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\title{Universal-cover splitting\\ for compact K\"ahler manifolds}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
Let the holomorphic tangent bundle of a compact connected K\"ahler manifold split into two positive-rank holomorphic subbundles, each closed under Lie bracket. Then the ordinary universal cover of the manifold is a product, and the two factor tangent bundles are the lifts of the two given summands. This is the two-summand form of Beauville's compatible splitting conjecture. The proof uses a Hartogs-type extension for maps adapted to the two foliations, a boundary-crossing estimate driven by a K\"ahler form, and a finite-grid assembly that turns these local facts into a biholomorphism of universal covers. We also show that when the canonical bundle is nef and big, every such splitting is automatically integrable.
\end{abstract}

\section{The question}

Let $X$ be a connected complex manifold, and let $T_X$ be its \emph{holomorphic tangent bundle}: over each point $x \in X$ sits the vector space of holomorphic tangent directions at $x$. If $X$ is itself a product $X = Y_1 \times Y_2$, then a tangent direction on $X$ is the same as a pair (a direction on $Y_1$, a direction on $Y_2$). Bundle-theoretically,
\[
  T_X \;=\; \pr_1^* T_{Y_1} \,\oplus\, \pr_2^* T_{Y_2},
\]
where $\pr_i$ is the projection onto the $i$-th factor.

\begin{definition}
A \emph{holomorphic splitting} of $T_X$ is a decomposition
$T_X = E_1 \oplus E_2$ into holomorphic subbundles of positive rank.
A subbundle $E \subset T_X$ is \emph{integrable} (or \emph{involutive}) if the Lie bracket of any two local holomorphic sections of $E$ is again a section of $E$.
\end{definition}

Beauville's starting question is the converse of the product observation above:

\begin{quote}
\itshape
Suppose we are given a specific holomorphic splitting $T_X = E_1 \oplus E_2$ into positive-rank subbundles. When is it induced, on the ordinary universal cover, by a product structure on $X$?
\end{quote}

The problem is not vacuous even as a local statement. Integrability is the right analytic condition. Holomorphic Frobenius's theorem (see Section~\ref{sec:prelim}) says that if both $E_1$ and $E_2$ are integrable, then every point of $X$ has a neighbourhood $\D^{r_1} \times \D^{r_2}$ in which $E_1$ and $E_2$ are the two coordinate directions. Consequently, a holomorphic splitting with both summands integrable \emph{always} has a local product structure. The hard question is whether these local pieces fit together globally.

The answer is not obviously ``yes.'' Two geometric obstructions can appear. First, the local product charts on different parts of $X$ may be glued by holonomies that fail to close up globally: the leaf space of $E_1$ may be non-Hausdorff, the holonomy of one foliation along leaves of the other may be nontrivial. Second, even if $X$ is itself a product, a \emph{given} splitting need not be compatible with \emph{that} product structure. Both obstructions are real; the first is where the compact-K\"ahler hypothesis and the analytic estimates enter, the second is what Beauville's example rules out.

\subsection*{A warning: product does not imply compatible}

Take $X = A \times \Pp^1$, where $A$ is a complex abelian surface and $\Pp^1$ is the Riemann sphere. On the abelian surface, choose two independent translation-invariant vector fields $U, V$; these have $[U, V] = 0$. On the sphere, choose two vector fields $S, T$ with $[S, T] \neq 0$. On the product, define two rank-one subbundles by
\[
  E_1 \;=\; \mathrm{rank}\text{-}2 \text{ span of } \{U + S,\; V + T\}, \qquad E_2 \;=\; T_{\Pp^1}.
\]
The bundle $E_1$ has rank two at each point and complements $E_2$. But $E_1$ is \emph{not} integrable. Indeed,
\[
  [\,U + S,\; V + T\,] \;=\; [U,V] + [S,T] \;=\; [S,T] \;\in\; T_{\Pp^1},
\]
and $[S, T] \neq 0$. The bracket of two $E_1$-sections reaches into $E_2$, so $E_1$ is not closed under Lie bracket.

The moral: knowing that $X$ is a product is not the same as knowing that a given splitting comes from that product. Integrability of the specific subbundles has to be an explicit hypothesis, not a consequence of ambient product structure.

\subsection*{What we prove}

The main theorem says that on a compact K\"ahler manifold, integrability of both summands \emph{is enough} to make the local product structure global, up to replacing $X$ by its ordinary universal cover.

\begin{theorem}[Main theorem]\label{thm:main}
Let $X$ be a compact connected K\"ahler manifold of complex dimension $n \geq 2$, and let
\[
  T_X \;=\; E_1 \oplus E_2
\]
be a specified holomorphic splitting into integrable subbundles of positive ranks $r_1, r_2$. Let $\pi : \widetilde X \to X$ be the ordinary universal covering map. Then there are connected simply connected complex manifolds $Y_1, Y_2$ and a biholomorphism
\[
  \Phi : \widetilde X \xrightarrow{\;\sim\;} Y_1 \times Y_2
\]
such that
\[
  \dd\Phi(\pi^* E_i) \;=\; \pr_i^* T_{Y_i} \qquad (i = 1, 2).
\]
In particular, $\dim_\C Y_i = r_i$.
\end{theorem}

Here $\dd\pi$ identifies $T_{\widetilde X}$ with $\pi^* T_X$, so $\pi^* E_i$ is a holomorphic subbundle of $T_{\widetilde X}$. The important content of the equation $\dd\Phi(\pi^* E_i) = \pr_i^* T_{Y_i}$ is that the two \emph{prescribed} subbundles become the two coordinate directions on the product. It is more than the existence of some product structure on $\widetilde X$.

\begin{remark}
The conclusion is about the ordinary universal cover. It does not claim that any finite cover of $X$ is a product. The two factors $Y_1, Y_2$ the theorem produces are the universal covers of the two foliation leaves through a chosen basepoint; in particular, they can be noncompact. If one summand has rank zero, the corresponding factor is a point, so the positive-rank hypothesis merely excludes the empty case.
\end{remark}

\subsection*{The nef-and-big application}

Theorem~\ref{thm:main} has a clean corollary when the ambient manifold is projective and its canonical bundle is positive in the sense of the minimal model program. For a smooth projective variety $X$, the canonical line bundle $K_X$ is \emph{nef} if it has nonnegative degree on every curve, and \emph{big} if its sections grow at the maximal asymptotic rate. ``$K_X$ nef and big'' is the hypothesis of a canonical contraction $f : X \to Z$, and it implies a number of strong structures: $K_X$ is semiample, $Z$ has canonical singularities, $K_X = f^* K_Z$, and the smooth locus of $Z$ pulls back to a dense open subset $X^0 \subset X$ over which the limiting K\"ahler--Einstein metric is smooth.

\begin{corollary}[Nef-and-big corollary]\label{thm:nef-big}
Let $X$ be a smooth connected projective complex manifold of dimension $n \geq 2$ with $K_X$ nef and big. Every specified holomorphic splitting $T_X = E_1 \oplus E_2$ into positive-rank subbundles is integrable and induces a compatible product decomposition of the ordinary universal cover as in Theorem~\ref{thm:main}.
\end{corollary}

There is an earlier route to this corollary through H\"oring's integrability theorem \cite{Horing07}, but the paper gives a direct analytic proof, and we follow that route. The key point is that a splitting whose summands have complementary curvature integrates to a parallel splitting in the presence of a canonical metric, and the second fundamental forms of the summands are then forced to vanish by a first-Chern-class identity. This is the content of Section~\ref{sec:canonical}.

A second, independent proof of Corollary~\ref{thm:nef-big} is known, via the geometry of collapsed canonical fibres and rational chains; it uses a different continuation mechanism from the present one. This paper follows the Hartogs-type analytic-continuation route throughout, because it is self-contained and does not need the extra machinery of collapsed fibres. We mention the alternative only for context.

\begin{remark}[\quad]
\upshape
The compact K\"ahler condition enters Theorem~\ref{thm:main} at exactly two places: it makes a K\"ahler form available for the boundary-crossing estimate of Section~\ref{sec:boundary}, and it puts a Hermitian metric on $T_X$ whose projections $P_i$ onto $E_i$ have a uniform operator-norm bound. The second is closed under complexification and therefore no simplification is gained by assuming projective; the first uses closedness of the form $\omega$, not its metric positivity after that. See the discussion at the end of Section~\ref{sec:boundary}.
\end{remark}


\section{Product charts and boxes}\label{sec:prelim}

Throughout this section and the next, $X$ is a compact connected K\"ahler manifold of complex dimension $n \geq 2$, and $T_X = E_1 \oplus E_2$ is a fixed holomorphic splitting into integrable subbundles of ranks $r_1, r_2$. Let $\D = \{z \in \C : |z| < 1\}$ be the open unit disk.

Fix a K\"ahler form $\omega$ on $X$ and its associated Hermitian metric
\[
  g(a, b) \;=\; \omega(a, Jb)
\]
on the real tangent bundle, where $J$ is the complex structure. We identify the real tangent bundle with the holomorphic tangent bundle as complex vector bundles, so multiplication by $i$ on real tangent vectors means applying $J$. The projections $P_i$ onto $E_i$ are then complex-linear endomorphisms of the real tangent bundle. No orthogonality of $E_1$ and $E_2$ is assumed: the metric $g$ need not make them perpendicular. Compactness of $X$ does, however, give a uniform bound
\begin{equation}\label{eq:proj-bound}
  \sup_{x \in X} \; \norm{P_i(x)} \;<\; \infty \qquad (i = 1, 2),
\end{equation}
because the projection $P_i$ varies continuously over the compact base.

The integrability hypothesis has a very concrete local meaning. At each $x \in X$, the two summands $E_1(x)$ and $E_2(x)$ are transversal complex subspaces of $T_X(x)$. By holomorphic Frobenius (applied separately to $E_1$ and to $E_2$) there are holomorphic coordinates in which $E_1$ is the first coordinate direction and $E_2$ is the second.

\begin{lemma}[Product charts]\label{lem:product-charts}
Every point of $X$ has a holomorphic coordinate neighbourhood with coordinates $z = (z_1, z_2)$ and range $\D^{r_1} \times \D^{r_2}$ in which $E_i$ is tangent to the $z_i$-directions. Between two such charts, the transition map has, locally, the separated form
\begin{equation}\label{eq:separated-transition}
  z_1' \;=\; h_1(z_1), \qquad z_2' \;=\; h_2(z_2),
\end{equation}
with each $h_i$ a local biholomorphism.
\end{lemma}

\begin{proof}
Fix a point $x$, and choose a local holomorphic coordinate projection $\pi : V \to \C^{r_1}$ whose differential is nonsingular on the distribution $E_1$ near $x$. Lift the coordinate vector fields on the base to vector fields in $E_1$. Integrability of $E_1$ says the Lie brackets of these lifts remain sections of $E_1$, hence project to zero in the base. The projection is an isomorphism from $E_1$ to the tangent of the base, so those brackets must actually be zero. The commuting flows of the lifted fields give, near $x$, holomorphic coordinates $(z_1, z_2)$ with $E_1 = \ker(\dd z_2)$; this is the Frobenius coordinate theorem. Applying the same construction to $E_2$ and combining the two coordinate systems gives the desired pair, after restricting to a product of polydiscs. A chart change preserves both distributions, so the two off-diagonal blocks of its differential vanish on small product coordinate neighbourhoods. Because the chart change is invertible, the two diagonal blocks are themselves invertible.
\end{proof}

We call these \emph{product charts}. A \emph{first plaque} is a slice of a product chart with $z_2$ fixed (i.e., a small piece of an $E_1$-leaf); a \emph{second plaque} is a slice with $z_1$ fixed. A continuous path is a \emph{first} (or \emph{second}) \emph{plaque path} if each of its points has a neighbourhood on which the path lies in a first (respectively second) plaque. Transitions are required to be separated only \emph{locally}: we do not assume a single formula over a disconnected chart overlap.

\begin{definition}[Holomorphic box]\label{def:box}
Let $U \subset \C^a \times \C^b$ be open, with parameter blocks written $(s, t)$. A holomorphic map $f : U \to X$ is a \emph{holomorphic box} if, at every point of $U$,
\[
  \dd f(T\C^a \oplus 0) \subset E_1, \qquad \dd f(0 \oplus T\C^b) \subset E_2.
\]
The conditions require tangency only; $f$ need not be an immersion or even a local embedding.
\end{definition}

The definition is the cleanest way to state that $f$ respects the two foliations separately. In a product chart, the box condition reads $f(s,t) = (f_1(s), f_2(t))$ locally, and conversely this local separation implies the two tangencies. The scalar case $a = b = 1$ will be the first we handle; its parameters $s$ and $t$ retain their prescribed directions even when we later describe a domain using a different affine coordinate system.

All uses of the identity theorem below are for holomorphic maps or holomorphic bundle maps on connected domains. In particular, tangency persists under holomorphic extension: if the wrong-summand projections of the differential vanish on an open set, they vanish throughout the connected extension domain.

\subsection*{Conventions for fundamental groups and covers}

Throughout, ``universal cover'' and ``fundamental group'' mean the ordinary topological versions, not the algebraic-geometric \'etale versions. Leaves of the two foliations always carry the \emph{intrinsic} manifold topology in which plaque paths are continuous, not the subspace topology inherited from $X$. A path in a leaf means a continuous path in this intrinsic topology; a continuous path in $X$ that happens to lie in a single leaf need not be intrinsically continuous if the leaf is very badly embedded.


\section{Hartogs extension for boxes}\label{sec:hartogs}

The first real analytic input is a Hartogs-type extension theorem for boxes. The statement is local in the source, but uses the compact K\"ahler structure of the target: we have no ordinary mean of extending holomorphic \emph{maps} into $X$, because $X$ is not Stein. The mechanism is due to Ivashkovich (1992). A holomorphic map from a Hartogs figure into a compact K\"ahler manifold always extends meromorphically to the envelope of holomorphy, but the meromorphic extension may fail to be holomorphic on a proper analytic subset. Those obstructions are \emph{exceptional fibres of the graph}: curves mapping into a single point of the target. For a box, such a fibre is impossible: the two tangencies force the graph to avoid precisely the configuration that makes exceptional fibres possible.

\begin{proposition}[Hartogs extension for boxes]\label{prop:hartogs}
Let $(s,t)$ be the standard coordinates of $\C^2$, and let $(w,v)$ be an invertible complex-affine coordinate system (the affine coordinates defining the Hartogs figure need not be $(s,t)$ themselves). For real numbers $0 < e < a$ and $0 < c < b$, put
\[
  H \;=\; \{|w| < e,\ |v| < b\} \;\cup\; \{|w| < a,\ c < |v| < b\},
  \qquad
  Q \;=\; \{|w| < a,\ |v| < b\}.
\]
Every holomorphic box $f: H \to X$ with the tangencies $\dd f(\partial_s) \in E_1$, $\dd f(\partial_t) \in E_2$ extends uniquely to a holomorphic box $Q \to X$, and the extension satisfies the same tangencies.
\end{proposition}

\begin{proof}
We build the extension in four steps.
\\[2pt]
\emph{Step 1. Meromorphic extension.}
Scalar holomorphic functions on $H$ extend holomorphically to $Q$: expanding a function of the shell $\{c < |v| < b\}$ in a Laurent series in $v$, the coefficients of the negative powers are holomorphic functions of $w$ that vanish on the disk $|w| < e$ (because the shell meets that disk in the lower part of $H$), and by analytic continuation they vanish on $|w| < a$; Cauchy estimates on each intermediate circle $c < |v| < b$ give normal convergence on compact subsets of $Q$. Hence the envelope of holomorphy of $H$ is $Q$. Ivashkovich's meromorphic extension theorem (see \cite[Theorem 2.2]{Ivashkovich2008}, and also the original paper \cite{Ivashkovich1992}) applies to any meromorphic map from a domain in a Stein manifold into a compact K\"ahler manifold and extends the given map meromorphically to the envelope. Here $H \subset \C^2$ is a domain in a Stein manifold and $f$ is holomorphic, so $f$ extends meromorphically to $Q$.
\\[2pt]
\emph{Step 2. Setup for the graph argument.}
Let $\Gamma \subset Q \times X$ be the meromorphic graph, with the reduced analytic structure, and let $\rho : \Gamma \to Q$ be the projection. Then $\Gamma$ is a proper analytic subset of $Q \times X$ (so $\rho$ is proper), and there is a proper analytic subset $A \subset Q$, disjoint from the interior of $H$, such that over $U = Q \setminus A$ the graph $\Gamma$ is the ordinary graph of a holomorphic map $F : U \to X$. The ordinary graph is dense in $\Gamma$, and $U$ is connected because the complement of a proper analytic subset of a connected complex manifold is connected. Applying the identity theorem to the two holomorphic bundle-valued expressions $P_2 \dd F(\partial_s)$ and $P_1 \dd F(\partial_t)$ yields
\begin{equation}\label{eq:meromorphic-box-tangency}
  P_2 \dd F(\partial_s) \;=\; 0, \qquad P_1 \dd F(\partial_t) \;=\; 0 \qquad \text{on } U,
\end{equation}
where $P_i : T_X \to E_i$ are the projections determined by the given splitting.
\\[2pt]
\emph{Step 3. No positive-dimensional fibre of $\rho$.}
At each local irreducible surface branch of $\Gamma$, both source coordinates $s$ and $t$ are nonconstant: choosing 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 $Q$. It follows that for every $s_0$, the analytic set
\[
  \Gamma \cap \{s = s_0\}
\]
has local dimension at most one; and symmetrically for $t$.

Suppose some fibre over $(s_0, t_0)$ has positive dimension. Because the fibre is contained in $\{s = s_0\}$ and that reduced set has dimension at most one, the fibre contains a curve component $C$. Take a small piece of $C$ inside 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 curve piece, which lies in $\{(s_0, t_0)\} \times X$, would be a point.

Assume first that $h$ is a component of $z_1$. The curve piece is a component of the reduced set $\Gamma \cap \{s = s_0\}$, so its smooth points carry dimension exactly one. Choose a general point $q$ where the reduced set is smooth and $h$ is nonconstant. Then $(s, h)$ has an \emph{isolated} zero at $q$: locally the only point of $\Gamma$ with $(s, h) = (s_0, h(q))$ is $q$ itself. Work in ambient complex coordinates on $Q \times X$ near $q$, and choose a closed Euclidean ball $\overline B$ centered at $q$ inside these coordinates, small enough that $\Gamma$ meets the $(s,h)$-level of $\Psi(q) = (s_0, h(q))$ only at $q$. The image under $\Psi$ of $\Gamma \cap \partial B$ is compact and misses $\Psi(q)$. Choose points $q_j$ of the ordinary graph tending to $q$; for all sufficiently large $j$, the value $\Psi(q_j)$ misses the compact $\Psi$-image of $\Gamma \cap \partial B$. Therefore
\[
  K_j \;=\; \Gamma \cap \overline B \cap \Psi^{-1}\bigl(\Psi(q_j)\bigr)
\]
is a compact analytic subset of the \emph{open} ball $B$ (no points on $\partial B$). But near $q_j$, on the ordinary graph, equation~\eqref{eq:meromorphic-box-tangency} 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 set in Euclidean complex space has no positive-dimensional component (maximize the strictly plurisubharmonic squared coordinate norm on the component, and restrict to an analytic curve germ through the maximum point).

If instead $h$ is a component of $z_2$, the same argument uses $\Psi = (t, h)$ and varies $s$. In either case the given positive-dimensional fibre is impossible, so every fibre of $\rho$ has dimension zero. Because $\rho$ is proper, each fibre is finite.
\\[2pt]
\emph{Step 4. Singleton fibres, so a holomorphic extension.}
Fix $p \in Q$, and write its fibre as $\{(p, x_1), \dots, (p, x_k)\}$. Choose pairwise disjoint target neighborhoods $V_1, \dots, V_k$ with pairwise disjoint closures, and $x_i \in V_i$. Properness gives a small base ball $D$ about $p$ such that every point of $\rho^{-1}(D)$ has its target coordinate inside $\bigcup_i V_i$: otherwise a sequence of graph points over points tending to $p$, outside that union, would converge to a point of the central fibre, contradicting the definition of the $V_i$. The ordinary locus $D \setminus A$ is connected, so its image under $F$ sits in a \emph{single} $V_i$. Density of the ordinary graph then places every $x_j$ in $\overline V_i$; the disjoint closures force $k = 1$.

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, the image of $\widehat F$ lies in a target coordinate chart, so its coordinate functions are continuous and holomorphic off the proper analytic subset $A$. The removable-singularity theorem for locally bounded holomorphic functions across analytic subsets of a complex manifold makes them holomorphic everywhere. Hence $\widehat F$ is holomorphic, and the tangencies in~\eqref{eq:meromorphic-box-tangency} extend to all of $Q$ by the identity theorem. Uniqueness follows from the identity theorem on the connected bidisk $Q$.
\end{proof}

Two features of this proposition deserve emphasis. First, \emph{the holomorphic extension is obtained without any rank condition on the box}: $f$ need not be an immersion; it can have deficient rank. The argument only uses the separate tangencies. Second, \emph{the Hartogs figure need not be aligned with the box coordinates}: $(w, v)$ can be any invertible complex-affine change of $(s, t)$, so a box defined by tangency to $E_1, E_2$ can be extended across a Hartogs figure aligned with arbitrary affine coordinates. This is what will allow the boundary-crossing proposition of the next section to build the shell in an arbitrary affine frame.


\section{Extending a box across a mixed boundary}\label{sec:boundary}

The second analytic input extends a holomorphic box across a \emph{smooth boundary point whose complex tangent has both of its components nonzero}. Write the source locally as an open subset of $\C^2_{s,t}$. At a boundary point $p$, if the complex tangent line to $\partial B$ at $p$ has a vector with both components nonzero, then the local geometry is a ``mixed'' one: the boundary cuts across both directions $s$ and $t$. The extension requires detecting that the boundary point is close enough to the box's interior that a small shell can be built. The K\"ahler form makes this work: because it is closed, it controls the areas of shrinking slices, and the areas are too small at scales much larger than the surviving distance to the boundary, so the slices collapse to a point and a Hartogs shell becomes available.

The underlying compactness fact is a small-area lemma in the spirit of Brody's reparametrization argument.

\begin{lemma}[Small area]\label{lem:small-area}
Let $M$ be a compact Hermitian manifold with associated positive $(1,1)$-form $\omega_M$, and let $U \subset \C$ be a domain. Suppose $h_j : U \to M$ are holomorphic and
\[
  \int_U h_j^* \omega_M \longrightarrow 0.
\]
Then $\norm{\dd h_j}$ tends to zero uniformly on every compact subset of $U$. Here the source carries the Euclidean metric, and closedness of $\omega_M$ is \emph{not} required.
\end{lemma}

\begin{proof}
It suffices to work on a disk $D$ with $\overline D \subset U$. Put $d(z) = \operatorname{dist}(z, \partial D)$ and
\[
  M_j \;=\; \max_{z \in \overline D} \; d(z) \, \norm{\dd h_j(z)}.
\]
If $M_j \not\to 0$, pass to a subsequence with $M_j \geq \varepsilon > 0$ for some fixed $\varepsilon$. Let $z_j \in D$ be a maximizing point and put $a_j = \norm{\dd h_j(z_j)}$, $d_j = d(z_j)$, so $a_j d_j = M_j$. The maps
\[
  k_j(\zeta) \;=\; h_j(z_j + \zeta / a_j)
\]
are defined for $|\zeta| < \varepsilon / 4$. On this disk, $d(z_j + \zeta / a_j) \geq d_j / 2$, and maximality gives
\[
  \norm{\dd k_j(\zeta)} \leq 2, \qquad \norm{\dd k_j(0)} = 1.
\]
After a subsequence, $k_j(0)$ converges to a point of $M$. The derivative bound then places the images of a fixed smaller disk inside a relatively compact coordinate neighbourhood of that point, so 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 yet smaller disk. This contradicts the vanishing areas of $k_j$: those areas are bounded by the areas of $h_j$ and are unchanged by the affine reparametrization.

Hence $M_j \to 0$. 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 $U$.
\end{proof}

\begin{proposition}[Mixed-boundary extension]\label{prop:mixed}
Let $B \subset \C^2_{s,t}$ be a convex open set, and let $f : B \to X$ be 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 two components in the $s, t$ directions are both nonzero, then $f$ extends as a holomorphic box to a neighbourhood of $p$, agreeing with $f$ on its intersection with $B$.
\end{proposition}

\begin{proof}
Use the fixed K\"ahler form $\omega$, its Hermitian norm, and the projection bound~\eqref{eq:proj-bound}. Write $p = (s_*, t_*)$.
\\[2pt]
\emph{Affine coordinates at $p$.} Choose complex-affine coordinates $(u,v)$ with
\begin{equation}\label{eq:boundary-affine}
  (s,t) \;=\; (s_*, t_*) + (a_1, a_2)\, u + (b_1, b_2)\, v, \qquad b_1 b_2 \neq 0,
\end{equation}
where the $v$-direction is the complex tangent and the positive real $u$-direction points outward from $B$. All constants below may depend on $p$ and these coordinates, but not on any subsequent parameter.
\\[2pt]
\emph{Shrinking slices and bounded area.} Take a local defining function $\rho$ with $B = \{\rho < 0\}$ near $p$. For real $x$ and small $v$, Taylor's theorem gives
\[
  \rho(x, v) \;\leq\; c x + M(x^2 + |v|^2), \qquad c > 0,
\]
because both real derivatives of $\rho$ vanish in the complex tangent direction. Choose $k > 0$ small enough that for all $x < 0$ close to zero, the closed disks
\[
  D_x \;=\; \{u = x,\ |v| \leq r(x)\}, \qquad r(x) = k \sqrt{-x},
\]
are compactly contained in $B$.
\\[2pt]
Let $e_u = \partial_{\operatorname{Re} u}$ and $e_v = \partial_{\operatorname{Re} v}$. The box tangencies and complex linearity give the exact identity
\[
  \dd f(e_u) \;=\; \sum_{i=1}^2 \frac{a_i}{b_i} \, P_i \dd f(e_v).
\]
Consequently
\begin{equation}\label{eq:boundary-speed}
  \norm{\dd f(e_u)} \;\leq\; C\, \norm{\dd f(e_v)}, \qquad C = \sum_{i=1}^2 \left| \frac{a_i}{b_i} \right| \, \sup_X \norm{P_i}.
\end{equation}
Holomorphicity makes $\norm{\dd f(e_v)}$ equal to the speed in every unit real direction of the $v$-plane, including the radial direction.
\\[2pt]
\emph{Bounded-area estimate.} Set $\Omega = f^* \omega$ and
\[
  A(x) \;=\; \int_{D_x} \Omega.
\]
Because $\Omega$ is closed and $D_x$ depends smoothly on $x$, Stokes' theorem applied to the moving disks gives
\[
  A'(x) \;=\; \int_0^{2\pi}
    \Omega\bigl(e_u + r'(x) \partial_r,\, \partial_\theta\bigr)\big|_{u=x,\ v = r(x) e^{i\theta}} \, \dd\theta,
\]
where the boundary velocity is $e_u + r'(x) \partial_r$ and the circles are taken counterclockwise. On a boundary circle put $q = \norm{\dd f(e_v)}$. Since $\partial_\theta = r J \partial_r$,
\[
  \Omega(\partial_r, \partial_\theta) \;=\; r q^2, \qquad
  |\Omega(e_u, \partial_\theta)| \;\leq\; C r q^2.
\]
Hence
\[
  A'(x) \;\leq\; \int_0^{2\pi} r(x)\bigl(C + r'(x)\bigr) q^2 \, \dd\theta.
\]
Here $r'(x) = -k/(2\sqrt{-x})$, so choose $x_0 < 0$ sufficiently close to zero that $r'(x) \leq -C$ for $x_0 \leq x < 0$. We obtain the uniform bound
\begin{equation}\label{eq:boundary-area}
  0 \;\leq\; A(x) \;\leq\; A(x_0) \;<\; \infty \qquad (x_0 \leq x < 0).
\end{equation}
No product property of the K\"ahler metric has been used; only closedness.
\\[2pt]
\emph{A small-image annulus.} Let $x_j = -\delta_j$ with $\delta_j \downarrow 0$. Among the scales $\lambda = 5^\ell \delta_j^{3/4}$ with integer $\ell \geq 0$, all lying 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 as $j \to \infty$. For large $j$ they all fit inside $D_{x_j}$. By the uniform bound~\eqref{eq:boundary-area}, one such annulus, with a scale we call $\lambda_j$, has area tending to zero. The chosen scales satisfy the two inequalities
\begin{equation}\label{eq:boundary-scales}
  \frac{\delta_j}{\lambda_j} \;\leq\; \delta_j^{1/4} \longrightarrow 0, \qquad
  \frac{\lambda_j}{\sqrt{\delta_j}} \;\leq\; \delta_j^{1/8} \longrightarrow 0.
\end{equation}
Apply Lemma~\ref{lem:small-area} to $h_j(\zeta) = f(x_j, \lambda_j \zeta)$ on the annulus $1/2 < |\zeta| < 2$. The derivatives tend uniformly to zero on a neighbourhood of $3/4 \leq |\zeta| \leq 3/2$. This compact annulus has bounded intrinsic Euclidean diameter, so the image of
\[
  u = x_j, \qquad 3\lambda_j/4 < |v| < 3\lambda_j/2
\]
has diameter tending to zero. A finite cover of $X$ by product charts has a positive Lebesgue number; for large $j$ this entire image therefore lies inside a single product chart $z : V \to V_1 \times V_2$. Write the box's coordinates in that chart, restricted to the slice $u = x_j$, as $(g_1(v), g_2(v))$.
\\[2pt]
\emph{The Hartogs shell.} Put $w = u - x_j$ and $c_i = a_i/b_i$. Choose $\eta > 0$ independent of $j$, so that $\eta \max(1, |c_1|, |c_2|) < 1/8$. On the shell
\[
  S_j \;=\; \{|w| < \eta \lambda_j,\ \lambda_j < |v| < 5\lambda_j/4\}
\]
define
\begin{equation}\label{eq:boundary-shell}
  F_j(w, v) \;=\; z^{-1}\bigl(g_1(v + c_1 w),\; g_2(v + c_2 w)\bigr).
\end{equation}
Both shifted arguments stay in the annulus $3\lambda_j/4 < |v| < 3\lambda_j/2$, so $F_j$ is well defined. The range of $z$ is a product, so combining the two valid factor coordinates in~\eqref{eq:boundary-shell} needs no additional target-chart margin. Moreover,
\[
  v + c_1 w \;=\; \frac{s - s_* - a_1 x_j}{b_1}, \qquad
  v + c_2 w \;=\; \frac{t - t_* - a_2 x_j}{b_2},
\]
so $F_j$ is a box in the original $(s,t)$ directions.
\\[2pt]
The new map agrees with $f$ on an open germ at every point of $S_j$ with $w = 0$. To see this, continuity puts $f$ into the same chart $V$ on a neighbourhood of that point. There $f$'s separated expression is $(\varphi_1(s), \varphi_2(t))$. Restriction to the slice $u = x_j$ gives the local expressions for $g_i$, and the displayed identities show that~\eqref{eq:boundary-shell} reproduces $f$ on a full neighbourhood. This is stronger than agreement merely on the slice $w = 0$.
\\[2pt]
The closed slice disk $\{u = x_j,\ |v| \leq 5\lambda_j/4\}$ is contained in $B$, so there is $0 < e_j < \eta \lambda_j$ such that $f$ is 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 $X$ therefore extends the open-germ agreement to this whole overlap, so $f$ and $F_j$ together give a box on the Hartogs figure $C_j \cup S_j$.
\\[2pt]
\emph{Extension, and the conclusion.} By Proposition~\ref{prop:hartogs}, the box on $C_j \cup S_j$ extends to 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 $j$ by~\eqref{eq:boundary-scales}. This gives the required box extension across $p$.
\end{proof}

The mixed-boundary proposition, together with the Hartogs extension of Section~\ref{sec:hartogs}, is the engine that constructs boxes from prescribed axes in the next section. Two features of the proof are worth flagging for readers who want to see the geometry. First, \emph{no product property of $\omega$ is used}: any closed positive $(1,1)$-form would do. Second, \emph{the annulus scale $\lambda_j$ is chosen between two very different powers of $\delta_j$}: the upper inequality $\lambda_j/\sqrt{\delta_j} \to 0$ makes $\lambda_j$ fit well inside the disk $D_{x_j}$, and the lower inequality $\delta_j/\lambda_j \to 0$ makes the shell in the $u$-direction reach past the remaining distance $\delta_j$ to the boundary. The exponent $3/4$ is the geometric mean of these constraints; the specific constants are immaterial.


\section{Boxes with prescribed axes}\label{sec:boxes}

The mixed-boundary extension of the previous section lets us combine two holomorphic disks tangent to $E_1$ and $E_2$ into a single box on the \emph{entire} bidisk. Starting from the local box germ at the common initial point, boundary-crossing lets us extend it across both axes of the bidisk. The next result upgrades this to polydisk parameter spaces, where the parameter dependence is the technical heart: pointwise existence of a box on each pair of polyradii does not give a holomorphic family by itself.

\begin{proposition}[Disk axes]\label{prop:disk-axes}
Let $\alpha, \beta : \D \to X$ be holomorphic maps with $\dd\alpha(T\D) \subset E_1$, $\dd\beta(T\D) \subset E_2$, and $\alpha(0) = \beta(0)$. There is a unique holomorphic box $F : \D^2 \to X$ such that
\[
  F(s, 0) = \alpha(s), \qquad F(0, t) = \beta(t)
\]
for all $s, t \in \D$.
\end{proposition}

\begin{proof}
Product coordinates at the common initial point give a unique holomorphic box germ at $(0,0)$: take the first coordinate from $\alpha$ and the second coordinate from $\beta$. Any two holomorphic boxes extending this germ agree on a connected common domain by the identity theorem, so the germ determines any extension.
\\[2pt]
Fix an integer $m \geq 1$ and consider the convex open domains
\[
  B_m(R) = \{(s, t) \in \C^2 : |s|^{2m} + |t|^{2m} < R^{2m}\}, \qquad 0 < R \leq 1.
\]
Let $R_*$ be the supremum of radii to which the germ extends as a box. The nested extensions agree and combine to a box on $B_m(R_*)$; by the one-variable identity theorem, its values on the two axes are the given $\alpha$ and $\beta$. We suppose $R_* < 1$ and look for an extension past $B_m(R_*)$, contradicting maximality.
\\[2pt]
\emph{Boundary points with $s t \neq 0$.} At such a point of $\partial B_m(R_*)$, the complex tangent to the boundary has both components nonzero. Indeed, on $\partial B_m(R_*)$ the real defining function is $|s|^{2m} + |t|^{2m} - R^{2m}$, and at any point with $s t \neq 0$ its complex derivatives $\partial \rho/\partial s$ and $\partial \rho/\partial t$ are both nonzero, so the complex tangent direction $(s, t) \mapsto (\partial \rho/\partial t, -\partial \rho/\partial s)$ has both components nonzero. Proposition~\ref{prop:mixed} then extends the box across the boundary point.
\\[2pt]
\emph{Boundary points of the form $(s_*, 0)$.} Here the mixed-boundary proposition does not apply, because the complex tangent degenerates. We instead reconstruct the box on a small product neighbourhood using one coordinate already supplied by $\alpha$ and the other by a known interior point. Choose a disk $U$ around $s_*$ with $\overline U \subset \D$ and such that $\alpha(U)$ lies in a single product chart $z = (z_1, z_2)$. Fix a point $s' \in U$ with $|s'| < R_*$, so that the box $F$ is already defined at $(s', t)$ for small $t$. For $\epsilon > 0$ sufficiently small, the formula
\[
  G(s, t) = z^{-1}\bigl(z_1(\alpha(s)),\; z_2(F(s', t))\bigr), \qquad (s, t) \in U \times \D_\epsilon,
\]
is well defined and is a holomorphic box: the first coordinate comes from $\alpha$ and the second from $F$'s known values. The product range of $z$ means the combination needs no additional target-chart margin. Near $(s', 0)$, the map $G$ coincides with $F$ by local separated expression. Because $(U \times \D_\epsilon) \cap B_m(R_*)$ is convex, they agree on the whole intersection. This is a box extension across $(s_*, 0)$. The boundary points $(0, t_*)$ are treated symmetrically.
\\[2pt]
\emph{Gluing the local extensions.} 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 on the segment joining the centres; that segment lies in $\overline{B_m(R_*)}$ by convexity. Since the intersection is open, it also meets $B_m(R_*)$, and there the two extensions agree. They therefore agree on their whole connected intersection. A finite subcollection covers the compact boundary $\partial B_m(R_*)$, giving an extension to a neighbourhood of $\overline{B_m(R_*)}$. That neighbourhood contains $B_m(R')$ for some $R_* < R' < 1$, contradicting the definition of $R_*$.
\\[2pt]
Hence the germ extends to $B_m(1)$ for every $m$. The domains $B_m(1)$ increase with $m$ to the bidisk $\D^2$, and the resulting boxes agree on their connected overlaps. Their union is a holomorphic box $F : \D^2 \to X$. The initial uniqueness observation proves uniqueness on $\D^2$.
\end{proof}

We will also use continuous boxes, on products of closed intervals. A \emph{continuous box} on a product of closed real intervals is a continuous map into $X$ that is locally separated in product coordinates: every horizontal slice locally follows an $E_1$-plaque and every vertical slice locally follows an $E_2$-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 each connected slice.

\begin{lemma}[Uniqueness of continuous boxes]\label{lem:box-unique}
Two continuous boxes on $[0,1]^2$ with the same bottom edge and left edge are equal.
\end{lemma}

\begin{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 of the two candidate maps, 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 tile 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 of the two maps on the whole grid.
\end{proof}

\begin{proposition}[Polydisk axes]\label{prop:polydisk-axes}
Let $S$ and $T$ be unit polydisks of positive dimensions. Suppose $\alpha : S \to X$ and $\beta : T \to X$ are holomorphic maps tangent to $E_1$ and $E_2$, respectively, and suppose $\alpha(p_0) = \beta(q_0)$ for some $(p_0, q_0) \in S \times T$. There is a unique holomorphic box $F : S \times T \to X$ with
\[
  F(p, q_0) = \alpha(p), \qquad F(p_0, q) = \beta(q).
\]
\end{proposition}

\begin{proof}
Polydisk automorphisms reduce the problem to $p_0 = q_0 = 0$. For fixed $(p, q) \in S \times T$, the maps
\[
  \sigma \longmapsto \alpha(\sigma p) \quad (\sigma \in \D), \qquad
  \tau \longmapsto \beta(\tau q) \quad (\tau \in \D)
\]
are holomorphic on disks of radii greater than one; after rescaling, Proposition~\ref{prop:disk-axes} gives a holomorphic box whose restriction to the closed real square we denote $R_{p,q} : [0,1]^2 \to X$. Set
\[
  F(p, q) := R_{p,q}(1,1).
\]
The constant-axis cases and Lemma~\ref{lem:box-unique} give the prescribed axis values at $q_0$ and $p_0$. It remains to prove that $F$ is holomorphic and satisfies the two tangencies. Pointwise existence of $R_{p,q}$ alone does not give parameter continuity, so this is the real content.
\\[2pt]
Fix a reference pair $(p_*, q_*)$ and choose a finite grid on the parameter square such that the closed tile images of $R_{p_*, q_*}$ lie in product charts. We reconstruct this square, for pairs $(p, q)$ near $(p_*, q_*)$, using those fixed charts, and set its tile-by-tile values equal to $R_{p,q}$. Begin with its two prescribed axes and process tiles from the bottom and left. The induction maintains three properties: (a) each constructed tile depends jointly continuously on its real parameters $\sigma, \tau$ and on $(p, q)$; (b) each constructed tile is a continuous box in the real parameters; (c) at each fixed real parameter pair $(\sigma, \tau)$, the tile map is holomorphic in $(p, q)$ with $p$-directions in $E_1$ and $q$-directions in $E_2$. All three hold on the initial axes, since $\alpha$ and $\beta$ are holomorphic.
\\[2pt]
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 pair $(p_*, q_*)$, their images lie in the chosen product chart $z = (z_1, z_2)$. Joint continuity and compactness of those edge images put both images into the same chart, after shrinking the parameter neighbourhood of $(p_*, q_*)$. Define the new tile by
\begin{equation}\label{eq:parameter-tile}
  z^{-1}\bigl(z_1(B(\sigma; p, q)),\; z_2(L(\tau; p, q))\bigr).
\end{equation}
The two edges meet at a common corner, and plaque constancy on each edge shows that the formula reproduces both of them; properties (a) and (b) follow directly from the formula. For each fixed $(\sigma, \tau)$, the formula is holomorphic in $(p, q)$ because $z_1 \circ B$ and $z_2 \circ L$ are. The induction hypothesis gives
\[
  \partial_q (z_1 \circ B) = 0, \qquad \partial_p (z_2 \circ L) = 0,
\]
so the two parameter tangencies carry through. At the reference pair $(p_*, q_*)$, the reconstructed tile is exactly the original tile, so the induction continues. There are only finitely many tiles and hence only finitely many shrinkings of the parameter neighbourhood; replace the initial neighbourhood by their finite intersection.
\\[2pt]
The tile maps agree on common edges and form a continuous box across seams. To check the latter assertion at a seam or a vertex, use a product chart around the common image. On a small real neighbourhood, the unwanted coordinate along a horizontal or vertical slice is constant on each tile piece and agrees at the shared endpoint, so it is constant across the seam. Lemma~\ref{lem:box-unique} then says that 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 on all of $S \times T$. Finally, the axes determine the box's germ at $(0,0)$, so the identity theorem proves uniqueness.
\end{proof}

The construction is not merely existence of some box with the prescribed axes: it produces the box from \emph{full-polydisk representatives} of the edges. In the next section we exploit this, by transporting an edge along a path in such a way that its full-polydisk representative is preserved. That, rather than a bound on the size of its image, is what lets the finite grid be filled.


\section{Path rectangles and the universal cover}\label{sec:global}

Every analytic ingredient is now in place. The remaining work is to combine the polydisk box theorem of Section~\ref{sec:boxes} with a finite-grid assembly to produce, along \emph{arbitrary continuous paths}, either local product structure on the leaves through a basepoint or a continuation of the product germ along a path in $X$. Once this continuation is available on every path, monodromy on simply connected covers turns it into a biholomorphism.

The key feature of the previous section is not that boxes exist pointwise but that a transported edge retains a holomorphic parametrization on an \emph{entire polydisk}. Its image need not remain small, nor lie in a single original chart. This is what makes the finite-grid assembly work: after each tile, the outgoing edges still have the same rich holomorphic parametrization as the incoming ones, so there is no error to accumulate.

\begin{proposition}[Path rectangles]\label{prop:rectangles}
The following assertions hold.
\begin{enumerate}
\item If $\alpha, \beta : [0,1] \to X$ are continuous paths which locally follow $E_1$- and $E_2$-plaques, respectively, and $\alpha(0) = \beta(0)$, then there is a continuous box $R : [0,1]^2 \to X$ with $R(s,0) = \alpha(s)$ and $R(0,t) = \beta(t)$.
\item Every continuous path $\gamma : [0,1] \to X$ is the diagonal of a continuous box: there is a continuous box $R : [0,1]^2 \to X$ with $R(s,s) = \gamma(s)$ for all $s$.
\end{enumerate}
\end{proposition}

\begin{proof}
\emph{(1) Plaque paths.} 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 then has a representation
\[
  \alpha(s) = A_i(a_i(s)), \qquad s \in I_i,
\]
where $a_i : I_i \to \D^{r_1}$ is \emph{continuous} and $A_i : \D^{r_1} \to X$ is a \emph{holomorphic} plaque parametrization tangent to $E_1$. Similarly, $\beta(t) = B_j(b_j(t))$ for $t \in J_j$, with $b_j : J_j \to \D^{r_2}$ continuous and $B_j : \D^{r_2} \to X$ holomorphic and tangent to $E_2$. Here $r_i = \rank E_i$.
\\[2pt]
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 \emph{entire} polydisks $\D^{r_1}$ and $\D^{r_2}$ and tangent to the respective bundles, and they agree at their common corner. Apply Proposition~\ref{prop:polydisk-axes} with origins $a_i(u_{i-1})$ and $b_j(v_{j-1})$, obtaining a holomorphic box $G : \D^{r_1} \times \D^{r_2} \to X$. Its evaluation
\[
  R(s, t) = G(a_i(s),\, b_j(t))
\]
fills the tile. The opposite horizontal and vertical edges of the new tile 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 shrinking the edge domains.
\\[2pt]
\emph{(2) Diagonal of a general continuous path.} Subdivide so that each $\gamma(I_i)$, $I_i = [u_{i-1}, u_i]$, lies in a product chart with inverse $\theta_i : \D^{r_1} \times \D^{r_2} \to X$. Write $\gamma(s) = \theta_i(a_i(s), b_i(s))$ on that interval. Define the diagonal tile by
\[
  R(s, t) = \theta_i(a_i(s),\, b_i(t)), \qquad (s, t) \in I_i^2.
\]
These tiles agree at consecutive diagonal vertices, and their edges have the full-polydisk representations used above.
\\[2pt]
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~\ref{prop:polydisk-axes} 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 values already assigned, and its output edges retain their full-polydisk representatives.
\\[2pt]
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~\ref{prop:polydisk-axes}: 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.
\end{proof}

Fix a basepoint $o \in X$. For $i = 1, 2$, let $L_i$ be the leaf through $o$ of the foliation determined by $E_i$, endowed with its \emph{intrinsic} plaque topology in which plaque paths are continuous (see the convention at the end of Section~\ref{sec:prelim}). The next lemma establishes that these leaves are honest complex manifolds and that they admit universal covers.

\begin{lemma}[Intrinsic leaves]\label{lem:leaves}
Each $L_i$ is a connected, Hausdorff, second-countable complex manifold of dimension $r_i$, and its inclusion into $X$ is an injective holomorphic immersion. It admits a connected simply connected cover $\widetilde L_i$, also a second-countable complex manifold. A path that locally follows $E_i$-plaques is continuous in the intrinsic topology of its leaf.
\end{lemma}

\begin{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.
\\[2pt]
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.
\\[2pt]
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 last 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 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.
\end{proof}

A product chart at $o$ supplies mutually inverse germs
\[
  H_o : (L_1 \times L_2, (o,o)) \to (X, o), \qquad
  K_o : (X, o) \to (L_1 \times L_2, (o,o)).
\]
The first combines coordinates on the two plaques through $o$; the second projects onto them. By the separated chart transitions of Lemma~\ref{lem:product-charts}, these germs do not depend on the chosen product chart. The next result gives analytic continuation of both germs along paths.

\begin{proposition}[Transport of inverse germs]\label{prop:germ-transport}
Let $R : [0,1]^2 \to X$ be a continuous box with $R(0,0) = o$, and set
\[
  \alpha(r) = R(r, 0) \in L_1, \qquad \beta(r) = R(0, r) \in L_2, \qquad \gamma(r) = R(r, r) \in X.
\]
There are mutually inverse local biholomorphism germs $H_r$, $K_r$ at $(\alpha(r), \beta(r))$ and at $\gamma(r)$ respectively, which give analytic continuation of $H_o$ along the pair $(\alpha, \beta)$ and of $K_o$ along $\gamma$.
\end{proposition}

\begin{proof}
Transport a first-coordinate germ from the $L_1$-plaque at $\alpha(r)$ along the vertical path $u \mapsto R(r, ru)$, $0 \leq u \leq 1$. 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 the horizontal path $u \mapsto R(ru, r)$. 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.
\\[2pt]
We must show \emph{genuine} analytic continuation, not 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 neighbourhood 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 neighbourhoods, for all $r$ sufficiently close to $r_0$. Also $\alpha(r)$ and $\beta(r)$ remain in fixed intrinsic leaf plaques by Lemma~\ref{lem:leaves}.
\\[2pt]
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 neighbourhood, 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
\begin{equation}\label{eq:fixed-transport}
  (x, y) \longmapsto z^{-1}\bigl(A(a(x)),\, B(b(y))\bigr)
\end{equation}
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.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Every path from $(o,o)$ in $L_1 \times L_2$ consists of two intrinsically continuous leaf paths. Proposition~\ref{prop:rectangles} provides a box with these as its axes, and Proposition~\ref{prop:germ-transport} 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.
\\[2pt]
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
\[
  H : Y_1 \times Y_2 \to X, \qquad K : \widetilde X \to L_1 \times L_2
\]
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.
\\[2pt]
Since both source manifolds are simply connected, these maps lift to holomorphic maps
\[
  \widetilde H : Y_1 \times Y_2 \to \widetilde X, \qquad
  \widetilde K : \widetilde X \to Y_1 \times Y_2.
\]
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 $\widetilde X \simeq Y_1 \times Y_2$.
\\[2pt]
It remains to retain the specified splitting. Near the basepoint, the initial product chart gives
\[
  \dd\Phi(\pi^* E_1) \subset \pr_1^* T_{Y_1}, \qquad \dd\Phi(\pi^* E_2) \subset \pr_2^* T_{Y_2}.
\]
Equivalently, the two wrong-factor components of $\dd\Phi$ vanish there. They are holomorphic bundle maps, so they vanish on all of the connected manifold $\widetilde X$. Finally, $\dd\Phi$ is invertible and $\dim_\C Y_i = r_i$ by Lemma~\ref{lem:leaves}; hence the inclusions are equalities. The factors $Y_i$ are connected and simply connected by construction, proving the asserted compatible splitting.
\end{proof}

With Theorem~\ref{thm:main} proved, we turn to the nef-and-big corollary. The remaining task is to verify that in the presence of a canonical K\"ahler--Einstein metric, the summands of any holomorphic splitting are automatically parallel, hence automatically integrable. Once integrability is established, Theorem~\ref{thm:main} gives the corollary immediately.


\section{The nef-and-big case: automatic parallelism and integrability}\label{sec:canonical}

Throughout this section, $X$ is a smooth connected projective complex manifold of dimension $n \geq 2$ with $K_X$ nef and big, and $T_X = E_1 \oplus E_2$ is a holomorphic splitting into positive-rank subbundles. \emph{Integrability is not assumed.} The point of the section is that in this setting it holds automatically, so the main theorem of Section~\ref{sec:global} applies.

The mechanism is a curvature identity. When a canonical K\"ahler--Einstein metric exists, the Ricci curvature of $X$ is proportional to the metric. The summands of a splitting have curvature contributions that, after cancellation across the whole splitting, are bounded by the so-called second fundamental forms. A first-Chern-class identity forces the total $L^2$-mass of those second fundamental forms to vanish in the limit; hence they vanish identically, and the splitting is parallel.

\subsection{The canonical contraction}

Let $X$ be smooth, projective, with $K_X$ nef and big. The minimal model program for this situation (base-point-freeness) says that $K_X$ is semiample and defines a birational contraction to the canonical model. The specific statement we need is the following.

\begin{proposition}[Canonical contraction]\label{canon:contraction}
There is a projective birational morphism with connected fibres
\[
  f : X \to Z
\]
onto a normal projective variety with canonical singularities and ample $\mathbf{Q}$-Cartier canonical divisor. Moreover, $K_X = f^* K_Z$, and $f$ is an isomorphism over $Z_{\mathrm{reg}}$.
\end{proposition}

\begin{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 any integer $a > 1$ (see \cite[Theorem 3.3]{KM98}). The associated contraction, with its Stein factorization, has normal projective target and connected fibres. For some positive integer $m$ and ample line bundle $L$ on $Z$,
\[
  \mathcal O_X(m K_X) \;\simeq\; f^* L.
\]
Bigness makes $f$ generically finite; connected fibres 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 that open set, $L$ agrees with $\mathcal O_Z(m K_Z)$; reflexive extension therefore identifies $L$ with $\mathcal O_Z(m K_Z)$ everywhere. Thus $K_Z$ is $\mathbf{Q}$-Cartier and ample.
\\[2pt]
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.
\\[2pt]
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 together imply that this restriction is an isomorphism.
\end{proof}

Set $X^0 = f^{-1}(Z_{\mathrm{reg}})$. This is a connected dense open subset of $X$, and we identify it with $Z_{\mathrm{reg}}$. Our convention is
\[
  \Ric(\omega) = -\sqrt{-1}\, \partial\bar\partial \log \omega^n,
\]
so $[\Ric(\omega)] = 2\pi c_1(X)$.

\subsection{The canonical K\"ahler--Einstein metric}

Song's work on the Riemannian geometry of K\"ahler--Einstein currents gives the metric input.

\begin{lemma}[The canonical metric]\label{canon:metric-input}
Fix a K\"ahler form $\omega_A$ from a polarization of $X$. There are smooth K\"ahler metrics $g_\varepsilon$ on $X$, for $0 < \varepsilon < 1$, whose forms satisfy
\[
  \Ric(\omega_\varepsilon) = -\omega_\varepsilon + \varepsilon \omega_A, \qquad
  [\omega_\varepsilon] = 2\pi c_1(K_X) + \varepsilon [\omega_A].
\]
As $\varepsilon \to 0$, they converge smoothly on compact subsets of $X^0$ to a K\"ahler metric $g$ with $\Ric(g) = -g$. The completion of the intrinsic Riemannian distance of $(X^0, g)$ is naturally homeomorphic to $Z$.
\end{lemma}

\begin{proof}
Proposition~\ref{canon:contraction} makes $f : X \to Z$ a smooth projective crepant resolution of a canonical model. The completion statement is \cite[Theorem 1.2]{Song14}. The approximating metrics and their smooth convergence are \cite[Equation (4.28) and Lemma 4.1]{Song14}, with $\varepsilon = e^{-t}$. The cohomology identity follows directly from the Ricci equation.
\end{proof}

\subsection{The second-fundamental-form estimate}

The main estimate of this section shows that both summands of the splitting are parallel for the limiting metric on $X^0$, and hence involutive on all of $X$. The proof is a global $L^2$ identity for the second fundamental forms, whose vanishing follows from first-Chern-class cancellation. The method belongs to the subbundle-curvature theory of Enoki \cite{Enoki88} and is used for splittings by Guenancia \cite[Section 3.4, Step 4, Equations (3.12)--(3.13)]{Guenancia16}.

\begin{proposition}[Parallelism and integrability]\label{canon:parallel}
The connection of the limiting K\"ahler metric $g$ of Lemma~\ref{canon:metric-input} preserves both $E_1|_{X^0}$ and $E_2|_{X^0}$. Both subbundles are integrable on all of $X$. They admit simultaneous local holomorphic product coordinates.
\end{proposition}

\begin{proof}
Give $T_X$ the Hermitian metric induced by $g_\varepsilon$. Let $P_{j,\varepsilon}$ be the 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, and we do not assume orthogonality of $E_1$ and $E_2$ for any of the underlying metrics.
\\[2pt]
\emph{Subbundle curvature formula.} Write $F_V$ for the Chern curvature of a Hermitian bundle $V$. For the subbundle $E_j \subset T_X$ with the induced metric, the trace of the Chern curvature of $E_j$ against $\omega_\varepsilon$ satisfies
\[
  \tr\!\bigl(\sqrt{-1}\, \Lambda_{\omega_\varepsilon} F_{E_j}\bigr) \;=\;
  \tr\!\bigl(P_{j,\varepsilon}\, \sqrt{-1}\, \Lambda_{\omega_\varepsilon} F_{T_X}\bigr) \;-\; |B_{j,\varepsilon}|^2.
\]
This is the standard subbundle curvature formula: write the connection of $T_X$ in blocks for $E_j \oplus E_j^\perp$, whose off-diagonal blocks are $B_{j,\varepsilon}$ and $-B_{j,\varepsilon}^*$. The second fundamental forms on the right are measured in the complex Hermitian norm.
\\[2pt]
In the K\"ahler case, the mean-curvature endomorphism $\sqrt{-1}\, \Lambda_{\omega_\varepsilon} F_{T_X}$ is the Ricci endomorphism
\[
  H_\varepsilon = -I + \varepsilon A_\varepsilon, \qquad A_\varepsilon = \omega_\varepsilon^{-1} \omega_A > 0.
\]
\\[2pt]
\emph{Cancellation of total first Chern class.} Integrate the trace formula over $X$. Since $c_1(E_1) + c_1(E_2) = c_1(T_X)$, and writing $dV_\varepsilon = \omega_\varepsilon^n / n!$, the $E_j$-trace contributions combine to
\[
  \int_X \sum_{j=1}^2 |B_{j,\varepsilon}|^2 \, dV_\varepsilon
  \;=\; \int_X \tr(C_\varepsilon H_\varepsilon)\, dV_\varepsilon,
  \qquad
  C_\varepsilon = P_{1,\varepsilon} + P_{2,\varepsilon} - I.
\]
\\[2pt]
\emph{Bound on the right side.} The ranks satisfy $\rank P_1 + \rank P_2 = n = \rank I$, so $\tr C_\varepsilon = 0$. Moreover $-I \leq C_\varepsilon \leq I$ as self-adjoint endomorphisms, because each $P_{j,\varepsilon}$ is an orthogonal projection with spectrum in $\{0, 1\}$. Positivity of $A_\varepsilon$ therefore gives the estimate
\begin{align}
  0 \;\leq\; \int_X \sum_j |B_{j,\varepsilon}|^2 \, dV_\varepsilon
  \;&\leq\; \varepsilon \int_X \tr(A_\varepsilon)\, dV_\varepsilon
  \notag \\
  &=\; \frac{\varepsilon}{(n-1)!} \int_X \omega_A \wedge \omega_\varepsilon^{n-1}
  \;=\; O(\varepsilon).
  \label{eq:canon:energy}
\end{align}
The last integral is a bounded polynomial in $\varepsilon$, because $[\omega_\varepsilon] = 2\pi c_1(K_X) + \varepsilon [\omega_A]$ is a bounded combination of two fixed classes as $\varepsilon \to 0$. No angle bound between the summands is used anywhere.
\\[2pt]
\emph{Vanishing of the limiting second fundamental forms.} On compact subsets of $X^0$, the metrics, their connections, the orthogonal projections, and the second fundamental forms all converge smoothly to their limiting counterparts by Lemma~\ref{canon:metric-input}. The mass bound~\eqref{eq:canon:energy} says
\[
  \int_X \sum_j |B_{j,\varepsilon}|^2 \, dV_\varepsilon \to 0.
\]
The integrand is nonnegative and converges pointwise to $\sum_j |B_j|^2$ on compact subsets of $X^0$; the limit must therefore vanish identically there. So the limiting second fundamental forms $B_j$ vanish on $X^0$.
\\[2pt]
\emph{Integrability.} The connection of a K\"ahler metric is torsion-free. Hence on $X^0$, each subbundle is integrable: its local sections are closed under Lie bracket. The Frobenius obstruction
\[
  \Lambda^2 E_j \to T_X / E_j, \qquad
  u \wedge v \mapsto [u,v] \bmod E_j,
\]
is a holomorphic bundle morphism on all of $X$: it is an algebraic-geometric tensor that can be defined globally and is manifestly holomorphic. It vanishes on the dense open set $X^0$, hence everywhere. Thus $E_1$ and $E_2$ are integrable on all of $X$.
\\[2pt]
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, as in Lemma~\ref{lem:product-charts}. Transitions between such charts locally preserve the two factors.
\end{proof}

\subsection{Proof of the nef-and-big corollary}

We now prove Corollary~\ref{thm:nef-big}.

\begin{proof}[Proof of Corollary~\ref{thm:nef-big}]
A smooth projective complex manifold is a compact K\"ahler manifold. By Proposition~\ref{canon:parallel}, the two summands of the given splitting $T_X = E_1 \oplus E_2$ are integrable on $X$. Theorem~\ref{thm:main} therefore applies and gives precisely the compatible product claimed in the corollary.
\end{proof}

\begin{remark}
The argument just given uses the canonical metric only to establish integrability. No further input from the metric is needed. Once both summands are integrable, Theorem~\ref{thm:main} closes the argument. This is the shortest path from K\"ahler--Einstein theory to compatible splitting in the nef-and-big case.
\end{remark}


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\end{document}