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), 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]; see also [21]. Here \(H\) is a domain in the Stein manifold \(\mathbb C^2\), 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 \[ P_2\mathrm dF(\partial_s)=0, \qquad P_1\mathrm dF(\partial_t)=0 \quad\hbox{on }U. \tag{3}\] 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]. 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 \[ \Gamma\cap\{s=s_0\} \tag{4}\] 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 (4), 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 (4), since that set has dimension at most one. Choose a general point \(q\) of the curve piece such that the reduced set (4) 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 \[(s,h)^{-1}\bigl(s_0,h(q)\bigr)\subset\Gamma.\] 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 \[\Psi=(s,h)\] 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 \[K_j=\Gamma\cap\overline B\cap\Psi^{-1}\bigl(\Psi(q_j)\bigr)\] 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) 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) 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 \[M_j=\max_{\overline D}d(z)\left\lVert\mathrm dh_j(z)\right\rVert.\] 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 \(a_j=\left\lVert\mathrm dh_j(z_j)\right\rVert\) and \(d_j=d(z_j)\), so \(a_jd_j=M_j\). The maps \[k_j(\zeta)=h_j(z_j+\zeta/a_j)\] are defined for \(|\zeta|<\varepsilon/4\). On this disk, \(d(z_j+\zeta/a_j)\ge d_j/2\), and maximality gives \[\left\lVert\mathrm dk_j(\zeta)\right\rVert\le2, \qquad \left\lVert\mathrm dk_j(0)\right\rVert=1.\] 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 (1).

Write \(p=(s_*,t_*)\). Choose complex-affine coordinates \((u,v)\) with \[ (s,t)=(s_*,t_*)+(a_1,a_2)u+(b_1,b_2)v, \qquad b_1b_2\ne0, \tag{5}\] 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.

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 \[B_m(R)=\{(s,t):|s|^{2m}+|t|^{2m}<R^{2m}\},\qquad 0<R\leq1.\] 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 8 therefore extends the box across that point. At a boundary point \((s_*,0)\), choose a disk \(U\) around \(s_*\), with \(\overline U\subset\mathbb D\), 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 \[G(s,t)=z^{-1}\bigl(z_1(\alpha(s)),z_2(F(s',t))\bigr), \qquad (s,t)\in U\times\mathbb D_\epsilon,\] is defined and is a box. Here \(\mathbb D_\epsilon=\{|t|<\epsilon\}\). The product range of the chart makes the formula well-defined. Local separation shows that \(G=F\) near \((s',0)\). Their common domain \((U\times\mathbb D_\epsilon)\cap B_m(R_*)\) 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 \(\mathbb D^2\), and the resulting boxes agree on their connected overlaps. Their union is the required box. The initial uniqueness observation proves uniqueness on \(\mathbb D^2\). ◻

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 \[\sigma\longmapsto\alpha(\sigma p),\qquad \tau\longmapsto\beta(\tau q)\] are holomorphic on disks of radii greater than one. After rescaling, Proposition 9 gives a box whose restriction to the closed real square is denoted by \(R_{p,q}\). Set \[F(p,q)=R_{p,q}(1,1).\] The constant-axis cases and Lemma 10 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 \[ z^{-1}\bigl(z_1(B(\sigma;p,q)),z_2(L(\tau;p,q))\bigr). \tag{10}\] 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 \[\partial_q\bigl(z_1\circ B\bigr)=0, \qquad \partial_p\bigl(z_2\circ L\bigr)=0.\] 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 10, 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 \[\alpha(s)=A_i(a_i(s)),\qquad s\in I_i,\] where \(a_i:I_i\to\mathbb D^{r_1}\) is continuous and \(A_i:\mathbb D^{r_1}\to X\) is a holomorphic plaque parametrization tangent to \(E_1\). Similarly write \(\beta(t)=B_j(b_j(t))\), with \(b_j:J_j\to\mathbb D^{r_2}\) continuous and \(B_j\) holomorphic and tangent to \(E_2\). Here \(r_i=\mathop{\mathrm{rank}}E_i\).

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 \(\mathbb D^{r_1}\) and \(\mathbb D^{r_2}\), tangent to the respective bundles. They agree at their common corner. Apply Proposition 11 with origins \(a_i(u_{i-1})\) and \(b_j(v_{j-1})\), obtaining a holomorphic box \(G:\mathbb D^{r_1}\times\mathbb D^{r_2}\to X\). Its evaluation \[R(s,t)=G(a_i(s),b_j(t))\] fills the tile. The opposite horizontal and vertical edges have the holomorphic representatives \[a\longmapsto G(a,b_j(v_j)),\qquad b\longmapsto G(a_i(u_i),b),\] 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 \(\theta_i:\mathbb D^{r_1}\times\mathbb D^{r_2}\to X\). Write \(\gamma(s)=\theta_i(a_i(s),b_i(s))\) on that interval. Define the diagonal tile by \[R(s,t)=\theta_i(a_i(s),b_i(t)),\qquad (s,t)\in I_i^2.\] 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 11 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 11: 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 13.

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 \[ (x,y)\longmapsto z^{-1}\bigl(A(a(x)),B(b(y))\bigr) \tag{11}\] 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. Every path from \((o,o)\) in \(L_1\times L_2\) consists of two intrinsically continuous leaf paths. Proposition 12 provides a box with these as its axes, and Proposition 14 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 \[H:Y_1\times Y_2\longrightarrow X, \qquad K:\widetilde X\longrightarrow L_1\times L_2\] 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 \[\widetilde H:Y_1\times Y_2\longrightarrow\widetilde X, \qquad \widetilde K:\widetilde X\longrightarrow Y_1\times Y_2.\] 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 \[\mathrm d\Phi(\pi^*E_1)\subset\mathop{\mathrm{pr}}_1^*T_{Y_1},\qquad \mathrm d\Phi(\pi^*E_2)\subset\mathop{\mathrm{pr}}_2^*T_{Y_2}.\] Equivalently, the two wrong-factor components of \(\mathrm d\Phi\) vanish there. They are holomorphic bundle maps, so they vanish on all of the connected manifold \(\widetilde X\). Finally, \(\mathrm d\Phi\) is invertible and \(\dim_\mathbb CY_i=r_i\) by Lemma 13; 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]. 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 \(Z\), \[\mathcal O_X(mK_X)\simeq f^*L.\] 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 15 makes \(f\) a smooth projective crepant resolution of a canonical model. The completion statement is [29]. The approximating metrics and their smooth convergence are [29], 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]. 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 \[\mathop{\mathrm{tr}}\!\left(\sqrt{-1}\Lambda_{\omega_\varepsilon}F_{E_j}\right) = \mathop{\mathrm{tr}}\!\left( P_{j,\varepsilon}\sqrt{-1}\Lambda_{\omega_\varepsilon}F_{T_X} \right)-|B_{j,\varepsilon}|^2.\] 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 \[H_\varepsilon=-I+\varepsilon A_\varepsilon, \qquad A_\varepsilon=\omega_\varepsilon^{-1}\omega_A>0.\]

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 \[\int_X\sum_{j=1}^2|B_{j,\varepsilon}|^2\,dV_\varepsilon = \int_X\mathop{\mathrm{tr}}(C_\varepsilon H_\varepsilon)\,dV_\varepsilon, \qquad C_\varepsilon=P_{1,\varepsilon}+P_{2,\varepsilon}-I.\] The ranks add to \(n\), so \(\mathop{\mathrm{tr}}C_\varepsilon=0\). Moreover \(-I\leq C_\varepsilon\leq I\), since each \(P_{j,\varepsilon}\) is an orthogonal projection. Positivity of \(A_\varepsilon\) therefore gives \[\begin{align*} 0 &\leq\int_X\sum_j|B_{j,\varepsilon}|^2\,dV_\varepsilon \leq\varepsilon\int_X\mathop{\mathrm{tr}}(A_\varepsilon)\,dV_\varepsilon \\ &=\frac{\varepsilon}{(n-1)!} \int_X\omega_A\wedge\omega_\varepsilon^{n-1} =O(\varepsilon). \tag{12}\end{align*}\] The last integral is a bounded polynomial in \(\varepsilon\), by Lemma 16. 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 (12). 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 \[\Lambda^2E_j\longrightarrow T_X/E_j,\qquad u\wedge v\longmapsto [u,v]\bmod E_j,\] 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 3. A smooth projective complex manifold is compact Kähler. By Proposition 17, the supplied summands are integrable. Theorem 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 \[h(v,w)=g(Q_1v,Q_1w)+g(Q_2v,Q_2w).\] Proposition 17 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 \[ag\leq h\leq bg.\] 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]. 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 16 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].

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]. This consequence is also used explicitly in [25]. Here \(X\) is smooth and \(Z\) is klt by Proposition 15, 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 \[ c_{s,t}(x,y)=w^{-1}\bigl(h_1(z_1(x)),h_2(\zeta_2(y))\bigr). \tag{16}\] Chart changes cancel in this formula, and (15) 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 (16) 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 (14). 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], 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]. 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 \[s\circ\mu-s_0=x^a U(x,y),\qquad t\circ\mu-t_0=x^b V(x,y),\] where \(a,b>0\) and \(U,V\) are units. The holomorphic vector fields \[W_s=-xU_y\partial_x+(aU+xU_x)\partial_y, \qquad W_t=-xV_y\partial_x+(bV+xV_x)\partial_y\] annihilate \(\mathrm d(s\circ\mu)\) and \(\mathrm d(t\circ\mu)\), respectively. Off the exceptional locus, slice tangency gives \[\mathrm dH(W_s)\in E_2,\qquad \mathrm dH(W_t)\in E_1.\] These inclusions extend holomorphically to \(C\). There both vector fields are nonzero multiples of \(\partial_y\), whence \[\mathrm dH(T_C)\subset E_1\cap E_2=0.\] 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 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 \(T=\mathbb P^1\). 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 23 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 \(\alpha:\mathbb P^1\to M\) is tangent to \(E_1\), \(\beta:\mathbb P^1\to M\) 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 23 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 (16) gives precisely the same germs. ◻