribed axis values; parameter dependence remains.
Fix (p*, q*) and a finite grid on the parameter square such that closed tile images of R_(p*, q*) lie in product charts. Reconstruct the square for nearby (p, q) using these fixed charts, setting tile-by-tile values equal to R_(p, q). Begin with the axes and process tiles from bottom and left. Maintain three properties: (a) each constructed tile depends jointly continuously on its real parameters (sigma, tau) and on (p, q); (b) each is a continuous box in the real parameters; (c) at each fixed (sigma, tau), the tile is holomorphic in (p, q) with p-directions in E1 and q-directions in E2. These hold on the initial axes because alpha and beta are holomorphic.
To fill one tile, write B(sigma; p, q) and L(tau; p, q) for the constructed bottom and left edges. At (p*, q*) their images lie in the chosen product chart z = (z1, z2). Joint continuity and compactness of the edge images put both images in the same chart after shrinking the parameter neighbourhood. Define the new tile by
z^{-1}(z1(B(sigma; p, q)), z2(L(tau; p, q))).
The edges meet at a common corner; plaque constancy on each edge shows the formula reproduces both. Joint continuity and separation in (sigma, tau) follow. For fixed (sigma, tau) the formula is holomorphic in (p, q), and the induction hypothesis gives
partial_q (z1 o B) = 0, partial_p (z2 o L) = 0,
so the parameter tangencies carry through. At (p*, q*) the reconstructed tile is the original one, so the induction continues. Only finitely many tiles, hence finitely many shrinkings.
Tiles agree on common edges and form a continuous box across seams: in a product chart around the common image, the unwanted coordinate along a horizontal or vertical slice is constant on each tile piece and agrees at shared endpoints, hence constant across the seam. Uniqueness of continuous boxes identifies the reconstruction with R_(p, q) for every nearby parameter pair; its upper-right value is F(p, q). F is a holomorphic box near (p*, q*) and hence everywhere. Uniqueness follows from the axes determining the box germ at (0, 0) together with the identity theorem.
The construction produces the box from full-polydisk representatives of its edges, not only existence. A transported edge retains its holomorphic parametrisation on an entire polydisk; that property, not a size bound on the edge image, is what permits the finite-grid fill in the composition argument.
Path rectangles. (1) If alpha, beta: [0, 1] -> X are continuous and locally follow E1-, E2-plaques with alpha(0) = beta(0), then there is a continuous box R: [0, 1]^2 -> X with R(s, 0) = alpha(s), R(0, t) = beta(t). (2) Every continuous path gamma: [0, 1] -> X is the diagonal of a continuous box: there is a continuous box R: [0, 1]^2 -> X with R(s, s) = gamma(s).
Proof. (1) Subdivide the horizontal interval into pieces I_i = [u_{i-1}, u_i] and the vertical into J_j = [v_{j-1}, v_j] on which the paths lie in single plaques. Each horizontal piece reads alpha(s) = A_i(a_i(s)) for s in I_i, with a_i: I_i -> D^r1 continuous and A_i: D^r1 -> X holomorphic tangent to E1. Similarly beta(t) = B_j(b_j(t)) for t in J_j, b_j: J_j -> D^r2 continuous and B_j holomorphic tangent to E2; r_i = rank E_i.
Fill tiles I_i x J_j from bottom and left. Inductively the known edges have the forms A(a_i(s)) and B(b_j(t)) with A, B holomorphic on the entire polydisks D^r1, D^r2 and tangent to their bundles, and agreeing at the common corner. Polydisk axes at origins a_i(u_{i-1}) and b_j(v_{j-1}) gives a holomorphic box G: D^r1 x D^r2 -> X. Its evaluation R(s, t) = G(a_i(s), b_j(t)) fills the tile. The opposite horizontal and vertical edges have holomorphic representatives
a -> G(a, b_j(v_j)), b -> G(a_i(u_i), b)
on the whole original polydisks, so the induction needs no shrinking of edge domains.
(2) Subdivide so each gamma(I_i) lies in a product chart with inverse theta_i: D^r1 x D^r2 -> X. On that interval write gamma(s) = theta_i(a_i(s), b_i(s)). Define the diagonal tile by R(s, t) = theta_i(a_i(s), b_i(t)) for (s, t) in I_i^2. These tiles agree at diagonal vertices and their edges have full-polydisk representatives.
Fill the remaining tiles I_i x I_j in increasing |i - j|. If i < j use the filled bottom and right edges; if i > j use the top and left edges. Polydisk axes works from any corner because its origins are arbitrary. The two input edges have matching corner values: for |i - j| = 1 this is the prescribed diagonal value; for |i - j| > 1 it follows from the intervening tile closer to the diagonal. Tiles filled at the same stage share no edge; shared vertices lie on prescribed input edges. Every tile is consistent and its output edges retain full-polydisk representatives.
In either construction the finite tile collection glues continuously. Box behaviour holds across seams: in a product chart near a seam point, the unwanted coordinate is constant on consecutive slice pieces and agrees at common endpoints; the resulting map is a continuous box.
Intrinsic leaves. Fix a basepoint o in X. For i = 1, 2 let L_i be the leaf through o of the foliation determined by E_i, with its intrinsic plaque topology in which plaque paths are continuous (not the subspace topology from X). Each L_i is connected, Hausdorff, second-countable, and a complex manifold of dimension r_i; its inclusion into X is an injective holomorphic immersion. It admits a connected simply connected cover L_i~, also a second-countable complex manifold, and any path that locally follows E_i-plaques is continuous in the intrinsic topology of its leaf.
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 coincide and their coordinate change is holomorphic. The inclusion is an injective continuous immersion; distinct points are separated by inverse images of disjoint ambient open sets, so the leaf is Hausdorff. Path connectedness is by construction; plaque paths are intrinsically continuous.
Second countability. Take a countable ambient product atlas. Start with a plaque through o. For each reached plaque and each ambient chart the intersection has at most countably many components (open subset of a second-countable plaque), and each component lies in a single plaque of the new chart because the transverse coordinate is locally constant. Adjoin these plaques and repeat. Only countably many plaques appear at finite stages, and every plaque path from o is covered by a finite chain of them. 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. For second countability of the universal cover, refine the countable atlas to simply connected coordinate disks; there are countably many components of pairwise overlaps; a based loop is described up to homotopy by a finite sequence of these charts and components; the fundamental group is countable; the lifted countable chart basis has only countably many sheets and is a countable basis for the universal cover. Lifting the holomorphic charts gives the complex structure.
A product chart at o supplies mutually inverse germs
H_o: (L1 x L2, (o, o)) -> (X, o), K_o: (X, o) -> (L1 x L2, (o, o)).
The first combines coordinates on the two plaques through o; the second projects onto them. Separated chart transitions make these germs independent of the chosen product chart.
Transport of inverse germs. Let R: [0, 1]^2 -> X be a continuous box with R(0, 0) = o, and set alpha(r) = R(r, 0) in L1, beta(r) = R(0, r) in L2, gamma(r) = R(r, r) in X. There are mutually inverse local biholomorphism germs H_r, K_r at (alpha(r), beta(r)) and gamma(r) respectively, giving analytic continuation of H_o along (alpha, beta) and of K_o along gamma.
Proof. Transport a first-coordinate germ from the L1-plaque at alpha(r) along the vertical path u -> R(r, r u), 0 <= u <= 1. Subdivide into finitely many subsegments lying in product charts and pick 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, because separated transitions are independent of the second coordinate; it is therefore constant along the connected subsegment, and common refinements identify any two constructions. Similarly, transport a second-coordinate germ from the L2-plaque at beta(r) along u -> R(r u, r). Combining the two transported coordinates in one product chart at gamma(r) gives the local biholomorphism germ H_r; let K_r be its inverse.
Genuine analytic continuation. Fix r0. Choose finite chart chains for the two reference transport paths, a fixed small overlap neighbourhood at each junction, and a common final chart z around gamma(r0). Joint continuity of R keeps the same compact subsegments in the same charts and the same junctions in the same overlap neighbourhoods for all r near r0. Also alpha(r), beta(r) remain in fixed intrinsic leaf plaques.
Use coordinates a and b on the two starting plaques. Every first-coordinate transition in the vertical chain is one fixed holomorphic map independent of the second coordinate; after shrinking the initial coordinate neighbourhood, their finite composition is a single local biholomorphism A near a(alpha(r0)) that describes the transport for every nearby r. The horizontal chain similarly gives one fixed second-coordinate local biholomorphism B. Then one holomorphic map
(x, y) -> z^{-1}(A(a(x)), B(b(y)))
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 nearby K_r. At r = 0 both transport paths are constant, so these germs are precisely H_o and K_o.
Proof of the main theorem. Every path from (o, o) in L1 x L2 consists of two intrinsically continuous leaf paths. Path rectangles give a box with these as axes, and transport of inverse germs continues H_o along that path. Conversely every path from o in X is the diagonal of a box, so the same statements continue K_o along every such path.
Let pi: Xtilde -> X and q_i: Y_i = L_i~ -> L_i be the based connected simply connected covers, and set q = q1 x q2. The product Y1 x Y2 is the universal cover of L1 x L2. The monodromy theorem, applied on these simply connected source covers, gives holomorphic maps
H: Y1 x Y2 -> X, K: Xtilde -> L1 x L2
with the prescribed initial germs. Uniqueness follows from the identity theorem; finite chains of local representatives are stable under small path variations, so homotopic paths with fixed endpoints give the same terminal germ.
Both source manifolds are simply connected, so these maps lift to holomorphic maps
Htilde: Y1 x Y2 -> Xtilde, Ktilde: Xtilde -> Y1 x Y2.
Choose lifts that send the based points to one another. Their germs there are mutually inverse because H_o and K_o were; hence Htilde o Ktilde and Ktilde o Htilde agree with the identity near their basepoints. The identity theorem on connected sources makes both compositions identities everywhere. Thus Phi := Ktilde is a biholomorphism Xtilde ~ Y1 x Y2.
Retaining the splitting. Near the basepoint the initial product chart gives
dPhi(pi^* E1) in pr1^* T_Y1, dPhi(pi^* E2) in pr2^* T_Y2.
The two wrong-factor components of dPhi vanish there; being holomorphic bundle maps, the components vanish on all of the connected Xtilde. Since dPhi is invertible and dim_C Y_i = r_i by the intrinsic-leaves lemma, the inclusions are equalities. The factors Y_i are connected and simply connected by construction.
Nef and big canonical bundle. X is a smooth connected projective complex manifold of dimension n >= 2 with K_X nef and big, and T_X = E1 (+) E2 a holomorphic splitting into positive-rank subbundles. Integrability is not assumed; it is proved.
Canonical contraction. There is a projective birational morphism f: X -> Z with connected fibers onto a normal projective variety Z with canonical singularities and ample Q-Cartier canonical divisor; K_X = f^* K_Z, and f is an isomorphism over Z_reg.
./kollar-mori-1998-birational-geometry-of-algebraic-varieties.pdf
Proof. The base-point-free theorem applies to D = K_X: D is nef and a D - K_X = (a - 1) K_X is nef and big for any integer a > 1 (Kollar-Mori 1998, Theorem 3.3). The associated contraction with Stein factorization has normal projective target and connected fibers. For some positive integer m and ample line bundle L on Z,
O_X(m K_X) = f^* L.
Bigness makes f generically finite; connected fibers 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 that open set L agrees with O_Z(m K_Z); reflexive extension identifies L with O_Z(m K_Z) everywhere. Hence K_Z is Q-Cartier and ample.
Choose compatible canonical divisors. The divisor K_X - f^* K_Z is exceptional and Q-linearly trivial. A principal exceptional divisor is zero: the corresponding rational function has no divisor on Z and is constant. Hence K_X = f^* K_Z. Discrepancies on any higher resolution are nonnegative because X is smooth, so Z has canonical (in particular klt) singularities.
Over Z_reg the relative Jacobian has zero divisor; it is nowhere zero and f is a local biholomorphism there. Properness and birationality make this restriction an isomorphism.
Set X^0 = f^{-1}(Z_reg). This is a connected dense open subset of X, identified with Z_reg. Fix Ric(omega) = -sqrt(-1) partial dbar log omega^n, so [Ric(omega)] = 2 pi c1(X).
Canonical metric. Fix a Kaehler form omegaA from a polarization of X. There are smooth Kaehler metrics g_eps on X, 0 < eps < 1, with forms
Ric(omega_eps) = -omega_eps + eps omegaA,
[omega_eps] = 2 pi c1(K_X) + eps [omegaA].
As eps -> 0 the metrics converge smoothly on compact subsets of X^0 to a Kaehler metric g with Ric(g) = -g; the completion of the intrinsic Riemannian distance of (X^0, g) is canonically homeomorphic to Z.
./song-2014-riemannian-geometry-of-kaehler-einstein-currents.pdf
Proof. The canonical contraction makes f a smooth projective crepant resolution of a canonical model. Completion is Song 2014, Theorem 1.2; approximating metrics and smooth convergence are Song 2014, Equation (4.28) and Lemma 4.1, with eps = e^{-t}. The cohomology identity follows from the Ricci equation.
Parallelism and integrability. The connection of the limiting Kaehler metric g preserves both E1 restricted to X^0 and E2 restricted to X^0. Both subbundles are integrable on all of X. The subbundles admit simultaneous local holomorphic product coordinates.
./guenancia-2016-semistability-of-the-tangent-sheaf-of-singular-varieties.pdf
"His strategy is to use approximate Kaehler-Einstein metric on a resolution X'. By considering them as approximate Hermite-Einstein metrics on Tx', he obtains an approximate slope inequality."
Guenancia 2016, Section 3.4, describing Enoki 1988.
Proof. Give T_X the Hermitian metric induced by g_eps. Let P_(j, eps) be the orthogonal projection onto E_j and B_(j, eps) its second fundamental form. Orthogonal projection need not be projection along the complementary summand, and E1, E2 are not assumed orthogonal.
Subbundle curvature formula. Write F_V for the Chern curvature of a Hermitian bundle V. For the subbundle E_j with the induced metric,
tr( sqrt(-1) Lambda_(omega_eps) F_(E_j) )
= tr( P_(j, eps) sqrt(-1) Lambda_(omega_eps) F_(T_X) ) - |B_(j, eps)|^2.
(The second fundamental forms have their complex Hermitian norms.) In the Kaehler case the mean-curvature endomorphism sqrt(-1) Lambda_(omega_eps) F_(T_X) is the Ricci endomorphism
H_eps = -I + eps A_eps, A_eps = omega_eps^{-1} omegaA > 0.
The formula comes from writing the connection in blocks for E_j (+) E_j^perp, with off-diagonal blocks B_(j, eps) and -B_(j, eps)^*.
First-Chern-class cancellation. Integrate the trace formula over X. Since c1(E1) + c1(E2) = c1(T_X) and dV_eps = omega_eps^n / n!,
int_X sum_j |B_(j, eps)|^2 dV_eps = int_X tr(C_eps H_eps) dV_eps,
where C_eps = P_(1, eps) + P_(2, eps) - I. The ranks add to n, so tr(C_eps) = 0, and -I <= C_eps <= I because each P_(j, eps) is an orthogonal projection. Positivity of A_eps gives
0 <= int_X sum_j |B_(j, eps)|^2 dV_eps
<= eps int_X tr(A_eps) dV_eps
= eps / (n - 1)! int_X omegaA ^ omega_eps^(n - 1) = O(eps).
The last integral is a bounded polynomial in eps because [omega_eps] = 2 pi c1(K_X) + eps [omegaA]. No angle bound between the summands is used.
Vanishing of the limiting second fundamental forms. On compact subsets of X^0, metrics, connections, orthogonal projections, and second fundamental forms converge smoothly (canonical-metric lemma). The mass bound says int_X sum_j |B_(j, eps)|^2 dV_eps -> 0. The integrand is nonnegative and converges pointwise to sum_j |B_j|^2 on compacta of X^0, so the limit vanishes identically there; the limiting B_j are zero on X^0.
Integrability. The connection of a Kaehler metric is torsion-free, so on X^0 each subbundle is integrable. The Frobenius obstruction
Lambda^2 E_j -> T_X / E_j, u ^ v -> [u, v] mod E_j,
is a holomorphic bundle morphism on all of X. It vanishes on the dense X^0, hence everywhere; E1 and E2 are integrable on X. Local Frobenius submersions for the two complementary foliations give holomorphic coordinates (z, w) as in the product-charts lemma; transitions preserve both factors.
Corollary. Let X be a smooth connected projective complex manifold of dimension n >= 2 with K_X nef and big. Every holomorphic splitting T_X = E1 (+) E2 into positive-rank subbundles is integrable and induces a compatible product decomposition of the ordinary universal cover as in the main theorem.
Proof. A smooth projective complex manifold is compact Kaehler. Parallelism and integrability make both summands integrable, so the main theorem gives the compatible product.
The canonical metric enters only to establish integrability. Once both summands are integrable, the main theorem closes the argument; this is the shortest path from the Kaehler-Einstein structure to compatible splitting in the nef-and-big case.
Alternative continuation via collapsed fibres. The nef-and-big corollary also has a proof that avoids the Hartogs extension and instead uses the canonical metric and rational chains on the canonical fibres. The mechanism is separate nonexpansion of ordinary crossing near an exceptional fibre, together with a finite-grid construction that controls box size by the canonical pseudodistance. Throughout this part X satisfies the nef-and-big hypotheses, f: X -> Z is the canonical contraction, and g is the limiting Kaehler-Einstein metric on X^0 = f^{-1}(Z_reg).
Pseudodistance. There is a real-analytic Kaehler metric h on X^0, uniformly equivalent to g, for which E1 (+) E2 is orthogonal and locally a Riemannian product. Its intrinsic distance completion is canonically homeomorphic to Z. With rho the resulting distance on Z,
d(x, x') = rho(f(x), f(x'))
is a continuous pseudodistance on X, and d(x, x') = 0 if and only if f(x) = f(x').
./src-deturckkazdan1981.pdf
Proof. Let Q_j: T_X -> E_j be projection along the other supplied summand and set h(v, w) = g(Q1 v, Q1 w) + g(Q2 v, Q2 w). Parallelism of g with respect to E1 and E2 gives grad^g Q_j = 0, hence grad^g h = 0; grad^g is the Levi-Civita connection of h. It preserves J, so h is Kaehler. The positive g-self-adjoint endomorphism representing h is parallel; its eigenvalues are constant on the connected X^0, giving global positive a, b with a g <= h <= b g.
In local product coordinates first- and second-factor real coordinate fields V, W commute. The equality grad_V W = grad_W V together with preservation of the complementary summands makes both sides zero. Since grad h = 0, the first block of h depends only on first coordinates and the second only on second coordinates. Real analyticity of a Kaehler-Einstein metric (DeTurck-Kazdan 1981, Theorem 6.1(c)) and holomorphicity of the Q_j give real analyticity of h.
Uniform equivalence makes the identity on X^0 bi-Lipschitz for intrinsic distances and extends it to a bi-Lipschitz homeomorphism of completions. The canonical-metric lemma identifies that completion with Z, extending the given identification on X^0. The assertions about d follow from continuity of f and rho being a distance.
Fibre topology. For every z in Z the reduced fibre P = f^{-1}(z)_red is a compact connected projective algebraic set. Any two points of P are joined inside P by a chain of images of rational curves, and pi_1(P) = 1 in the ordinary complex topology.
./hacon-mckernan-2007-on-shokurov-rational-connectedness-conjecture.pdf
./takayama-2003-local-simple-connectedness-of-resolutions.pdf
Proof. Projectivity and connectedness follow from the construction of f. Rational chain connectedness of the fibre of a resolution of a klt variety is Hacon-McKernan 2007, Corollary 1.6. Simple connectedness is Takayama 2003, Theorem 1.1 and the first paragraph on p. 829: the fibre of a proper birational morphism from a smooth complex variety to a klt variety is simply connected in the ordinary topology. X is smooth and Z is klt, so the theorem applies to f: X -> Z; replacing the fibre by its reduction does not change its topological space.
Rational chains supply continuation routes; ordinary simple connectedness removes their monodromy. The two play separate roles in the argument.
Rational boxes. Let M be a smooth projective complex manifold with a holomorphic splitting T_M = E1 (+) E2 into integrable subbundles. Let F_i be the corresponding regular foliations. Frobenius submersions for the two foliations give holomorphic product charts z = (z1, z2) in which E_i is the i-th coordinate distribution. Changes between such charts have, locally, the form
(z1, z2) -> (F(z1), G(z2)).
The local crossing kappa(x, y) = z^{-1}(z1(x), z2(y)) is independent, as a germ along the diagonal, of the product chart.
Definition. Let A, B be locally path connected. A continuous R: A x B -> M is a box if, near every parameter pair, there are a product neighbourhood A' x B' and a product chart on its image with
z(R(a, b)) = (r1(a), r2(b)).
First-parameter slices locally follow F1 plaques and second-parameter slices locally follow F2 plaques; the slices are continuous into their leaves with the leaf topology. A rational tree is a connected nodal curve with every irreducible component a copy of P^1 and dual graph a finite tree.
For continuous maps on products of rational trees, holomorphic on each product of components, the box condition follows from separate tangencies: in a product chart the wrong-coordinate function is constant on every branch of its parameter slice; continuity makes these constants agree at a node.
Holonomy. A continuous plaque-wise path in an F2 leaf transports first transverse coordinates by composing the first-coordinate changes in a finite chain of product charts, giving a biholomorphic germ between first transversals. Reversing the path gives the inverse germ. The germ is unchanged under a leafwise homotopy with fixed endpoints (subdivide the compact homotopy square into pieces mapping into foliation charts, and cancel transverse changes around their boundaries). The same holds with the two foliations interchanged.
Germs supplied by a box. Let T be a rational tree or an interval, R: T x T -> M a box, u(s) = R(s, s). There is a coherent family of holomorphic map germs
c_(s, t): (M x M, (u(s), u(t))) -> (M, R(s, t))
agreeing with ordinary crossing when s = t and satisfying
d_1 c_(s, t)(T_M) in E1, d_1 c_(s, t)(E2) = 0,
d_2 c_(s, t)(T_M) in E2, d_2 c_(s, t)(E1) = 0.
Coherence: near each parameter pair, one ambient holomorphic map represents all the nearby germs. The family is unique among coherent families agreeing with ordinary crossing on the diagonal.
Proof. Fix s, t and a path gamma in T from s to t. Set p = u(s), q = u(t), r = R(s, t), and choose product charts z, zeta, w at p, q, r. The path R(s, gamma) runs from p to r in an F2 leaf; its holonomy gives a first-coordinate germ h1: z1 -> w1. Similarly, R(gamma^{-1}, t) gives a second-coordinate germ h2: zeta2 -> w2. Define
c_(s, t)(x, y) = w^{-1}(h1(z1(x)), h2(zeta2(y))).
Chart changes cancel and the differential relations follow. 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 the path.
Coherence. Choose finite chains of product charts along R(s, gamma) and R(gamma^{-1}, t) realising h1, h2. For s' near s and t' near t, continuity on these compact paths lets 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 neighbourhoods; the resulting endpoint segments lie in the endpoint product charts, so introduce no additional transverse-coordinate change. The same holomorphic representative of c represents c_(s', t'). At a node one uses a small star neighbourhood and connectors through the node. When s = t the constant path gives ordinary crossing. Coherence and uniqueness of analytic continuation of ambient holomorphic germs close the argument.
Extension of a rational box. Put S = P^1 x P^1. A holomorphic box on a neighbourhood of the diagonal, or of the union
D = (P^1 x {t0}) union ({s0} x P^1),
extends uniquely to a holomorphic box on S.
./merker-porten-2009-hartogs-extension-on-n-1-complete-complex-spaces.pdf
./serre-1956-gaga.pdf
Proof. Each divisor has class (1, 1), a hyperplane section of the Segre embedding of S; its complement V is smooth affine, hence Stein. Shrink the given neighbourhood to a connected neighbourhood W of the divisor. Removing a complex hypersurface does not disconnect W, so W minus D is connected. The set K = S minus W is compact in V, and V minus K = W minus D. Meromorphic Hartogs extension on Stein surfaces (Merker-Porten 2009, Theorem 2.2) extends meromorphic functions from V minus 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 values on W; the equations of embedded M hold on W, hence identically, so the resulting meromorphic map F: S -> M is defined. Slice tangencies persist on the holomorphic locus by the identity theorem. The meromorphic graph is a closed analytic subset of S x M, so by Chow's theorem (Serre 1956, Proposition 13) F is rational. Clearing the poles of finitely many projective coordinate ratios expresses F by a finite-dimensional linear system on S; remove the fixed divisorial part and resolve the base ideal by point blowups. Successive blowups of remaining basepoints drop the local intersection number of two general members by the product of multiplicities. The finite sum of these intersection numbers forces termination. This gives mu: Shat -> S and a holomorphic H: Shat -> M.
Every exceptional curve is contracted by H. At a generic point of an exceptional component C over (s0, t0) choose coordinates (x, y) with C = {x = 0}. For local product coordinates s, t downstairs write
s o mu - s0 = x^a U(x, y), t o mu - t0 = x^b V(x, y),
a, b > 0, U, V units. The holomorphic vector fields
W_s = -x U_y partial_x + (a U + x U_x) partial_y,
W_t = -x V_y partial_x + (b V + x V_x) partial_y
annihilate d(s o mu) and d(t o mu). Off the exceptional locus, slice tangency gives d H(W_s) in E2 and d H(W_t) in E1; these extend holomorphically to C. On C both vector fields are nonzero multiples of partial_y, so
d H(T_C) in E1 cap E2 = 0.
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 neighbourhood downstairs whose inverse image maps into the chart. Its coordinate functions descend holomorphically because mu_* O_Shat = O_S, equivalently by removal of the isolated singularity downstairs. Hence F is holomorphic. Slice tangencies give the box condition; uniqueness follows from the identity theorem.
Rational boxes exist. For every holomorphic u: T -> M from a rational tree there is a unique continuous box R_u: T x T -> M with R_u(s, s) = u(s) and holomorphic on every product of irreducible components. If a rational tree T' contains T as a subtree and u': T' -> M extends u, then R_(u') restricted to T x T equals R_u, and the ambient germ families agree on this restriction.
Proof. Take T = P^1. Local crossing applied to u(s), u(t) gives coherent holomorphic germs along the diagonal of S; compactness lets them be glued on a neighbourhood of the diagonal (take a finite family of representatives, shrink their domains, shrink around the diagonal so overlapping representatives agree). The extension-of-a-rational-box lemma 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, and uniqueness on S follows.
Axes construction. Suppose alpha: P^1 -> M is tangent to E1, beta: P^1 -> M is tangent to E2, and alpha(s0) = beta(t0). Transport the germ of beta at t0 along alpha using F1 holonomy. A loop in the parameter sphere contracts and its image is a leafwise homotopy, so there is no path ambiguity. In product charts the resulting map near (s, t0) 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 a finite-chart argument. The symmetric construction along the other axis plus gluing across the axes (both maps are ordinary crossing near their common corner) gives a box on a neighbourhood of the axes. The extension-of-a-rational-box lemma extends the box over S. The two prescribed axes determine its germ at (s0, t0); the extension is unique. This includes constant curves.
General tree. Construct the box on the square of each component first. Proceed through ordered pairs of distinct components by increasing distance in the dual tree. On C_i x 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, are tangent to E1, E2 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 glue continuously, and their separate tangencies give the box condition at nodes.
Uniqueness comes from the same induction, first on component squares and then from the inward axes; restriction to a subtree is R_u. To compare ambient germs, compute the two holonomies along paths lying in T; paths and box values are unchanged in T', so the germ formula gives the same germs.
Neighborhood crossing. Let P in M be a nonempty connected projective algebraic set, simply connected, and joined by rational chains. There are a connected open neighbourhood U of P and a holomorphic map c: U x U -> M that agrees near the diagonal with ordinary crossing in product charts and satisfies throughout U x U:
c(x, x) = x,
d_1 c(T_M) in E1, d_1 c(E2) = 0,
d_2 c(T_M) in E2, d_2 c(E1) = 0,
with the image subbundles evaluated at c(x, y).
Proof. Put Q = P x P, fix p in P, and write q* = (p, p). All germs are holomorphic maps from M x M to M.
An attained germ over (a, b) in Q is the crossing germ of the germs-supplied-by-a-box statement, associated to a rational-tree box R_u at two distinct smooth marked points s, t with u(s) = a and u(t) = b. The ordinary crossing at q* is attained using a constant sphere. Nothing yet identifies germs arising from different trees.
Tree-path lemma. Let k: E -> Q be holomorphic from a rational tree with distinct smooth markings m1, m2. Every attained germ at k(m1) continues along the image of any path from m1 to m2; the terminal germ is attained and independent of the path in D.
Proof. Choose a realising tree T with markings s, t. Attach two separate copies of E 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 attachment points. Its box restricts to the original, including ambient germs, by uniqueness and branch compatibility. Move the two parameter points along the copies of a path in E; the germs-supplied-by-a-box statement gives exactly the asserted continuation along its image under k. The terminal markings are smooth and distinct, so the terminal germ is attained. All paths between markings are homotopic in the simply connected tree; leafwise holonomy is unchanged.
Stability of continuation. Suppose a germ continues along a continuous path gamma: [0, 1] -> N in a complex manifold N. There are a representative of its terminal germ and a uniform neighbourhood of gamma in the space of paths starting at gamma(0) such that continuation along every path in that neighbourhood exists and ends in that representative.
Proof. Choose a finite subdivision and holomorphic representatives on open neighbourhoods of the compact subpaths. Consecutive representatives agree on an open neighbourhood of each subdivision point. After shrinking the subpath neighbourhoods and these overlaps, every uniformly close path stays in the same domains and crosses through the same equality overlaps. The same finite chain works.
Continuation along a fixed path from a fixed germ is unique: two continuation chains that agree initially agree locally at the first possible separation point, by the identity theorem in overlapping ambient neighbourhoods. A path followed by its reverse returns the initial germ.
Compact family of rational routes. Embed Q in P^N. Use the standard properties of genus-zero stable maps: fixed-degree spaces with three markings have projective coarse moduli spaces; evaluations are algebraic; maps factoring through a fixed projective subscheme form a closed locus; and locally the projective-space moduli problem has finite-quotient charts with universal families of nodal curves (Fulton-Pandharipande 1997, Theorem 1, Proposition 3, and Sections 1.3, 4.1, 4.4, 5.1). Local families use the ambient target P^N, which is convex, and restrict to the locus of maps into Q.
./fulton-pandharipande-1997-notes-on-stable-maps-and-quantum-cohomology.pdf
Parameter space. There is a compact semialgebraic space S* of three-pointed rational-tree maps into Q whose first evaluation is q* and whose second evaluation e: S* -> Q is surjective.
Proof. For each degree d >= 0 take the closed locus S_d in the moduli space of genus-zero three-marked stable maps of degree d into P^N, of maps with values in Q and 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, 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; stabilize. This does not change the endpoint evaluations.
Write Q = Q1 union ... union Qr for the irreducible components. For each i,
Qi = union_{d >= 0} (Qi cap F_d).
If none of the F_d contained Qi, this would express the compact complex variety Qi as a countable union of closed nowhere dense sets, contradicting Baire category. Choose d_i with Qi in F_{d_i}. The disjoint union of the finitely many S_{d_i}, omitting repetitions, is the required S*.
Routes in a degenerating tree. Fix a stable-map parameter in S* and a path between its first two markings chosen along smooth component arcs and through the intervening nodes. For every sufficiently nearby parameter one can choose a path between its corresponding markings whose image in Q, after parametrisation on [0, 1], is uniformly as close as desired to the fixed image path.
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 nodes, finitely many relative smooth charts move the chosen arcs to nearby fibers with prescribed marked ends. At a node the family is locally xy = a, where a depends on the parameter; the incoming and outgoing nearby pieces can be joined inside an arbitrarily small plumbing neighbourhood when a is small, and likewise when a = 0. The family map is continuous, so all images in that neighbourhood lie close to the limiting nodal value. Parametrise the joining pieces on the corresponding small subintervals; the resulting paths have uniformly close images. Contracted components cause no difficulty because their limiting images are constant. Restricting the parameter locus to maps into Q keeps all these image paths inside Q.
Semialgebraic lift. Every continuous semialgebraic path sigma: [0, 1] -> Q has a finite subdivision such that on each closed subinterval it admits a continuous lift to S*. Lifts on consecutive subintervals need not agree at their common endpoint.
./coste-1999-an-introduction-to-o-minimal-geometry.pdf
./bochnak-coste-roy-1998-real-algebraic-geometry.pdf
Proof. Semialgebraic choice applied to
{(t, s) in [0, 1] x S* : e(s) = sigma(t)} -> [0, 1]
gives a semialgebraic section 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 them lift the correct endpoint values. Extend each interval lift by its one-sided limits (Coste 1999, Theorems 2.1, 3.1, 3.4; see also Bochnak-Coste-Roy 1998).
All-paths lemma. Every attained germ continues along every continuous semialgebraic path in Q from its basepoint, and the terminal germ is attained.
Proof. Use the semialgebraic lift 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 the tree-path lemma. For each t in [a, b] continue this fixed germ h forwards through the tree over t; 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 t0, the nearby-routes lemma gives uniformly close image routes from q*; stability of continuation supplies one ambient representative giving all g_t near t0. Thus the germs are analytic continuation along sigma on the whole interval, and each is attained by the tree-path lemma. Repeat from the terminal germ on the next interval. The intermediate germ h at q* need not be the original crossing; only its being fixed on the current interval is used.
Conclusion. The compact family of routes supplies continuation along every semialgebraic path, including through singularities and changes of component. Triangulate Q semialgebraically (Coste 1999, Theorem 4.4). P is simply connected, so Q is simply connected. 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 the all-paths lemma 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 and uniform continuity of the homotopy show that nearby slices have the same terminal germ; the terminal germ is then constant throughout the homotopy. Well-defined ambient germs exist at every point of Q.
Local coherence. Choose a small relative path-connected neighbourhood of an endpoint inside the domain of its representative. Appending small paths, piecewise linear in the fixed semialgebraic triangulation after subdivision, and lying in the neighbourhood, shows that the same representative gives the assigned germs there. Such neighbourhoods are available from the triangulation.
Representative on an actual neighbourhood of Q. Choose finitely many representatives c_i: V_i -> M giving the family on Q cap V_i, and smaller open sets W_i with compact closures in V_i 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 relatively closed. The compact sets
B_(i, j) = {x in closure(W_i) cap closure(W_j) : (c_i)_x != (c_j)_x}
are disjoint from Q. Removing their finite union and restricting to union_i W_i gives a neighbourhood of Q on which the representatives glue holomorphically. Equality here is equality of ambient germs, not only values on Q.
Compactness of P gives an open neighbourhood U of P with square in this domain. Because P is connected, take the connected component of U containing it. The glued map restricts to c: U x U -> M. Its differential identities hold near (p, p) and hence throughout connected U x U by holomorphicity. The diagonal identity extends throughout connected U. In a product chart about any x in U the differential identities make the first output coordinate depend only on the first coordinate of the first input, and the second only on the second coordinate of the second input; the diagonal identity identifies both functions with the corresponding coordinates. So c is ordinary crossing near every diagonal point.
Nonexpansion and finite grids. Return to the canonical contraction f: X -> Z. Put X^0 = f^{-1}(Z_reg) and let h be the orthogonal product metric from the pseudodistance construction. Established properties of h: real analytic, locally a product metric for E1 (+) E2, and its intrinsic length-metric completion canonically homeomorphic to Z. Write rho for the resulting distance on Z and d(x, x') = rho(f(x), f(x')). Then d is a continuous pseudodistance with zero-distance classes the fibers of f.
For each reduced fiber P = f^{-1}(z)_red, neighbourhood crossing gives a connected open neighbourhood U containing P and a holomorphic crossing map c: U x U -> X with the differential identities. Near a diagonal point c is the ordinary first-and-second factor extraction.
Nonexpansion lemma. For each fiber value z in Z choose a crossing map from the surrounding construction and r > 0 such that f^{-1}(B_rho(z, r)) is in U and
d(c(a, b), c(a', b)) <= d(a, a'),
d(c(a, b), c(a, b')) <= d(b, b'),
whenever the inputs in the relevant inequality belong to f^{-1}(B_rho(z, r/10)).
Proof. On
V = {(a, b) in U x U : a, b, c(a, b) in X^0}
the tensor identities
||d_1 c(v)||_{h, c(a, b)}^2 = ||v1||_{h, a}^2,
||d_2 c(w)||_{h, c(a, b)}^2 = ||w2||_{h, b}^2
hold for v = v1 + v2 and w = w1 + w2 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 x U, so V is connected; both sides of each identity are real analytic on V; the real-analytic identity principle proves the identities throughout V.
Properness of f and the topology induced by rho give r > 0 with f^{-1}(B_rho(z, r)) in U. For the first inequality, approximate (a, a', b) in X cubed by triples (a_m, a'_m, b_m) with all three inputs and both output points in X^0. Such triples are dense: each excluded condition is a proper analytic subset of U cubed, and properness of the output conditions follows from 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 (d on X^0 is its intrinsic length distance). Every point of the path lies over B_rho(z, r/3 + eta): initial point within r/9 of z, path length less than 2 r / 9 + eta. The path stays in U with a margin for perturbation.
For this fixed b_m the set
A_m = {x in U cap X^0 : c(x, b_m) not in X^0}
is a proper analytic subset (output at a_m is regular), hence real codimension at least two. Keeping endpoints fixed, perturb the path to avoid A_m with arbitrarily small length error: on a neighbourhood of the compact path the metric is smooth; general position relative to an analytic stratification gives such a perturbation. Along the perturbed path the tensor identities bound output speed by input speed, and integration followed by letting length errors tend to zero gives
d(c(a_m, b_m), c(a'_m, b_m)) <= d(a_m, a'_m).
Continuity of c and d gives the first inequality; the second follows by interchanging variables.
Compactness of Z supplies a uniform delta > 0 with the property: every set S in X with diam_d(S) <= 2 delta lies in a crossing domain U, and the Lipschitz inequalities hold for all inputs from S. Take a finite subcover of Z by the smaller balls in the nonexpansion lemma and choose 2 delta strictly below a Lebesgue number.
One-tile rule. Suppose two prescribed edges alpha: I -> X, beta: J -> X meet at a corner p, follow the first and second foliations respectively, and each have diameter at most delta. Their union has diameter at most 2 delta; choose a crossing map by the uniform bound and set R(s, t) = c(alpha(s), beta(t)). This is a box and agrees with both prescribed edges: for fixed s, varying the second input along the part of alpha joining p to alpha(s) is constant by d_2 c(E1) = 0, so c(alpha(s), p) = c(alpha(s), alpha(s)) = alpha(s); the other edge follows from d_1 c(E2) = 0. The argument applies to continuous plaque paths because the corresponding functions are locally constant on parameter intervals. The Lipschitz inequalities show every horizontal or vertical partial slice of this tile has diameter at most delta, so the same bound is available for its two new edges.
Boxes exist. (1) For every continuous path lambda: [0, 1] -> X there is a box R: [0, 1]^2 -> X with R(s, s) = lambda(s). (2) For continuous paths alpha, beta in leaves of the first and second foliations with alpha(0) = beta(0) there is a box on [0, 1]^2 with R(s, 0) = alpha(s), R(0, t) = beta(t). Continuity of prescribed leaf paths is in the intrinsic leaf topology.
Proof. (1) Subdivide so each lambda(I_i) has diameter at most delta. On each diagonal tile put R(s, t) = c_i(lambda(s), lambda(t)); diagonal is lambda and slices have diameter at most delta. Tiles agree at common vertices. Fill remaining tiles I_i x I_j in increasing |i - j|: above the diagonal the bottom and right edges are defined; below it, the top and left. In either case the two known edges are one horizontal and one vertical edge meeting at a corner. Apply the one-tile rule. The induction invariant: every partial slice of each completed tile has diameter at most delta; the tile rule gives agreement on old edges and preserves the invariant. Tiles at the same stage share no edge; a shared vertex is already fixed by an intervening tile from an earlier stage. The finite filling is consistent.
(2) Subdivide each prescribed axis so every edge segment has diameter at most delta. Fill the rectangular array in increasing order of the sum of the two indices; the bottom and left edges of a new tile are prescribed. The same rule and invariant apply.
In both constructions tile maps glue continuously. The box property holds across seams: near a seam value, continuity places adjacent path pieces in one foliation chart, and both follow the plaque through their common endpoint.
The construction uses only continuity and diameter control. In particular it applies to paths inside a collapsed fiber, whose pseudodiameter is zero, without assigning those paths any finite length for the limiting metric.
Product of universal covers, alternative route. Fix o in X and give the leaves L1, L2 through o their intrinsic complex-manifold structures. Let H_o: (L1 x L2, (o, o)) -> (X, o), K_o: (X, o) -> (L1 x L2, (o, o)) be inverse local product germs. Every path in X from o is the diagonal of a box by the diagonal-box construction; its axes lie continuously in the fixed intrinsic leaves, so transport of inverse germs continues K_o along that path. Conversely a path in L1 x L2 consists of two intrinsically continuous leaf paths; the axes-box construction supplies a box with those axes; transport of inverse germs continues H_o along the prescribed path. All continued germs are local biholomorphisms.
Under the nef-and-big hypotheses let Xtilde -> X be the universal cover and L_itilde -> L_i the universal covers of the leaves through o. Then Xtilde is biholomorphic to L_1tilde x L_2tilde, with the lifts of E_i identified with the respective tangent bundles of the factors.
Proof. All-path continuation and monodromy give holomorphic maps
F: Xtilde -> L1 x L2, G: L_1tilde x L_2tilde -> X.
With compatible basepoints, the covering lifting theorem gives holomorphic lifts
Ftilde: Xtilde -> L_1tilde x L_2tilde, Gtilde: L_1tilde x L_2tilde -> Xtilde.
Their initial germs are inverse. Each composite is the identity near its basepoint, hence everywhere on its connected source by the identity theorem. Wrong-factor components of their differentials are holomorphic bundle morphisms that vanish in the initial product chart, so the components vanish identically; this is the compatibility with the supplied splitting.
The alternative route uses separate nonexpansion of crossings around collapsed canonical fibres together with finite-grid diameter bounds. Its mechanism differs from the Hartogs-extension argument, but the conclusion is the same.