Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations

Abstract

We consider an embedded n-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d}$ having $C_n$ as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

Appendix A. \(L^2\) Bounds of Cohomology Solutions and Small Divisors

1.1 A Question of Donin

Let E be a holomorphic vector bundle on a compact complex manifold C. The main purpose of this section is to obtain $L^2$ and sup-norm bounds for the cohomology equation

$$\begin{aligned} \delta u=f \end{aligned}$$

(A.1)

where $f\in Z^1({\mathcal {U}},{\mathcal {O}}(E))$ and ${\mathcal {U}}$ is a suitable covering of C. Our goal is to show that if $f=0$ in $H^1(C,{\mathcal {O}}(E))$, then there is a solution u such that

$$\begin{aligned} \Vert u\Vert _{{\mathcal {U}}}\le K(E)\Vert f\Vert _{{\mathcal {U}}}. \end{aligned}$$

(A.2)

Here $\Vert \cdot \Vert _{{\mathcal {U}}}$ is the $L^2$-norm for cochains of the covering ${\mathcal {U}}$. The main assertion is that the solution u admits estimate on the original covering ${\mathcal {U}}$ without any refinement, which is important to the application in this paper. For this purpose, we will choose the covering ${\mathcal {U}}$ which consists of biholomorphic images of the unit polydisc and which are in the general position. The question on the existence of such an estimate and solutions was raised by Donin who asked the general question if ${\mathcal {O}}(E)$ is replaced by a coherent analytic sheaf ${\mathcal {F}}$ on C and f is any p-cocycle, with $p>0$, of a covering ${\mathcal {U}}$ [9]. The result in this appendix provides an affirmative answer to Donin’s question for $p=1$ and the sheaf of holomorphic sections of a holomorphic vector bundle. Furthermore, we will introduce the small divisor for (A.1) in (A.2). Although some of results in this appendix can be further developed for a general setting, we limit to the case of $H^1(C,{\mathcal {O}}(E'\otimes E''))$; this suffices applications in this paper. One may take $E''$ to be the trivial bundle to deal with a general vector bundle E. In the applications we have in mind, C is embedded into a complex manifold M and we will take $E''$ to be symmetric powers ${\text {Sym}}^\ell N_C^*$ of $N_C^*$, the dual of the normal bundle of C in M. In this paper, $S^\ell E$ denotes the symmetric power ${\text {Sym}}^\ell E$ of a vector bundle E over C. We are mainly concerned with how various bounds depend on $\ell$ as $\ell \rightarrow \infty$ when we employ the important Fisher metric on $S^\ell N_C^*$ for unitary the normal bundle $N_C$. This will be crucial in our applications.

To prove (A.2), we will first use the original estimate of Donin [9], without solving the cohomology equation. This serves as a smoothing decomposition in the sense of Grauert [15] by expressing

$$\begin{aligned} f=g+\delta u \end{aligned}$$

(A.3)

where g is defined on a larger covering while u is defined on a shrinking covering. We will then combine it with the proof of finiteness theorem of cohomology groups from Grauert–Remmert [15] to refine the decomposition (A.3) by expressing g in a base of cocycles. Finally, we will obtain (A.2) by avoiding shrinking of covering. This last step is motivated by a method of Kodaira–Spencer and Ueda [42]. We take a different approach by an essential use of the uniqueness theorem. This allows us to introduce the small divisors in (A.2) to the cohomology equation (A.1).

1.2 Bounds of Solutions of Cohomology Equations

We now start to introduce nested coverings of C. This will be an essential ingredient of the small divisors for the cohomology equation. We cover C by finitely many open sets $U_i, i\in {\mathcal {I}}$ such that there are open sets $V_i$ in M with $V_i\cap C=U_i$. We also assume that there are biholomorphic mappings $\Phi _i$ from $V_i$ onto the polydisc $\Delta _{n+d}^{r^*}$ of radius $r^*$, where n is the dimension of C and $n+d$ is the dimension of M. Assume further that $\Phi _i(U_i^{r^*})=\Delta _n^{r^*}\times \{0\}$ for $\varphi _i\times \{0\}=\Phi _i|_{U_i}$. Set ${\mathcal {U}}^r=\{U_i^r:i\in {\mathcal {I}}\}$ with $U_i^r= \varphi _i^{-1}(\Delta _n^r)$. We assume that $r^*<1$ and ${\mathcal {U}}^{r_* }$ with $r_*<r^*$, remains a covering of C. When $U_I^r:=U^r_{i_0}\cap \cdots U_{i_q}^r$ is non-empty, it is still Stein [15, p. 127].

Definition A.1

Let $\{U_j^r\}$ be an open covering of C for each $r\in [r_*,r^*]$. We say that the family of coverings $\{U_j^r\}$ is nested, if each connected component of $U_{k}^{\rho }\cap U_{j}^{r_*}$ intersects $U_{k}^{r_*}\cap U_{j}^{r_*}$ when $r_*\le \rho \le r^*$. In particular, $U_{k}^{r_*}\cap U_{j}^{r_*}$ is non-empty if and only if $U_{k}^{\rho }\cap U_{j}^{r_*}$ is non-empty.

Let $N(U_i^{ r^*})$ be the union of all $\overline{U_k^{r^*}}$ that intersect $\overline{U_{i}^{r^*}}$; as in [9] we will call the union the star of $U_i^{r^*}$. Refining ${\mathcal {U}}^{r^*}$ if necessary, we may assume that there is a biholomorphism $\varphi _i$ from a neighborhood of the star onto an open set in $\mathbf{C}^n$. If $E',E''$ are holomorphic vector bundles over C, we will fix a trivialization of $E'$ over $U_i$ by fixing a holomorphic basis $e'_k=\{e'_{k,1},\dots , e'_{k,m}\}$ in $\overline{U_k^{r^*}}$. We also fix a holomorphic base $e''_j=\{e''_{j,1},\dots , e''_{j,d}\}$ of $E''$ in $\overline{U_j^{r^*}}$. On $U_{ I}^{r^*}=U_{i_0}^{r^*}\cap \cdots \cap U^{r^*}_{i_q}$, it will be convenient to use the base

$$\begin{aligned} e_ {i_0\dots i_q}:=e_{i_0}'\otimes e_{i_q}'':=\{e'_{i_0,k}\otimes e''_{i_q,j}:1\le k\le m, 1\le j\le d\}. \end{aligned}$$

Throughout the paper $\Vert \cdot \Vert _D$ and $|\cdot |_D$ denote, respectively, the $L^2$ and sup norms of a function in D, when D is a domain in $\mathbf{C}^n$. If $f=(f_1,\dots , f_d)$ is a vector of functions, we define the $L^2$ norm, metric, and sup norms as follows:

$$\begin{aligned} \Vert f\Vert _{D}^2&:= \Vert f\Vert _{L^2(D)}^2:=\Vert f_1\Vert ^2_{D}+\cdots +\Vert f_d\Vert ^2_{D},\\ |f|_{D}^2&:=\sup _{z\in D}|f_1(z)|^2+\cdots +|f_d(z)|^2, \\ |f|_{\infty , D}&:=\sup _{z\in D}\max \{|f_1(z)|, \dots , |f_d(z)|\}. \end{aligned}$$

For a $d\times d$ matrix t of functions on D, denote by $|t|_D, \Vert t\Vert _D$, $|t|_{\infty ,D}$, respectively, the operator norms defined by

$$\begin{aligned} |t|_D=\sup _{|f|_D=1}|tf|_D, \quad \Vert t\Vert _D=\sup _{\Vert f\Vert _D=1}\Vert tf\Vert _D, \quad |t|_{\infty ,D}=\sup _{|f|_{\infty ,D}=1}|tf|_{\infty ,D}. \end{aligned}$$

Therefore, $\Vert t\Vert _D\le |t|_D$ as $\Vert tf\Vert _D\le (\sup _{z\in D }|t(z)|)\Vert f\Vert _D=|t|_D\Vert f\Vert _D$.

Then we define the $L^2$ norm for $f\in C ^q({\mathcal {U}}^{r},{\mathcal {O}}(E'\otimes E''))$ by

$$\begin{aligned} a_ Ie_{ I}:= & {} \sum _{\mu =1}^{md}a_ I^{\mu } e_{ I,\mu },\\ \Vert f\Vert _{{\mathcal {U}}^{r}}:= & {} \max _{I=(i_0,\dots , i_q)\in {\mathcal {I}}^ {q+1}}\left\{ \Vert a_{ I}\circ \varphi _{i_q}^{-1}\Vert _{ \varphi _{i_q}(U_I)}:f_ i=a_{ I} e_{ I}\ \text {in}\, U_{ I}\right\} . \end{aligned}$$

Sometimes we denote $\Vert f\Vert _{{\mathcal {U}}^{r_*}}$ by $\Vert f\Vert$ for abbreviation. We define similarly the metric norm $|f|_{{\mathcal {U}}^{r_*}}$, or |f|, and the sup-norm $|f|_{\infty ,{\mathcal {U}}^{r_*}}$ or $\sup |f|$. It is obvious that

$$\begin{aligned}&||f||\le C |f|, \quad \sup |f|\le \Vert f\Vert \le C\sqrt{{\text {rank}}(E'\otimes E'')}\sup |f|, \\&|t|_{\infty }\le |t|\le C{\text {rank}}(E'\otimes E'') |t|_{\infty }, \end{aligned}$$

where C does not depend on $E',E''$.

The first result of this appendix is to find a way to obtain solutions to (A.1) that have certain bounds on the original covering, if a solution with a bound exists on a shrinking covering. This relies on the nested coverings defined above. We first study the $L^2$ norms case.

Lemma A.2

Let ${\mathcal {U}}^r=\{U_i^r:i\in {\mathcal {I}}\}$ with $r_*\le r\le r^*$ be a family of nested finite coverings of C. Suppose that $f\in C^1({\mathcal {U}}^{r^*},E'\otimes E'')$ and $f=0$ in $H^1({\mathcal {U}}^{r^*},E'\otimes E'')$. Assume that there is a solution $v\in C^{0}({\mathcal {U}}^{r_*})$ such that

$$\begin{aligned} \delta v=f, \quad \Vert v\Vert _{{\mathcal {U}}^{r_*}}\le K\Vert f\Vert _{{\mathcal {U}}^{r^*}}. \end{aligned}$$

(A.4)

Then there exists a solution $u\in C^{0}({\mathcal {U}}^{r^*})$ such that $\delta u=f$ on ${\mathcal {U}}^{r^*}$ and

$$\begin{aligned} \Vert u\Vert _{{\mathcal {U}}^{r^*}}\le C( |\{t'_{kj}\}|_{{\mathcal {U}}^{r^*}}+ K|\{t'_{kj}\}|_{{\mathcal {U}}^{r^*}}|\{t''_{kj}\}|_{{\mathcal {U}}^{r^*}})\Vert f\Vert _{{\mathcal {U}}^{r^*}}, \end{aligned}$$

(A.5)

where $t_{kj}',t_{kj}''$ are the transition matrices of $E',E''$, respectively, and C depends only on the number $|{\mathcal {I}}|$ of open sets in ${\mathcal {U}}^{r^*}$ and transition functions of C. In particular, C does not depend on $E',E''$.

Proof

By assumptions, we have

$$\begin{aligned}&f_{jk}=(\delta v)_{jk}, \quad U_j^{r_*}\cap U_k^{r_*}, \end{aligned}$$

(A.6)

$$\begin{aligned}&\Vert v\Vert _{{\mathcal {U}}^{r_*}}\le K\Vert f\Vert _{{\mathcal {U}}^{r^*}}. \end{aligned}$$

(A.7)

Take any $v^*\in C^0({\mathcal {U}}^{r^*},E'\otimes E'')$ such that $\delta v^* =f$. Then $(\delta v^*-\delta v)_{jk}=0$ in $U^{r_*}_j\cap U^{r_*}_k$, because $(\delta v^*)_{jk}=f_{jk}$ on the larger set $U_j^{r^*}\cap U_k^{r^*}$. Since $\{U_j^{r_*}\}$ is a covering of C then $w:= v_j-v^*_j$ is a global section of $E'\otimes E''$. This shows that $v_j$, via $v^*_j$, extends to a holomorphic section in $U_j^{r^*}$. In fact, $v_j$ is the restriction of $u_j=v^*_j+w$ defined on $U_j^{r_*}$.

We now derive the bound for $u_j$. Suppose that $U_j^{r^*}\cap U_k^{r_*}$ is non-empty. By the assumptions, each component of $U_j^{r^*}\cap U_k^{r_*}$ intersects $U_j^{r_*}\cap U_k^{r_*}$. We have $u_j= u_k+f_{jk}$ on $U_j^{r_*}\cap U_k^{r_*}$ and hence the uniqueness theorem implies that it holds on $U_j^{r^*}\cap U_k^{r_*}$ too. And on $U_{j}^{r^*}\cap U_k^{r_*}$, we have $u_k=v_k$ and $u_j=v_k-f_{kj}$. We express the identity in coordinates

$$\begin{aligned} u_j={\tilde{u}}_j e_j, \quad v_k={\tilde{v}}_ke_k={\hat{v}}_{kj}e_{j}, \quad f_{kj}={\tilde{f}}_{kj}e_{kj}={\hat{f}}_{kj}e_{jj}. \end{aligned}$$

Let $t_{kj}', t_{kj}''$, respectively, be the transition matrices of $e_{j}', e_{j}''$ for $E', E''$. Then ${\tilde{t}}_{kj}=t_{kj}'\otimes t_{kj}''$ are the transition matrices of $e_{kj}$ for $E'\otimes E''$. Then we have

$$\begin{aligned} {\hat{v}}_{kj}= t_{jk}'\otimes t_{jk}''{\tilde{v}}_k, \quad {\hat{f}}_{kj}=t_{jk}'\otimes I_{d}{\tilde{f}}_{kj}. \end{aligned}$$

Thus, ${\tilde{u}}_j={\hat{v}}_{kj}-{\hat{f}}_{kj}= t_{jk}'\otimes t_{jk}''{\tilde{v}}_k -t_{jk}'\otimes I_{d}{\tilde{f}}_{kj}$. We have

$$\begin{aligned} \Vert {\tilde{u}}_j\Vert _{L^2(U_j^{r^*}\cap U_k^{r_*})}&=\Vert {\tilde{u}}_j\circ \varphi _j^{-1}\Vert _{L^2( \varphi _j(U_j^{r^*}\cap U_k^{r_*}))}\\&\le \Vert ( t_{jk}'\otimes t_{jk}''{\tilde{v}}_k)\circ \varphi _j^{-1}\Vert _{L^2( \varphi _j( U_j^{r^*}\cap U_k^{r_*}))}\\&\quad + \Vert (t_{jk}'\otimes I_{d}{\tilde{f}}_{kj})\circ \varphi _j^{-1}\Vert _{L^2( \varphi _j(U_j^{r^*}\cap U_k^{r^*}))}. \end{aligned}$$

Here $t_{jk}\circ \varphi _j^{-1}=t_{jk}\circ \varphi _k^{-1} \circ \varphi _{kj}$. By the properties of operator norm and $\Vert t_{kj}'\otimes t_{kj}''\Vert _D\le |t_{kj}'\otimes t_{kj}''|_D\le |t_{kj}'|_D| t_{kj}''|_D$ for $D= \varphi _j( U_j^{r^*}\cap U_k^{r_*})$, we have

$$\begin{aligned}&\Vert (t_{jk}'\otimes t_{jk}''{\tilde{v}}_k) \circ \varphi _j^{-1}\Vert ^2_{D}\le C_* |t'_{jk}|_D^2 \times |t''_{jk}|_D^2\times \Vert {\tilde{v}}_k\Vert ^2_{ \varphi _k( U_j^{r^*}\cap U_k^{r_*})}, \end{aligned}$$

where the constant $C_*$ comes from the Jacobian of $z_k= \varphi _{kj}(z_j)$. By (A.7), we have

$$\begin{aligned} \Vert {\tilde{v}}_k\circ \varphi _k^{-1}\Vert _{L^2}^2\le K^2\Vert f\Vert _{L^2}^2. \end{aligned}$$

We also have

$$\begin{aligned} \Vert ( t_{jk}'\otimes I_{d} {\tilde{f}}_{kj})\circ \varphi _j^{-1}\Vert _{ \varphi _j( U_j^{r^*}\cap U_k^{r_*})}\le |t_{jk}'\circ \varphi _j^{-1}|_{ \varphi _j( U_j^{r^*}\cap U_k^{r_*})}\times \Vert f\Vert _{ \varphi _j( U_j^{r^*}\cap U_k^{r_*})}. \end{aligned}$$

Since $U_j^{r^*}$ is covered by $\{U_{j}^{r^*}\cap U_k^{r_*}\}$, we get the desired bound from

$$\begin{aligned} \Vert {\tilde{u}}_j\Vert _{L^2(U_j^{r^*})}\le \sum _k\Vert {\tilde{u}}_j\Vert _{L^2(U_j^{r^*}\cap U_k^{r_*})}. \end{aligned}$$

$\square$

The argument for the norm $|\cdot |$ is verbatim and we can take the above constant $C_*$ to be one.

Corollary A.3

With notations and assumptions in Lemma A.2, the solution u also satisfies

$$\begin{aligned} |u|_{\infty ,{\mathcal {U}}^{r^*}}\le C( |\{t'_{kj}\}|_{{\mathcal {U}}^{r^*}}+ K|\{t'_{kj}\}|_{{\mathcal {U}}^{r^*}}|\{t''_{kj}\}|_{{\mathcal {U}}^{r^*}})\sqrt{{\text {rank}}(E'\otimes E'')}|f|_{\infty ,{\mathcal {U}}^{r^*}}, \end{aligned}$$

where C does not depend on $E',E''$.

The above lemma leads us to the following proposition and definition.

Proposition A.4

Let ${\mathcal {U}}^r=\{ U_i^r:i\in {\mathcal {I}}\}$ with $r_*\le r\le r^*$ be a family of nested coverings of a compact complex manifold C. Let $E'$ (resp. $E'')$ be a holomorphic vector bundle over C with bases $\{e_j'\}$ (resp. $\{e_j''\})$ and transition matrices $t_{kj}'$ (resp. $\{t_{kj}''\})$. Suppose that there is a finite number K such that for any $f\in C^1({\mathcal {U}}^{r^*},E'\otimes E'')$ with $f=0$ in $H^1({\mathcal {U}}^{r^*},E'\otimes E'')$, there is a solution $v\in C^{0}({\mathcal {U}}^{r_*},E'\otimes E'')$ satisfying (A.4). Then there is a possible different solution $v\in C^0({\mathcal {U}}^{r_*}, E'\otimes E'')$ satisfying (A.4) in which K is replaced by

$$\begin{aligned}&K_*(E'\otimes E'')=\sup _{u_1}\inf _{u_0}\bigl \{\Vert u_0\Vert _{{\mathcal {U}}^{r_*}}:\delta u_0=\delta u_1 \,\text {on}\, {\mathcal {U}}^{r_*}, \nonumber \\&\quad \Vert \delta u_1 \Vert _{{\mathcal {U}}^{r^*}}=1, u_i\in C^0({\mathcal {U}}^{r_i},E'\otimes E'')\bigr \}. \end{aligned}$$

(A.8)

Proof

By the assumption, $K_*=K_*(E'\otimes E'')$ is well-defined and $K_*\le K$. Fix $u_1\in C^0({\mathcal {U}}^{r_i},E'\otimes E'')$. Suppose that $\delta u_1=f$ and $\Vert f\Vert _{{\mathcal {U}}^{r^*}}=1$. By the definition (A.8), there exists $u_0^j$ such that $\delta u_0^m=f$ on ${\mathcal {U}}^{r_*}$ and $\Vert u_0^m\Vert _{{\mathcal {U}}^{r_*}}\le K_*+1/m$. By the Cauchy formula on polydiscs, $(u_0^m)_j\circ \varphi _j^{-1}$ is locally bounded in $\varphi _j(U_j)$ in sup-norm. We may assume that as $m\rightarrow \infty$, $(u_0^m)_j$ converges uniformly to $u_0^\infty$ on each compact subset of $U_j$ for all j. This shows that $\Vert (u_0^\infty )_j\circ \varphi _j^{-1}\Vert _{L^2(E)}\le K_*$ for any compact subset E of $\varphi _j(U_j)$. Since E is arbitrary, we obtain $\Vert u_0^\infty \Vert _{U^{r_*}}\le K_*$. By the uniform convergence, we also have $\delta u_0^\infty =f$ on ${\mathcal {U}}^{r_*}$. $\square$

Definition A.5

Let $E', E'', e_j',e_j'', t_{kj}',t_{kj}''$ be as in Proposition A.4. Let $t_{kj}''(S^mE'')$ be the transition matrices of the symmetric power $S^mE''$ induced by $t_{kj}''$. For $m=2,3,\dots$, we shall call

$$\begin{aligned} K(E'\otimes S^mE'')&= |\{t'_{kj}(E')\}|_{{\mathcal {U}}^{r^*}}\\&\quad + K_*(E'\otimes S^m E'')|\{t'_{kj}(E')\}|_{{\mathcal {U}}^{r^*}}|\{t''_{kj}( S^mE'')\}|_{{\mathcal {U}}^{r^*}} \end{aligned}$$

the generalized small divisors of $E'\otimes E''$ with respect to $e_j'', t_{kj}''$.

1.3 Donin’s Smoothing Decomposition

Grauert’s smoothing decomposition for cochains of analytic sheaves is an important tool. Here we will follow an approach of Donin [9], by specializing for vector bundles.

We first need to introduce coverings by analytic polydiscs.

Lemma A.6

Let C be a compact complex manifold. Let $\{U_i^{r_*}:i\in {\mathcal {I}}\}$ be a finite open covering of C, and let $\varphi _j$ map $U_j^r$ biholomorphically onto $\Delta ^n_{r}$ for $r_*<r<r^*<1$. Assume further that $\varphi _i$ is a biholomorphism defined in a neighborhood of the star $N(U_i^{r^*})$ onto a domain in $\mathbf{C}^n$. Suppose that $r_*<r_i'<r_i<r^*$, and

$$\begin{aligned} U_I^{r'}:=U_{i_0}^{r_0'}\cap \cdots \cap U_{i_q}^{r_q'}\ne \emptyset . \end{aligned}$$

Then for constant $c_n \in (0,1)$ depending only on n,

$$\begin{aligned}&{\text {dist}}\left( \partial ( \varphi _{i_q}(U_I^r)),\partial ( \varphi _{i_q}(U_I^{r'}))\right) \ge c_n\kappa \min _j(r_j-r_j'), \end{aligned}$$

(A.9)

$$\begin{aligned}&\kappa := \inf \left\{ 1,\frac{| \varphi _{i_q}\circ \varphi _{i_\ell }^{-1}(z')- \varphi _{i_q}\circ \varphi _{i_\ell }^{-1}(z)|}{|z'-z|}:z,z'\in \Delta _{ r^*}^n,\forall U^{r^*}_{i_0\dots i_q}\ne \emptyset \right\} . \nonumber \\ \end{aligned}$$

(A.10)

Proof

Note that for sets in $\mathbf{C}^n$, if $A\subset A'$, $B\subset B'$, and A, B are non-empty, then

$$\begin{aligned} {\text {dist}}( A,B)\ge {\text {dist}}(A',B'). \end{aligned}$$

Recall that $\varphi _{i_q}$ is a diffeomorphism from a neighborhood V of the star $N(U_{i_q})$ onto a subset ${\hat{V}}$ of $\mathbf{C}^n$. We have $\partial \varphi _{i_q}(U_I^{r})\subset \cup _j\partial \varphi _{i_q}(U_{i_j}^{r})$. Thus

$$\begin{aligned} {\text {dist}}(\partial \varphi _{i_q}(U_I^{r}), \varphi _{i_q}(U_I^{r'}))&\ge \min _j{\text {dist}}(\partial \ \varphi _{i_q}(U_{i_j}^{r}), \varphi _{i_q}(U_I^{r'}))\\&\ge \min _j{\text {dist}}(\partial \varphi _{i_q}(U_{i_j}^{r}), \varphi _{i_q}(U_{i_j}^{r'})). \end{aligned}$$

We have ${\text {dist}}(\partial ( \varphi _{i_q}(U_{i_j}^{r}), \varphi _{i_q}(U_{i_j}^{r'}))={\text {dist}}(\partial ( \varphi _{i_q}\circ \varphi _{i_j}^{-1}(\Delta ^{n}_{r})), \varphi _{i_q}\circ \varphi _{i_j}^{-1}(\Delta ^{n}_{r'}))$. Recall that $\varphi _{i_q}$ is defined on $N(U_{i_q})\supset U^{r_*}_{i_j}$. Then the distance is attained for some $z'\in \partial \Delta _{r'}^n$ and $z\in \partial \Delta _r^n$. By the definition of $\kappa$, we get the desired estimate. $\square$

We will recall the following smoothing decomposition of Donin [9]. Here we restrict to the case of $H^1$ and the holomorphic vector bundle to indicate the specific bounds in the estimates.

Theorem A.7

(Donin [9]). Let C be a compact complex manifold and let ${\mathcal {U}}^{r}$ $(r_*<r<r^*<1)$ be a family of open coverings of C as in Lemma A.6. Let $E'\otimes E''$ be a holomorphic vector bundle of rank m over C and fix a holomorphic base $e'_j$ (resp. $e_j'')$ for $E'$ (resp. $E'')$ over $U_j$. Let $r_*<r''<r'<r<r^*$, and

$$\begin{aligned} r'-r''\le r^*-r. \end{aligned}$$

Assume that

$$\begin{aligned} U_{kj}^{ r_*} \ne \emptyset , \quad \text {whenever}\, U_{kj} ^{r^*}\ne \emptyset . \end{aligned}$$

(A.11)

Let $\{f_{jk}\}\in Z^1({\mathcal {U}}^{r'},{\mathcal {O}}(E'\otimes E''))$. Then there exist $g\in Z^1({\mathcal {U}}^{r},{\mathcal {O}}(E'\otimes E''))$ and $u\in C^0({\mathcal {U}}^{r''},{\mathcal {O}}(E'\otimes E''))$ such that

$$\begin{aligned}&f=g+\delta u,\quad \text {in}\, C^1({\mathcal {U}}^{r''},{\mathcal {O}}(E'\otimes E'')), \end{aligned}$$

(A.12)

$$\begin{aligned}&\Vert u\Vert _{{\mathcal {U}}^{r''}}+\Vert g\Vert _{{\mathcal {U}}^{r}}\le \frac{ C_{n} |\{t'_{kj}\}||\{t_{kj}''\}| }{(r'-r'')\kappa }\Vert f\Vert _{{\mathcal {U}}^{r'}}, \end{aligned}$$

(A.13)

where $\kappa$ is defined (A.10). The constant $C_n$ is independent of $E',E''$. Furthermore, $f\mapsto g=Lf$ and $f\mapsto u=Sf$ are $\mathbf{C}$-linear.

Proof

With $f^{r'}_{ij}=f_{ij}$ we are given a cocycle $\{f^{r'}_{ij}\}$ of holomorphic sections of $E'\otimes E''$ over the covering ${\mathcal {U}}^{r'}$. Recall that $r_*<r''<r'<r<r^*$ and ${\mathcal {U}}^{r''}$ is an open covering of C.

As in [9], we will apply $L^2$-theory for (0, 1)-forms on a bounded pseudoconvex domain in $\mathbf{C}^n$. In our case the domain is actually a polydisc. Fix a holomorphic base $e_{k}'=(e'_{k,1},\dots ,e'_{k,m})$ for the vector bundle $E'$ in $U_k^{r^*}$ with transition functions $t_{kj}^{\prime }(z_j)$. Analogously, let $t_{kj}^{\prime \prime }(z_j)$ be the transition matrices for basis $e_k''$ of $E''$ for ${\mathcal {U}}^{r^*}$. For brevity, we write $t_{kj}$ for $t_{kj}(z_j)$.

We can write

$$\begin{aligned} f^{r'}_{ij}= {\tilde{f}}^{r'}_{ij}e_{ij} =t_{ki}^{\prime }\otimes t_{kj}^{\prime \prime }{\tilde{f}}_{ij}^{r'}e_{kk}:= {\hat{f}}_{ ij;k }^{r';r^*} e_{ kk}, \quad \text {on}\, U_i^{r'}\cap U_j^{r'}\cap U_k^{r^*}. \end{aligned}$$

(A.14)

The $U_k^{r^*}$ is covered by ${\mathcal {U}}_k^{r';r^*}:=\{U^{r'}_{i}\cap U^{r^*}_k\}_{ i}$, while $\{{\hat{f}}_{ij;k}^{r';r^*}\}\in Z^1({\mathcal {U}}_k^{r';r^*}, {\mathcal {O}}^{md})$. Now $\{{\hat{f}}_{ij;k}^{r';r^*}\circ \varphi _k^{-1}\}\in Z^1( \varphi _k({\mathcal {U}}_k^{r';r^*}),{\mathcal {O}}^{md})$, where $\varphi _k({\mathcal {U}}_k^{r';r^*})$ is a covering of the polydisc $\Delta ^n_{r^*}$. By Lemma A.6, we have

$$\begin{aligned} c_{i;k}:={\text {dist}}(\partial ( \varphi _k(U_{i}^{r'}\cap U_{k}^{r^*})), \varphi _k(U_{i}^{r''}\cap U_{k}^{r}))\ge c_n \kappa (r'-r''). \end{aligned}$$

(A.15)

Let $d_{ i;k}(z)$ be the distance to $\varphi _k(U_{i}^{r''}\cap U_{k}^{r})$ from $z\in \mathbf{C}^n$. Let $\chi$ be a non-negative smooth function in $\mathbf{R}$ so that $\chi (t)=1$ for $t<3/4$ and $\chi (t)=0$ for $t>7/8$. By smoothing the Lipschitz function $\chi (\frac{1}{c_{i;k}}d_{i;k}(z))$, we obtain a non-negative smooth function $z\rightarrow {{\tilde{\phi }}}_{i;k}^{r'';r'}(z)$ that equals 1 when $d_{i;k}(z)\le \frac{1}{2}c_{i;k}$ and by (A.15) it has compact support in $\varphi _k(U_{i}^{r'}\cap U_{k}^{r^*})$. Note that we can achieve

$$\begin{aligned} |\nabla {{\tilde{\phi }}}_{i;k}^{r'';r'}|< C_nc_{i;k}^{-1}\le c_n C_n \kappa ^{-1} /{(r'-r'')}. \end{aligned}$$

(A.16)

Then ${{\tilde{\phi }}}_{i;k}^{r'';r'}\circ \varphi _k$ is a non negative function with compact support in $U_i^{r'}\cap U_k^{r^*}$ such that for ${{\tilde{\phi }}}_k^{r'';r'}:=\sum {{\tilde{\phi }}}_{i;k}^{r'';r'}$, we have ${{\tilde{\phi }}}_k^{r'';r'}\circ \varphi _k>1/2$ in $U_k^{r}=\bigcup _i(U_i^{r''}\cap U_k^r)$ since $\chi (\frac{1}{c_{i;k}}d_{i;k})=1$ on $\varphi _k(U_{i}^{r''}\cap U_{k}^{r})$. Then by the mean-value theorem and the first inequality of (A.16), we get

$$\begin{aligned} {{\tilde{\phi }}}_k^{r'';r'}( \varphi _k(x))>1/4, \quad \text {if}\, {\text {dist}}( \varphi _k(x), \varphi _k(U_k^r))< \min _{i}c_{i,k}/{C_*}, \end{aligned}$$

(A.17)

for some suitable $C_*$. Recall that $c_n\le 1$ and $\kappa _n\le 1$. Since ${\text {dist}}( \varphi _k(U_k^{r}), \varphi _k(\partial U_k^{r^*})) =r^*-r'\ge c_n\kappa (r'-r'')$, there is a smooth function ${\hat{\phi }}_k^{r;r^*}:\varphi _k(U_k^{r^*})\rightarrow [0,1]$ with compact support such that ${\hat{\phi }}_k^{r;r^*}=1$ in $\varphi _k(U_k^{r})$, and

$$\begin{aligned} {\hat{\phi }}_k^{r;r^*}(x)<3/4, \quad \text {if}\, {\text {dist}}( \varphi _k(x), \varphi _k(U_k^{r}))> \min _{i}c_{i,k}/{C_*}. \end{aligned}$$

(A.18)

Note that the latter can be achieved with

$$\begin{aligned} |\nabla {\hat{\phi }}_k^{r;r^*}|<{\tilde{C}}_1/{\min _{i}c_{i,k}}\le C_2 \kappa ^{-1}/{(r'-r'')}. \end{aligned}$$

In $U_k^{r^*}$, define a non-negative smooth function

$$\begin{aligned} \phi _{i;k}^{r'';r'} =\left\{ \frac{{{\tilde{\phi }}}_{i;k}^{r'';r'}}{1-{\hat{\phi }}_k^{r;r^*}+{{\tilde{\phi }}}_{k}^{r'';r'}}\right\} \circ \varphi _k, \end{aligned}$$

where the smoothness follows from the denominator being bigger than 1/4 by (A.17) and (A.18). Thus, $\phi _{i;k}^{r'';r'}$ has compact support in $U_i^{r'}\cap U_k^{r^*}$ and $\sum _i \phi _{i;k}^{r'';r'}=1$ in $U_k^{r}=\bigcup _i(U_i^{r''}\cap U_k^r)$, as ${\hat{\phi }}_k^{r;r^*}=1$ on $U_k^r$. We can verify that

$$\begin{aligned} |\nabla (\phi _{i;k}^{r'';r'}\circ \varphi _k^{-1})|< C'\kappa ^{-1} /{(r'-r'')}. \end{aligned}$$

(A.19)

Consider the expression

$$\begin{aligned} w_{j;k}= \sum _\ell \phi _{\ell ;k}^{r'';r'}{\hat{f}}_{ \ell j;k}^{r';r^*}. \end{aligned}$$

(A.20)

Recall that $\phi _{\ell ;k}^{r'';r'}$ has compact support in $U_\ell ^{r'}\cap U_k^{r^*}$. Thus it is smooth on $\omega :=U_j^{r'}\cap U_k^{r^*}\cap U_{\ell }^{r'}$ and vanishes on an open set D containing $U_{j}^{r'}\cap U_k^{r^*}\,{\setminus }\,\omega$. On the other hand, ${\hat{f}}^{r';r^*}_{\ell j;k}$ is holomorphic in $\omega$. Hence the product $\phi _{\ell ;k}^{r'';r'}{\hat{f}}_{ \ell j;k}^{r';r^*}$ is smooth in $U_j^{r'}\cap U_k^{ r^*}$. Then $v_{j;k}=\overline{\partial }w_{j;k}$ is a smooth (0, 1) form in $U_j^{r'}\cap U_k^{ r^*}$.

Let ${\mathcal {A}}$ denote the sheaf of smooth functions on C. We now pull back the forms from the polydisc $\Delta ^n$ via $\varphi _k$. For each fixed k, we have $\{w_{j;k}\}_{j}\in C^0({\mathcal {U}}_k^{r';r^*},{\mathcal {A}}^m)$. Let us denote $t_{kj}^{\prime }\otimes I$ by $t_{kj}'$. By $f_{ij}=f_{ik}-f_{jk}$ and (A.14), we have

$$\begin{aligned} t_{ki}^{\prime }\otimes t_{kj}^{\prime \prime }{\tilde{f}}_{ij}^{r'}=t_{ki}^{\prime } {\tilde{f}}_{ik}^{r'}- t_{kj}^{\prime } {\tilde{f}}_{jk}^{r'}. \end{aligned}$$

Since $\sum _i\phi _{i;k}^{r'';r'}=1={\hat{\phi }}_k^{r;r^*} \circ \varphi _k$ on $U_k^r$, then by $\delta f=0$ and (A.14), we get on $U_i^{r'}\cap U_k^{r}\cap U_j^{r'}$

$$\begin{aligned} w_{i;k}-w_{j;k}&= \sum _\ell \phi _{\ell ;k}^{r'';r'}({\hat{f}}_{ \ell i;k}^{r';r^*}-{\hat{f}}_{ \ell j;k}^{r';r^*})= \sum _\ell \phi _{\ell ;k}^{r'';r'}(t_{k\ell }^{\prime }\otimes t_{ki}^{\prime \prime }{\tilde{f}}_{\ell i}^{r'} -t_{k\ell }^{\prime }\otimes t_{kj}^{\prime \prime }{\tilde{f}}_{\ell j}^{r'}) \\&=\sum _\ell \phi _{\ell ;k}^{r'';r'}( t_{kj}^{\prime }{\tilde{f}}_{jk}^{r'} - t_{ki}^{\prime }{\tilde{f}}_{ik}^{r'}) =t_{kj}^{\prime }{\tilde{f}}_{jk}^{r'} - t_{ki}^{\prime }{\tilde{f}}_{ik}^{r'}. \end{aligned}$$

The latter is holomorphic. Thus $(\delta v)_{ij;k}=\overline{\partial }(\delta w)_{ij;k} =0$ on $U_i^{r'}\cap U_k^{ r^*}\cap U_j^{r'}$. This shows that

$$\begin{aligned} v_k:=v_{j;k} \end{aligned}$$

is actually a $\overline{\partial }$-closed (0, 1) form in $U_{k}^{ r^*}$. Thus $( \varphi _k^{-1})^*v_k$ is a $\overline{\partial }$-closed (0, 1)-form on the polydisk $\Delta _{ r^*}^n$. By the $L^2$ theory [21, Thm. 4.4.3] applied to each component of $v_k=\sum _{\ell =1}^m {\tilde{v}}_{k}^{\ell }e_{kk,\ell }$, we have a bounded linear operator $S:v_k\rightarrow u_k$ such that $\overline{\partial }( ( \varphi _k^{-1})^* u_k)=( \varphi _k^{-1})^*v_k$. Returning to the complex manifold via $\varphi _k$, we have

$$\begin{aligned} \Vert u_k\Vert _{U_k^{ r^*}}&= \Vert u_k\circ \varphi _k^{-1}\Vert _{L^2(\Delta ^n_{r^*})}\le C\Vert v_k\circ \varphi _k^{-1}\Vert _{L^2(\Delta ^n_{r^*})}\\&\le \frac{{\tilde{C}}\kappa ^{-1} |\{t'_{kj}\}||\{t_{kj}''\}|}{r'-r''}\Vert f\Vert _{L^2({\mathcal {U}}^{r^*})}. \end{aligned}$$

Here we have used (A.20), estimate (A.19) and the definition of norm (A.4). Note that the ${\tilde{C}}$ is independent of the rank since we applied the $L^2$ norm componentwise. Set ${\hat{g}}_{j;k}^{r';r}=w_{j;k}-u_{k}$ on $U_j^{r'}\cap U_k^r$. We obtain

$$\begin{aligned}&{\hat{g}}_{i;k}^{r',r} -{\hat{g}}_{j;k}^{r',r} ={\hat{f}}_{ij;k}^{r';r^*}, \quad U_{i}^{r'}\cap U_k^{r}\cap U_{j}^{r'}, \end{aligned}$$

(A.21)

$$\begin{aligned}&\max _j\Vert {\hat{g}}_{j;k}^{r';r}\Vert _{U_j^{r'}\cap U_k^r}\le \frac{C\kappa ^{-1} |\{t'_{kj}\}||\{t_{kj}''\}|}{r'-r''}\Vert f\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

(A.22)

We have obtained (A.13).

To verify (A.12), we will use the same base $e_k$ and take the product of (A.21) with $e_k$ to obtain on $U_i^{r''}\cap U_j^{r''}\cap U_k^{r}\cap U_\ell ^{r}$

$$\begin{aligned} g_{i;k}^{r';r}-g_{j;k}^{r';r}={\hat{f}}_{i j;k}^{r';r^*}e_k=f_{ij}^{r'}={\hat{f}}_{ij;\ell }^{r';r^*}e_{\ell }= g_{i;\ell }^{r';r} - g_{j;\ell }^{r';r} \end{aligned}$$

and thus

$$\begin{aligned} g_{j;\ell }^{r';r} - g_{j;k}^{r';r} = g_{i;\ell }^{r';r} - g_{i;k}^{r';r}, \quad \text {on}\, U_i^{r''}\cap U_j^{r''}\cap U_k^{r}\cap U_\ell ^{r}. \end{aligned}$$

(A.23)

Then we have a (well-defined) holomorphic section

$$\begin{aligned} g_{k\ell }^{r}:=g_{i;\ell }^{r';r}-g_{i;k}^{r';r}, \quad U_k^{r}\cap U_\ell ^{r}. \end{aligned}$$

We verify that $\{g_{k\ell }^{r}\}\in Z^1({\mathcal {U}}^{r},{\mathcal {O}}^m)$. Set $u_i^{r''}:=g_{i;i}^{r'';r}$. Since $r'\le r$ we actually have $\{u_i^{r''}\}\in C^0({\mathcal {U}}^{r'},E'\otimes E'')$. However, only on $U_{i}^{r''}\cap U_j^{r''}$, we can verify via (A.23) that

$$\begin{aligned} g_{ij}^{r}-f_{ij}^{r'}=(g_{i;j}^{r'';r}-g_{j;j}^{r'';r})-(g_{i; j}^{r'';r}-g_{i;i}^{r'';r})=u_i^{r''}-u_j^{r''}. \end{aligned}$$

$\square$

The above result is a type of Grauert’s smoothing decomposition, which can also be obtained by open mapping theorem. See for instance [15, p. 200]. However, this yields an unknown bound in the estimates.

1.4 Finiteness Theorem with Bounds

The above smoothing decomposition does not provide a solution to the cohomology equations, i.e. if $f=0$ in $H^1({\mathcal {U}}^{r'},{\mathcal {O}}(E'\otimes E''))$, then there exists $u\in C^0({\mathcal {U}}^{r''},{\mathcal {O}}(E'\otimes E''))$ such that $\delta u=f$ on ${\mathcal {U}}^{r''}$, for some $r''\le r'$. We will follow [15] to derive the finiteness theorem with explicit bounds. In particular, this provides solutions of first cohomology equations with bounds on shrinking domains.

We first recall the resolution atlases from [15, p. 194], specializing them for the vector bundles. Assume that we have coordinate charts

$$\begin{aligned} \varphi _k:U_k^{r^*}\rightarrow P_k:= \varphi _k(U_k^{r^*})=\Delta ^{r^*}_n. \end{aligned}$$

Define $U_I^{r^*}=U_{i_0}^{r^*}\cap \dots \cap U^{r^*}_{ i_q}$ for $I\in {\mathcal {I}}^{q+1}$. Then $\varphi _I=( \varphi _{i_0},\dots , \varphi _{i_q})$ is defined on $U_I^{r^*}$ with range ${\hat{U}}_I^{r^*}$. Unless otherwise stated, we omit the superscript $r^*$ in $U_I^{r^*}$. We can define a proper embedding

$$\begin{aligned} \varphi _{I}:U_{I}\rightarrow {\hat{U}}_I\hookrightarrow P_{I}:=\Delta ^{r^*}_{ n_q}, \quad n_q=n(q+1). \end{aligned}$$

Then the push-forward of the vector bundle $E'\otimes E''|_{U_{ I}}$ defines a coherent analytic sheaf $( \varphi _I)_*(E'\otimes E'')$ over $P_I$ by trivial zero extension; see [15, p. 5, p. 195] and [14, p. 239]. A section $f\in \Gamma (U_I, E'\otimes E'')$ yields a section ${\hat{f}}_I$ of $( \varphi _I)_*( E'\otimes E'' )$ over $P_I$ by

$$\begin{aligned} {\hat{f}}_I\circ \varphi _I(x)=(f_I (x),\dots , f_I(x)), \quad {\hat{f}}_I|_{P_I\setminus {\hat{U}}_I}=0. \end{aligned}$$

Note that $\overline{U^{ r^*}}$ has a Stein neighborhood. Then following notation in [15, p. 196] we have an epimorphism by Cartan’s Theorem A:

$$\begin{aligned} \epsilon _I:{\mathcal {O}}^{\ell }|_{\Delta ^{ r^*}_{n_q}}\rightarrow ( \varphi _I)_*(E'\otimes E'' )|_{U_I}, \quad \ell \ge {\text {rank}}(E'\otimes E'') , \end{aligned}$$

where $\epsilon _I$ is defined by finitely many global sections defined in a neighborhood of $\overline{P_I}$. When $E'\otimes E''$ is a vector bundle, we take $\ell$ to be the minimal value, the rank of $E'\otimes E''$, and specify the above $\epsilon _I$ by taking

$$\begin{aligned} \epsilon _I:g_I\rightarrow {\tilde{g}}_I:=( \varphi _I)_*\{g_I\circ \varphi _I e_{I}\}. \end{aligned}$$

Here we want to obtain a more general description without restricting to a vector bundle. Define

$$\begin{aligned} C^q({\mathcal {U}}):=\prod _{I\in {\mathcal {I}}^{q+1}}{\mathcal {O}}^\ell (P_{I}). \end{aligned}$$

(Set ${\mathcal {O}}^\ell (P_{I})=0$ when $U_I^{ r^*}$ is empty.) We recall that $P_I=\Delta ^{ r^*}_{n_q}$ is independent of the order of multi-indices. Thus

$$\begin{aligned} C^q({\mathcal {U}})\cong ({\mathcal {O}}(\Delta ^{ r^*}_{n_q}))^{L}:={\mathcal {O}}^{L}(\Delta ^{ r^*}_{n_q}). \end{aligned}$$

Here $L\le |{\mathcal {I}}^{q+1}|\ell$. Let ${\mathcal {O}}_h(\Delta ^{r}_{n_q})$ be the space of holomorphic functions on $\Delta ^{r}_{n_q}$ with finite $L^2$ norm on $\Delta ^{r}_{n_q}$. Set $P_I^r=\Delta ^{r}_{n_q}$ for $I\in {\mathcal {I}}^{q+1}$. We define a Hilbert space

$$\begin{aligned} C^q_h({\mathcal {U}}^{r}):=\prod _{I\in {\mathcal {I}}^{q+1}}{\mathcal {O}}_h^\ell (P^{r}_{I}):={\mathcal {O}}_h^{ L}(\Delta ^{r}_{n_q}), \end{aligned}$$

which is a subspace of $C^q({\mathcal {U}}^{r})$.

Using the collection $\epsilon =\{\epsilon _I:I\in {\mathcal {I}}^{q+1}\}$, we define

$$\begin{aligned} {\mathcal {C}}_h ^q({\mathcal {U}}^{r},E'\otimes E'' ):=\epsilon (C_h ^q({\mathcal {U}}^{r}))\cong C_h ^q({\mathcal {U}}^{r})/(\ker \epsilon \cap C_h ^q({\mathcal {U}}^{r})), \end{aligned}$$

which is the vector space of q-cochains, equipped with the standard coboundary operator $\delta$.

Remark A.8

Our cochains are not necessary alternating. As in [15, p. 35], we let ${\mathcal {C}}_a ^q({\mathcal {U}},E'\otimes E'' )$ denote alternating cochains. For the isomorphism of the two kinds of Cečh cohomology groups; see [15, p. 35] and Serre [39]. Since we are interested in the cohomological solutions with bounds, we fix our notation without requiring that the cochains be alternating.

Let $\Vert \cdot \Vert _{\Delta _{n_q}^r}$ be the Hilbert space norm on ${\mathcal {C}}_h ^q({\mathcal {U}}^{r})$ and set

$$\begin{aligned} \Vert \zeta \Vert _{{\mathcal {U}}^{r}}^\bullet =\inf \{\Vert v\Vert _{\Delta _{n_q}^{r}}:v\in C_h ^q({\mathcal {U}}^{r}),\epsilon (v)=\zeta \}, \quad \zeta \in C_h ^q({\mathcal {U}}^{r},E'\otimes E'' ). \end{aligned}$$

The inclusion ${\mathcal {C}}_h ^q({\mathcal {U}}^{r},E'\otimes E'' )\hookrightarrow {\mathcal {C}} ^q({\mathcal {U}}^{r},E'\otimes E'' )$ is continuous and compact ([15, Thm. 3, p. 197]). We also define

$$\begin{aligned}&Z_h ^q({\mathcal {U}}^{r}):=\epsilon ^{-1}(Z_h ^q({\mathcal {U}}^{r}, E'\otimes E'' )),\\&\Vert \zeta \Vert _{{\mathcal {U}}^{r}}:=\inf \{\Vert v\Vert _{\Delta ^{n_q}_{r}}:v\in Z_h ^q({\mathcal {U}}^{r}),\epsilon (v)=\zeta \}, \quad \forall \zeta \in Z_h ^q({\mathcal {U}}^{r},E'\otimes E'' ),\\&\overline{v}:=\epsilon (v). \end{aligned}$$

Then $Z_h ^q({\mathcal {U}}^{r},E'\otimes E'' )$ is an isometric subspace of ${\mathcal {C}}_h ^q({\mathcal {U}}^{r},E'\otimes E'' )$ via inclusion. Let $\{g_0,g_1,\dots \}$ be a monotone orthogonal base of $Z_h^1({\mathcal {U}}^r)$ ([15, p. 141, p. 201]). An important feature of the monotone base is that the vanishing orders of $g_j$ at the origin satisfy

$$\begin{aligned} {\text {ord}}_0 g_0\le {\text {ord}}_0g_1\le \cdots , \quad \lim _{i\rightarrow \infty }{\text {ord}}_0g_i=\infty . \end{aligned}$$

By [15, Thm. 1, p. 192 and p. 201], for a given $\nu$ there is an $\mu$ such that

$$\begin{aligned}&g_i(Z)=O(|Z|^\nu ),\quad i>\mu , \quad Z\in \Delta _r^{n_q}. \end{aligned}$$

(A.24)

In fact, let the index set be ${\mathcal {I}}=\{1,\dots , L\}$. Set $\omega ((f_1,\dots , f_L))=\min \{(\alpha , Q):f_{\alpha ,Q}\ne 0\}$ by using order < on ${\mathcal {I}}\times \mathbf{N}^m$ defined by $(\alpha , P)<(\beta ,Q)$ if $| P|<|Q|$, or if $| P|=|Q|$ and there is an $\ell$ such that $p_\ell <q_\ell$ and $p_{\ell '}=q_{\ell '}$ for all $\ell '>\ell$, or if $P=Q$ and $\alpha <\beta$. Then the basis $\{g_j\}$ satisfies

$$\begin{aligned} \omega (g_j)<\omega (g_{j+1}). \end{aligned}$$

We now return to the case $q=1$ with $n_q=2n$. In the sequel, $\{|t_{kj}'|\}=\{|t_{kj}'|\}_{{\mathcal {U}}^{r^*}}$ and $\{|t_{kj}''\}|=\{|t_{kj}''|\}_{{\mathcal {U}}^{r^*}}$.

Theorem A.9

(Donin-Grauert-Remmert). Let C be a compact complex manifold and let ${\mathcal {U}}^{r}$ $(r_*<r<r^*<1)$ be a family of open coverings of C as in Lemma A.6 such that (A.11) holds for all k, j. Let $E=E'\otimes E''$ be a holomorphic vector bundle of positive rank m over C and fix a holomorphic base $e_j'$ (resp. $e_j'')$ for $E'$ (resp. $E'')$ over $U_j^{ r^*}$. Suppose that $r_*<r''<r'<r< r^*$ and $r'-r''\le r^*-r$. Let $\theta =r'/r$. Let $\{g_0,g_1,\dots \}$ be a monotone orthogonal base of $Z_h^1({\mathcal {U}}^r)$ as above. Assume that $\mu ,\nu$ satisfy (A.24) and

$$\begin{aligned}&t:=\frac{C_{n}\kappa ^{-1} }{(r'-r'')(r-r')^{2n}}\theta ^{\nu }<1/2. \end{aligned}$$

(A.25)

There exist $\overline{g_{m_0}},\dots , \overline{g_{m_{\mu ^*}}}$ such that their equivalence classes in $H^1({\mathcal {U}}^r,E)$ form a $\mathbf{C}$-linear basis of subspace spanned by $\overline{g_{0}},\cdots , \overline{g_{\mu }}$ in $H^1({\mathcal {U}}^r,E)$. For any $f\in Z_h^1({\mathcal {U}}^{r'},E)$ there exists $v\in C_h^0({\mathcal {U}}^{r''},E )$ satisfying $f=\delta v+\sum _0^{ \mu ^*} c_i \overline{g_{m_i}}$ with

$$\begin{aligned}&|c_i| \le \frac{ C_{n}\kappa ^{-1}A_{r}(E)}{r-r'}\Vert f\Vert _{{\mathcal {U}}^{r'}}, \end{aligned}$$

(A.26)

$$\begin{aligned}&\Vert v\Vert _{{\mathcal {U}}^{r''}}\le \frac{C_{n}\kappa ^{-1} B_{ r_-}(E)}{r-r'}\Vert f\Vert _{{\mathcal {U}}^{r'}}, \quad \forall r_- \in [r',r), \end{aligned}$$

(A.27)

$$\begin{aligned}&g_j=\sum _{i=0}^{\mu ^*} c_{ji}\overline{g_{m_i}}+\delta \eta _j^*, \quad \eta _j^*\in C^0({\mathcal {U}}^r,E), \end{aligned}$$

(A.28)

$$\begin{aligned}&A_r(E)= |\{t_{kj}'\}||\{t_{kj}''\}| \max _{0\le i\le \mu _*}\sum _{j=0}^{ ^{\mu }} |c_{ji}|, \quad B_{ r_-}(E)= |\{t_{kj}'\}||\{t_{kj}''\}| \sum _{j=0}^{ \mu } \Vert \{ \eta _j^*\}\Vert _{{\mathcal {U}}^{ r_-}}. \nonumber \\ \end{aligned}$$

(A.29)

Furthermore, all $c_j=0$ when $f=0$ in $H^1(C, E)$.

Remark A.10

The solution operator $f\rightarrow v$ may not be linear. See a proof by Donin [9] to get a linear solution operator for which the constant $C_*$ results from a lemma of Schwartz.

Remark A.11

The previous theorem gives a solution v, defined on a smaller domain, to the equation $f=\delta v$ (i.e cohomological equations) whenever f is 0 in the first cohomology group. It also provides a bound of the solution in terms of the data. We emphasize that this bound depends on the bundle $E'\otimes E''$. In the applications we have in mind, we will have to consider a sequence of bundles $\{ S^m E''\}_m$, and we will need to control the growth of these bounds as m goes to infinite, similarly to the small divisors appearing in local dynamical systems.

Proof

Recall that $q=1$ and $n_1=2n$. We may assume that $\Vert g_j\Vert _{\Delta _{2n}^r} =1$. By the definition of $\mu ,\nu$ and the monotone basis, we have for any $v\in Z^1_h({\mathcal {U}}^r)$,

$$\begin{aligned} \Vert v-\sum _{j=0}^{\mu }(v,g_j)g_j\Vert _{\Delta _{2n}^{r'}}\le \frac{C_n}{(r-r')^{2n}}(r'/r)^{\nu }\Vert v\Vert _{\Delta _{2n}^r} \end{aligned}$$

(A.30)

where $C_n(r-r')^{-2n}$ is the constant M in [15, Thm. 6, p. 191].

Replacing the smoothing lemma in [15, p. 200] by Theorem A.7, we derive some estimates following the proof of the finiteness lemma in [15, p. 201]. By assumption, we have

$$\begin{aligned} t=\frac{ C_{n}\kappa ^{-1} }{ (r'-r'')(r-r')^{2n}}\theta ^{\nu }<1/2,\quad \theta =\frac{r'}{r}<1. \end{aligned}$$

Let $\zeta _0 :=f\in Z_h^1({\mathcal {U}}^{r'},E'\otimes E'' )$. By Theorem A.7, we have for some $\xi _0\in Z_h^1({\mathcal {U}}^{r},E'\otimes E'' )$

$$\begin{aligned}&\zeta _0=\xi _0+\delta \eta _0,\\&\Vert \xi _0\Vert _{{\mathcal {U}}^{r}}\le t'\Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}, \quad \Vert \eta _0\Vert _{{\mathcal {U}}^{r''}}\le t'\Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}, \end{aligned}$$

with $t':=\frac{ C_{n} |\{t_{kj}'\}||\{t_{kj}''\}| }{\kappa (r'-r'')}$. Let $\overline{v}$ denote $\epsilon (v)$. Then $\xi _0=\overline{v}_0$ for some $v_0$ satisfying $\Vert v_0\Vert _{\Delta _{2n}^r}=\Vert \xi _0\Vert _{{\mathcal {U}}^r}$; see [15, p. 198]. Consider

$$\begin{aligned} w_1=v_0-\sum _{j=0}^{\mu }(v_0,g_j)_{\Delta ^r_{2n}}g_j, \quad \zeta _1=\overline{w}_1. \end{aligned}$$

According to (A.30), we have

$$\begin{aligned} \Vert \zeta _1\Vert _{{\mathcal {U}}^{r'}}\le \Vert w_1\Vert _{{\mathcal {U}}^{r'}}\le \frac{C_n}{(r-r')^{2n}}(r'/r)^{\nu }\Vert v_0\Vert _{\Delta ^r_{2n}}\le t\Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \zeta _0=\sum _{j=0}^{\mu }(v_0,g_j)_{\Delta ^r_{2n}}\overline{g}_j+\delta \eta _0+\zeta _1. \end{aligned}$$

In general, we have

$$\begin{aligned}&\zeta _\ell =\sum _{j=0}^{\mu }(v_\ell ,g_j)_{\Delta ^r_{2n}}\overline{g}_j+\delta \eta _\ell +\zeta _{\ell +1},\\&\Vert v_\ell \Vert _{\Delta ^r_{2n}}=\Vert \xi _\ell \Vert _{{\mathcal {U}}^r}\le t't^\ell \Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}},\\&\Vert \zeta _{\ell +1}\Vert _{{\mathcal {U}}^{r'}}\le t\Vert \zeta _\ell \Vert _{{\mathcal {U}}^{r'}}\le t^{\ell +1}\Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}},\\&\Vert \eta _\ell \Vert _{{\mathcal {U}}^{r''}}\le t' t^\ell \Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

Then we have

$$\begin{aligned}&f=\zeta _0=\sum _{j=0}^{\mu }\sum _{\ell =0}^\infty (v_\ell ,g_j)_{\Delta ^r_{2n}}\overline{g}_j+\delta \sum _{\ell =0}^\infty \eta _\ell ,\\&\sum _{\ell =0}^\infty |(v_\ell ,g_j)|\le \sum _{\ell =0}^\infty \Vert v_\ell \Vert _{\Delta ^r_{2n}}\le \frac{t'}{1-t}\Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}},\\&\sum _{\ell =0}^\infty \Vert \eta _\ell \Vert _{{\mathcal {U}}^{r''}}\le \frac{t'}{1-t} \Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

So far we have followed the proof of the finiteness lemma in [15, p. 201]. We now finish the proof of the theorem. Let us first find the linearly independent elements $\overline{g_{i_0}},\dots , \overline{g_{i_{\mu _*}}}$. Assume first that all $\overline{g_{i}}=0$ in $H^1:=H^1({\mathcal {U}}^r,E'\otimes E'')$. Then $\delta \eta _j=\overline{g_j}$ with $\eta _j\in C^0({\mathcal {U}}^r,E)$. Assume now that $\overline{g_{m_0}}\ne 0$ in $H^1$ for some $m_0$. Then we have two cases again: either $\overline{g_i}=c_{i0}\overline{g_{m_0}}+\delta \eta _{i}$ on ${\mathcal {U}}^r$ for all $i\in \{0,\dots , \mu \}\setminus m_0$, or it fails for some $m_1$. We repeat this to exhaust all elements so that

$$\begin{aligned} \overline{g_{j}}=\delta \eta _j^*+\sum _{i=0}^{\mu _*}c_{ji}\overline{g_{m_i}}, \quad \eta _{j}^*\in {\mathcal {C}}^0({\mathcal {U}}^r,E),\quad 0\le j\le \mu \end{aligned}$$

(A.31)

while $\overline{g_{m_0}},\dots ,\overline{g_{m_{\mu _*}}}$ are linearly independent in $H^1$. (Note that the above expression means the trivial identity $\overline{g_j}=\overline{g_j}$ when j is not in $\{m_0,\dots , m_{\mu ^*}\}$.) We have obtained (A.28) with the decomposition

$$\begin{aligned}&f=\sum _{j=0}^{\mu ^*}c_j\overline{g_{m_j}}+\delta v,\\&c_j=\sum _{\ell =0}^\infty (v_\ell ,g_j)_{\Delta ^r_{2n}}+\sum _{i=0}^\mu c_{ij}\sum _{\ell =0}^\infty (v_\ell ,g_i)_{\Delta ^r_{2n}},\\&v=\sum _{i=0}^\mu \sum _{\ell =0}^\infty (v_\ell ,g_i)_{\Delta ^r_{2n}}\eta ^*_i +\sum _{\ell =0}^\infty \eta _\ell . \end{aligned}$$

The solution $\eta _j^*$ in (A.31) can be bounded in $U^{r_-}$ for any $r_-<r$. Of course we need to estimate $\eta _j^*$ on ${\mathcal {U}}^{r'}$. Thus, $r_-\ge r'$. We have

$$\begin{aligned}&\sum _{j=0}^\mu \sum _{\ell =0}^\infty |(v_\ell ,g_j)_{\Delta ^r_{2n}}c_{ji}|\le \frac{t'}{1-t}\sum _{j=0}^\mu |c_{ji}| \Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}},\\&\left\| \left\{ \sum _{\ell =0}^\infty \eta _\ell +\sum _{\ell =0}^\infty \sum _{j=1}^{\mu } (v_\ell ,g_j)_{\Delta ^r_{2n}}\eta ^*_j\right\} \right\| _{{\mathcal {U}}^{r_-}}\le \frac{t'}{1-t} \left\{ 1+\sum _{j=0}^\mu \Vert \eta _j^*\Vert _{{\mathcal {U}}^{r_-}} \right\} \Vert \zeta _0\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

Set $A_r(E)= |\{t_{kj}'\}||\{t_{kj}''\}| \max _{i=0}^{\mu ^*}\sum _{j=0}^\mu |c_{ji}|$ and $B_{r_-}(E)= |\{t_{kj}'\}||\{t_{kj}''\}| (1+\sum _{j=0}^\mu \Vert \eta _j^*\Vert _{{\mathcal {U}}^{r_-}})$. We have obtained the required estimates.

Finally, let us assume that $f=0$ in $H^1(C,E)$ to show that all $c_j=0$ and thus $f=\delta v$. Since each $U^{r''}$ is Stein, we also have $f=0$ in $H^1({\mathcal {U}}^r,E)$. Thus $f=\delta {\tilde{v}}$ with ${\tilde{v}}\in {\mathcal {C}}^0(U^{r''},E)$. We get $\delta ({\tilde{v}}-v)=\sum _{j=0}^{\mu ^*} c_j\overline{g_{m_j}}$. By the linear independence, we conclude that $c_j=0$. We are done.$\square$

Theorem A.12

Let C be a compact complex manifold and let ${{\mathcal {U}}}^{r}$ $( r_*\le r\le r^*<1)$ be nested coverings of C as in Proposition A.19. Let $\mu ,\nu , r,r',r'', r_*,r^*$ be given in Theorem A.9, which satisfy (A.25). Let $f\in Z^1({\mathcal {U}}^{r'}, E'\otimes E'')$. Suppose that $f=0$ in $H^1(C,E'\otimes E'')$. Then there exists a solution $\{u_j\}\in C^0({\mathcal {U}}^{r'}, E'\otimes E'')$ such that $\delta u=f$ and

$$\begin{aligned}&\Vert u\Vert _{{\mathcal {U}}^{r'}}\le K(E'\otimes E'') \Vert f\Vert _{{\mathcal {U}}^{r'}}, \end{aligned}$$

(A.32)

$$\begin{aligned}&K(E'\otimes E''):=C(|\{t'_{kj}\}|_{{\mathcal {U}}^{r'}}+K_*(E'\otimes E'') |\{t'_{kj}\} |_{{\mathcal {U}}^{r'}}|\{t''_{kj}\}|_{{\mathcal {U}}^{r'}}), \end{aligned}$$

(A.33)

where $K_*(E'\otimes E'')$, defined by (A.8), satisfies

$$\begin{aligned} K_*(E'\otimes E'') \le \frac{C_nB_{ r_-}(E'\otimes E'')}{(r-r')\kappa }, \end{aligned}$$

(A.34)

where $\kappa$ and $B_{r_-}$ are defined by (A.10) and (A.29). The same conclusion holds if both sides are in sup norms $|\cdot |_{{\mathcal {U}}^{r'}}$, when $(r-r')\kappa$ is replaced by $((r-r')\kappa )^n$.

Remark A.13

The main conclusion is that (A.32) holds without shrinking the covering $\{U_i^{r'}\}$ on which f is defined. The solution operator $f\mapsto u$ may not be linear. The small divisor conditions are carried by $B_{ r_-}$ which is determined by (A.25) and (A.29), while the bounds in Theorem A.7 as smoothing lemma do not involve small divisors.

Proof

By the Leray theorem, we know that $[f]=0$ in $H^1({\mathcal {U}}^{r'},E)$. By Theorem A.9, we have a solution $u\in C^0({\mathcal {U}}^{r''},E)$ so that

$$\begin{aligned}&f_{jk}=(\delta u)_{jk}, \quad U_j^{r''}\cap U_k^{r''},\\&\Vert u\Vert _{{\mathcal {U}}^{r''}}\le K\Vert f\Vert _{{\mathcal {U}}^{r'}}. \end{aligned}$$

Then the conclusion follows from Lemma A.2.

When the super norm is used, we first obtain a solution $u=\{u_k\}$ for ${\mathcal {U}} ^{r^*}$ for $r^*=(r''+r')/2$, while (A.34) takes the form

$$\begin{aligned} \Vert u\Vert _{{\mathcal {U}}^{r^*}}\le K\Vert f\Vert _{{\mathcal {U}}^{r'}}\le ( \sqrt{\pi }r')^nK|f|_{{\mathcal {U}}^{r'}}. \end{aligned}$$

By ${\text {dist}}( \varphi _k(U_k^{r''}),\partial \varphi _k(U_k^{r^*}))= r^*-r''$ and power series expansion, we have $|u|_{{\mathcal {U}}^{r''}}\le (\sqrt{\pi }(r^*-r''))^{-n}\Vert u\Vert _{{\mathcal {U}}^{r^*}}.$ Then the conclusion follows from Lemma A.2 again. $\square$

1.5 Existence of Nested Coverings

In this subsection, our main goal is to construct nested coverings using transversality theorems and analytic polyhedrons. We recall that $C_n$ is an n-dimensional compact complex manifold. We shall omit to mention its dimension in what follows.

We first deal with the transversality for a piecewise smooth boundary of an analytic polyhedron and we then define the general position property of several analytic polyhedrons.

Definition A.14

• (a) Let $M_j$ be a $C^1$ real hypersurface defined by $r_j=0$, where $r_j$ is a $C^1$ function in an open set $\omega _j$ of a complex manifold C and $dr_j\ne 0$ on $M_j$. We say that $M_1,\dots , M_N$ are in the general position, if $dr_{i_0}\wedge \cdots \wedge dr_{i_q}\ne 0$ at each point of $M_{i_0}\cap \cdots \cap M_{i_q}$ for any $1\le i_0<\dots <i_q\le N$.

• (b) Let $\omega$ be a proper open set of a complex manifold C and let $f\in {\mathcal {O}}^N(\omega )$. We say that

  $$\begin{aligned} Q:=Q_N(f,\omega ):=\{z\in \omega \ \vert \;|f(z)|:=\max \{ |f_1(z)|,\dots ,| f_N(z)|\}< 1\}\nonumber \\ \end{aligned}$$

  (A.35)

  is an analytic N-polyhedron in $\omega$ if Q is non-empty and relatively compact in $\omega$, and Q does not contain any compact connected component. We say that Q is generic, if

  $$\begin{aligned} (d|f_{i_1}|\wedge \cdots \wedge d|f_{i_\ell }|)(x)\ne 0 \quad \forall x\in \{|f_{i_1}|=\cdots =|f_{i_\ell }|=1\}\cap \partial Q\nonumber \\ \end{aligned}$$

  (A.36)

  for all $i_1<\dots <i_\ell$ and $1\le \ell \le N$.

We will apply transversality theorems. This requires us to use open submanifolds in $\mathbf{C}^n$ which may not be closed in $\mathbf{C}^n$. Since $Q_N=Q_N(f,\omega )$ does not contain compact connected component, the closure of each connected component of $Q_N$ must intersect some $Q_N^i:=\{|f_i|=1\}\cap \omega$. We will call $Q_N^i$ a face of $Q_N$. Removing each $Q_N^i$ from $\omega$ if it does not intersect $\overline{Q}_N$, we get a new $\omega$ such that $\overline{Q}_N$ intersects each $Q_N^i$. Applying the same procedure to $Q_N^{i_1\dots i_k}:=Q_N^{i_1}\cap \cdots \cap Q_N^{i_k}$, we may assume that the non-empty intersection of any number of $Q_N^1,\dots Q_N^N$ intersects $\overline{Q}_N$. By (A.36), the closed set $\overline{Q}_N$ does not intersect the closed subset of $\omega$ defined by

$$\begin{aligned} (d|f_{i_1}|\wedge \cdots \wedge d|f_{i_\ell }|)(x)=0 \quad |f_{i_1}|(x)=\cdots =|f_{i_\ell }|(x)=1. \end{aligned}$$

Removing the above sets from $\omega$, we find a neighborhood $\omega ^*$ of $\overline{Q}_N$ such that if $Q_N^{i_1\dots i_k}$ with $i_1<i_2<\cdots <i_k$ intersects $\omega ^*$, then it intersects $\overline{Q}_N$ and it is a codimension k smooth submanifold in $\omega ^*$. For brevity we will call $\omega ^*$ a neat neighborhood of Q. We will take $\omega =\omega ^*$ without specifying $\omega ^*$.

Definition A.15

Let $\omega _i$ be open sets in C. For $i=0,\dots , p$, assume that $\phi _i\in {\mathcal {O}}^{N_i}(\omega _i)$ and $Q_{N_i}:=Q_{N_i}(\phi _i,\omega _i)$ is an analytic polyhedron in $\omega _i$. We say that they are in the general position, if all faces $Q_{N_i}^j$ of $Q_{N_i}$ for $1\le j\le N_i$ and $0\le i\le p$ are in general position. More precisely, $\omega ^*_{N_i}\cap Q_{N_i}^j$ are in the general position, where each $\omega ^*_i$ is a neat neighborhood of $\overline{Q_{N_i}}$.

Let us describe some elementary properties of generic analytic polyhedrons. If $Q_N(f,\omega )$ is defined in $\omega$ by (A.35), we denote for $\rho =(\rho _1,\dots ,\rho _N)$

$$\begin{aligned} Q_N^\rho (f,\omega ):=\{z\in \omega :|f_j(z)|<\rho _j, j=1,\dots ,N\}. \end{aligned}$$

Lemma A.16

Let $Q_{N_i}=Q_{N_i}(\phi _i,\omega _i)$ be generic polyhedrons in C for $0\le i\le p$. Suppose that $Q_{N_0},\cdots , Q_{N_p}$ are in the general position. Then

$$\begin{aligned} Q_{N_0+\cdots +N_p}((\phi _0,\dots ,\phi _p),\omega _0\cap \cdots \cap \omega _{p})= Q_{N_0}\cap \cdots \cap Q_{N_p}, \end{aligned}$$

if non-empty, is a generic $N_0+\cdots +N_p$ analytic polyhedron in $\omega _{0\cdots p}:=\omega _{0}\cap \cdots \cap \omega _{p}$.

Proof

Let $N=N_0+\cdots +N_p$. It is clear that $Q:=Q_{N_0}\cap \cdots \cap Q_{N_p}=Q_{N}((\phi _0,\dots , \phi _{p}), \omega _{i_0\cdots i_p})$. Since $\overline{Q}\subset \cap \overline{Q_{N_i}}$, then $\overline{Q}$ is compact in $\omega _{0\cdots p}$. Write $(\phi _0,\dots , \phi _p)=(\psi _1,\cdots , \psi _N)$. Suppose that $x\in \partial Q$. Since $\overline{Q}$ is compact in $\omega$, then there exist $\mu _1<\cdots <\mu _m$ with $m\ge 1$ such that $|\psi _{\mu _i}(x)|=1$ and $|\psi _{j}(x)|<1$ for $j\ne \mu _\ell$. By the assumption of the general position, we see that the faces of Q are in the general position. $\square$

Let X, Y be smooth real manifolds without boundary and W a smooth submanifold of Y. Following [11, p. 50], we say that a smooth mapping $h:X\rightarrow Y$ is transversal to W at $x\in X$, denoted by at x, if either $h(x)\not \in W$ or

$$\begin{aligned} T_{h(x)}W+dh(T_xX)=T_{h(x)}Y. \end{aligned}$$

Denote on A if at each $x\in A\subset X$. When h is the inclusion, we denote on A by on A. Finally, extending Definition A.14 (a), we say that smooth real submanifolds $W_0,\dots , W_k$ in Y are in the general position if for any $0\le i_1<\cdots <i_m\le k$ we have

$$\begin{aligned} \bigwedge _{\ell =1}^{k}\bigwedge _{j=1}^{d_{i_\ell }} dr_{i_\ell ,j} (y)\ne 0, \quad \forall y\in W_{i_1}\cap \cdots \cap W_{i_m}, \end{aligned}$$

(A.37)

where $W_i\subset \omega _i$ is defined by $r_{i,1}=\cdots =r_{i,d_i}=0$ with $dr_{ i,1}\wedge \cdots \wedge dr_{i,d_i}\ne 0$ at each point of $W_i$. Thus $d_i$ is the codimension of $W_i$ in $\omega _i$. It is clear that (A.37) holds if and only if

(A.38)

For an analytic N-polyhedron $Q_N$ in $\omega$ with faces $Q_N^1,\dots , Q_N^N$, we call $Q_N^{i_1\cdots i_k}=Q_N^{i_1}\cap \cdots \cap Q_N^{i_k}$ with $i_1<\cdots <i_k$ and $k\ge 1$ an edge of Q. When $Q_N$ is generic, a nonempty edge $Q_N^{i_1\cdots i_k}$ is a codimension k submanifold in $\omega$. Let $\{Q_N^1\cdots , Q_N^{N'}\}$ be the set of all edges, with the first N edges being the faces.

Proposition A.17

Let $Q_{N_i}=Q_{N_i}(\phi _i,\omega _i)$ be generic polyhedrons in C for $0\le i\le p$ with $\omega _i$ being a neat neighborhood of $\overline{Q}_{N_i}$. Then $Q_{N_0},\dots , Q_{N_p}$ are in the general position if and only if for all $0\le i_1<\dots <i_k\le p$ and $1\le j_\ell \le N'_{i_\ell }$, the edges $Q^{j_1}_{N_{i_1}}, \cdots , Q^{j_k}_{N_{i_k}}$ are in the general position. Equivalently, each edge $Q_{N_\ell }^s$ intersects transversally with each edge of the intersection of any number of $Q_{N_0},\dots , Q_{N_{\ell -1}}$, for $\ell =1,\dots , p$.

Proof

Since each edge of a polyhedron is the intersection of its faces, it is clear that if $Q_{N_0},\dots , Q_{N_p}$ are in the general position, then the edges $Q^{j_1}_{N_{i_1}}, \cdots , Q^{j_k}_{N_{i_k}}$ are in the general position for $0\le i_1<\cdots <i_k\le p$.

Conversely, let $\phi _i=(\phi _{i,1},\dots , \phi _{i,N_i})$ and let $\psi _1,\dots , \psi _m$ be a subset of $\phi _{0,1},\dots ,$ $\phi _{0, N_0}, \dots ,$ $\phi _{p,1},\dots$, $\phi _{p,N_p}$. We emphasize that we do not assume that the latter are distinct functions, although $\phi _{i,1}, \dots , \phi _{i, N_i}$ are distinct by the general position property of the faces of $Q_{N_i}$. Suppose that $\psi _\ell$ is in $\{\phi _{i_\ell ,1},\dots , \phi _{i_\ell ,N_{i_\ell }}\}$. We need to show that

$$\begin{aligned} d|\psi _1|\wedge \cdots \wedge d|\psi _m|(x)\ne 0 \end{aligned}$$

(A.39)

if for all $\ell$, $|\psi _\ell |(x)=1$ and $x\in \overline{Q}_{N_{i_\ell }}$. Without loss of generality, we may assume that $i_1\le i_2\le \cdots \le i_m$. Thus

$$\begin{aligned} (\psi _1,\dots , \psi _m)=({{\tilde{\psi }}}_{\alpha _1},\dots , {{\tilde{\psi }}}_{\alpha _\ell }), \quad \alpha _1<\alpha _2<\cdots <\alpha _\ell \end{aligned}$$

with ${{\tilde{\psi }}}_{\alpha _\beta }$ being a non-empty subset of components of $\phi _{\alpha _\beta }$. Without loss of generality, we may assume that ${{\tilde{\psi }}}_{\alpha _\beta }=(\phi _{\alpha _\beta ,1},\dots ,\phi _{\alpha _\beta ,\gamma _\beta })$ with $\gamma _\beta >0$. Thus $|\phi _{\alpha _\beta ,1}|=\cdots =|\phi _{\alpha _\beta ,\gamma _\beta }|=1$ define an edge $W_{\alpha _\beta }$ of $Q_{\alpha _\beta }$. Then (A.39) is equivalent to

$$\begin{aligned} \left( \bigwedge _{\delta =1}^{\gamma _\ell }d|\phi _{\alpha _\ell , \delta }|\right) \wedge \left( \bigwedge _{\ell '=1}^{\ell -1}\bigwedge _{\delta =1}^{\gamma _{\ell '}}d|\phi _{\alpha _{\ell '},\delta }|\right) (x)\ne 0. \end{aligned}$$

The equivalence of (A.37) and (A.38) implies that (A.39) follows from the assumption that , for $\alpha _1<\alpha _2<\cdots <\alpha _\ell$. $\square$

Lemma A.18

(Golubitsky-Guillemin [11, p. 53]). Let X, B,  and Y be smooth manifolds with W a submanifold of Y. Let $\psi :B\rightarrow C^\infty (X, Y)$ be a mapping (not necessarily continuous) and define $\Psi :X \times B \rightarrow Y$ by $\Psi (x, b) = \psi (b)(x)$. Assume that $\Psi$ is smooth and that . Then the set is dense in B.

Proposition A.19

Let C be a compact complex manifold of dimension n. Let $\{U_i:i=1,\dots , m\}$ be a finite open covering of C. Assume that $\varphi _j$ is a biholomorphism from a neighborhood $\omega _j$ of the star $N(U_j)$ of $U_j$ onto ${\hat{\omega }}_j\subset \mathbf{C}^n$ such that $U_j=\varphi _j^{-1}(\Delta _n)=Q_n(\varphi _j,\omega _j)$. There exists $\delta >0$ satisfying the following:

• (a) For each j, there are a relatively compact open set ${{\tilde{\omega }}}_j$ (resp. ${\tilde{U}}_j)$ in $\omega _j$ (resp. ${{\tilde{\omega }}}_j)$ and a dense open set $A_j$ of $\Delta ^\delta _n$ such that if $c_j\in A_j$, then ${{\tilde{\varphi }}}_j:=\varphi _j-c_j$ is a biholomorphic mapping from ${\tilde{U}}_j$ onto $\Delta _n$, and ${\tilde{U}}_1:=Q_n({{\tilde{\varphi }}}_1,{{\tilde{\omega }}}_1),\dots ,$ ${\tilde{U}}_m:=Q_n({{\tilde{\varphi }}}_m,{{\tilde{\omega }}}_m)$ are generic n-polyhedrons in the general position, where $\{{\tilde{U}}_1,\ldots {\tilde{U}}_m\}$ remains an open covering of C and ${{\tilde{\omega }}}_j$ is a neighborhood of $N({\tilde{U}}_j)$. In particular each ${{\tilde{\varphi }}}_j$, a translation of $\varphi _j$, is injective on ${{\tilde{\omega }}}_j$.

• (b) There is $0<r_*<1$ such that if $r_*\le \rho _i\le 1$, then ${\tilde{U}}_{i_0}^{\rho _0},\dots , {\tilde{U}}_{i_q}^{\rho _q}$ are generic n-polyhedrons in the general position, where ${\tilde{U}}_i^\rho :={{\tilde{\varphi }}}_i^{-1}(\Delta ^\rho _n)$.

Proof

(a) We will apply the transversality theorem for real submanifolds in $\mathbf{C}^n$. Therefore, we will use old coordinate charts $\varphi _j$ to map edges of polyhedrons $Q_j(\varphi _j,\omega _j)$ into $\mathbf{C}^n$. Set $c_1=0,{{\tilde{\varphi }}}_1=\varphi _1,{\tilde{U}}_1=U_1$. Let ${\hat{W}}_1,\dots , {\hat{W}}_{L_0}$ be all edges of $\Delta _n$. Let ${\tilde{U}}_1^1,\dots , {\tilde{U}}_1^{N'}$ be all edges of ${\tilde{U}}_1$. Set ${\widetilde{W}}^\ell _1=\varphi _2( \omega _2\cap {\tilde{U}}_{1}^{\ell })$. Define

$$\begin{aligned} \Psi :\mathbf{C}^n\times \Delta ^\delta _n\rightarrow Y:=\mathbf{C}^n \end{aligned}$$

with $\Psi (x,b)=x+b$ and $\psi ^b(x)=\Psi (x,b)$. Let $\psi ^b|_{{\widehat{W}}_{\ell '}}$ be the restriction of $\psi ^b$ to ${\hat{W}}_{\ell '}$. Applying Lemma A.18, mainly the density assertion in the lemma, finitely many times in which $W={\widetilde{W}}^\ell _1$, we can find $b_2\in \Delta _n^\delta$ such that

where $\omega _2'$ is a relatively compact open subset of $\omega _2$ which is independent of $\delta$, and $\overline{U_2}\subset \omega _2'$. We also remark that (A.18) can be applied for finitely many times since $\varphi _2(\overline{{\tilde{U}}_1}\cap \overline{\omega _2'})$ is compact. Since $\overline{{\tilde{U}}_1}\cap \overline{U_2}$ is compact, then

(A.40)

when $|c_2-b_2|$ is sufficiently small. Applying $\varphi _2^{-1}$ to (A.40) yields

(A.41)

With $c_2$ being determined, set

$$\begin{aligned} {{\tilde{\varphi }}}_2^{-1}=\varphi _2^{-1}({\text {I}}+c_2). \end{aligned}$$

Thus ${{\tilde{\varphi }}}_2=\varphi _2-c_2$. When $\delta$ and $|c_2-b_2|$ are sufficiently small, we have $\overline{{\tilde{U}}_2}={{\tilde{\varphi }}}_2^{-1}(\overline{\Delta _n})\subset \omega _2'$. Therefore, (A.41) implies that every edge of ${\tilde{U}}_2$ intersects each edge of ${\tilde{U}}_1$. We have determined ${\tilde{U}}_{2}={{\tilde{\varphi }}}_2^{-1}(\Delta _n)$.

We have verified (a) when $m=2$. Let us assume that it also holds for $m\ge j$. By Lemma A.16, each edge of a non-empty intersection of any number of ${\tilde{U}}_1,\dots , {\tilde{U}}_j$ is a smooth submanifold. We remark the above transversality argument mainly uses the fact that $\varphi _2$ is a biholomorphism, while each edge of ${\tilde{U}}_1$ is a smooth submanifold.

To repeat the above argument for $m=2$ in details, we list all edges of all possible intersections of ${\tilde{U}}_1,\dots , {\tilde{U}}_{j}$ as $W'_1,\dots , W'_L$ so that each $W_j$ is an edge of some analytic polyhedron $U_j'$, where $U_j'$ is the intersection of some of ${\tilde{U}}_1,\dots , {\tilde{U}}_{j'}$ which are in general position by the induction hypothesis as mentioned above. Therefore, by Lemma A.16, each $U_\ell '$ is generic. Now we are in the situation of $m=2$ by considering the sets of two analytic polyhedrons $\{ U_\ell ', U_{j+1}\}$ one by one for $\ell =1,\dots , j'$. Here $U_{j+1}=\varphi _{j+1}^{-1}(\Delta _n)$ with $\varphi _{j+1}$ being biholomorphic in a neighborhood $N(U_{j+1})$ of $U_{j+1}$. Therefore, we can find ${{\tilde{\varphi }}}_{j+1}=\varphi _{j+1}-c_{j+1}$ such that each edge of ${\tilde{U}}_{j+1}$ intersects each $W_\ell '$ transversally on $\overline{{\tilde{U}}_{j+1}}\cap \overline{U_\ell '}$.

The above argument shows the existence of $c_1,\dots , c_N$ in $\Delta ^\delta _n$ when $\delta$ is sufficiently small. The openness property on $A_j$ is clear, since by shrinking ${{\tilde{\omega }}}_j$ slightly the general position and generic properties are preserved under small perturbation of $c_j$. Then density of $A_j$ when $\delta$ is sufficiently small can also be achieved; indeed when $c_j$ is sufficiently small, we may shrink $\omega _j$ slightly and apply the above argument by replacing $\varphi _j-c_j$ with $\varphi _j$. Finally, $\{{\tilde{U}}_1,\dots , {\tilde{U}}_N\}$ still covers C when $\delta$ is sufficiently small. We have verified (a).

The assertion (b) follows from (a) and Proposition A.17. Indeed, we first note that when $r_*$ is less than 1, but it is sufficiently close to 1, the $\partial Q^\rho ({{\tilde{\varphi }}}_j)$ is in a given neighborhood of $\partial Q( {{\tilde{\varphi }}}_j,{{\tilde{\omega }}}_j)$, as $Q^\rho ({{\tilde{\varphi }}}_j,{{\tilde{\omega }}}_j)$ does not have any compact connected component. By the relative compactness of $Q_n({{\tilde{\varphi }}}_i,{{\tilde{\omega }}}_i)$, the condition (A.36) with $f_j$ being replaced by $f_j/{\rho _j}$ and the general position condition remain true when $\rho _j$ are in $[r_*,1]$ when $r_*<1$ is sufficiently close to 1. The proof is complete. $\square$

The following is a basic property of a generic analytic polyhedron.

Proposition A.20

Let C be a compact complex manifold of dimension n. Let $Q_N(f,\omega )$ be a generic analytic N-polyhedron C defined by (A.35) and (A.36). There exists $r_*\in (0,1)$ satisfying the following.

• (a) If $\rho =(\rho _1,\dots ,\rho _N)$ and $\rho '=(\rho _1',\dots ,\rho _N')$ satisfy $r_*\le \rho _i'\le \rho _i\le 1$, every connected component of $Q_N^{\rho }(f,\omega )$ intersects $Q_N^{\rho '}(f,\omega )$ and the latter is non-empty.

• (b) There are finitely many open sets $\omega _j''$ in C and smooth diffeomorphisms $\phi _j$ sending $\omega _j''$ onto ${\hat{\omega }}_j''$ in $\mathbf{R}^{2n}$ such that $\{\omega _j''\}$ covers $\partial Q_N(f,\omega )$, and for any $p_0,p_1\in \phi _j(\omega _j''\cap Q_N^{\rho }(f,\omega ))$ there is a smooth curve $\gamma$ in $\phi _j(\omega _j''\cap Q_N^{\rho }(f,\omega ))$ connecting $p_0$ and $p_1$ with length $|\gamma |\le C|p_1-p_0|$, where C depends only on $\phi _j$ and $\omega _j''$.

Proof

(a) Set $Q=Q_N(f,\omega )$ and $Q^\rho =Q_N^\rho (f,\omega )$. For each $x\in \partial Q$, we find $\mu _1<\dots <\mu _m$ with $m\le N$ such that

$$\begin{aligned}&|f_{\mu _i}(x)|=1, \quad i\le m; \quad |f_j(x)|<1, \quad j\ne \mu _1,\dots , \mu _m. \end{aligned}$$

(A.42)

Note that $\{\mu _{1},\dots ,\mu _{m}\}$ is uniquely determined by x. By the transversality condition (A.36), we have $m\le 2n$. Choose an open set $\omega '$ such that $x\in \omega '\subset \omega$ and

$$\begin{aligned} |f_i(z)|<1, \quad \forall z\in \overline{\omega '}, \ i\ne \mu _{1},\dots ,\mu _{m}. \end{aligned}$$

In particular, we have

$$\begin{aligned} Q\cap \omega '=\{z\in \omega ':|f_{\mu _i}(z)|<1, \quad i=1,\dots , m\}. \end{aligned}$$

By (A.36), we can take $( |f_{\mu _1}|,\dots , |f_{\mu _m}|)$ to be the first m components of a smooth diffeomorphism $\varphi :\omega '\rightarrow {\hat{\omega }}$, shrinking $\omega '$ if necessary. Taking a smaller open subset $\omega ''$ of $\omega '$ with $x\in \omega ''$, we may assume that

$$\begin{aligned} t\zeta \in {\hat{\omega }}, \quad \forall \zeta \in {\hat{\omega }}'':=\varphi (\omega ''), \quad 1-\delta \le t\le 1, \end{aligned}$$

for some $\delta \in (0,1]$.

Since $\partial Q$ is compact, there exists $\{x_j,\omega _j'', \omega _j':j=1,\dots , k\}$ satisfying the following:

• (a) The k is finite. For each j, we have that $x_j\in \omega ''_j\subset \omega _j'\subset \omega$, $x_j\in \partial Q$, and $\omega _j'$ is an open subset of $\omega$. For each j, we have $m_j$ and $\mu _{j,1}<\ldots <\mu _{j,m_j}$, which are the numbers associated to $x_j$, so that (A.42) holds for $x=x_j$. $\{\omega _1'',\dots \omega _k''\}$ is an open covering of $\partial Q$.

• (b) $|f_{\mu _{j,\ell }}(x_j)|=1$ for $\ell =1,\dots ,m_{j}$ and

  $$\begin{aligned}&M_j:=\sup _{z\in \overline{\omega _j'}}\{|f_i(z)|:i\ne \mu _{j,1}, \dots , \mu _{j,m_j}\}<1, \nonumber \\&\omega _j'\cap Q=\{z\in \omega _j':|f_{\mu _{j,\ell }}(z)|<1,\ell =1,\dots , m_j\}. \end{aligned}$$

  Here we set $M_j=0$ if $m_j=N$.

• (c) The $( |f_{\mu _{j,1}}|,\dots ,| f_{\mu _{j,m_j}}|)$ are the first $m_j$ components of a smooth diffeomorphism $\phi _{j}$ from $\omega _{j}$ onto a subset ${\hat{\omega }}_{j}$ of $\mathbf{C}^n$. There exists $\delta ^*>0$ such that ${\hat{\omega }}_{j}'':=\phi _j(\omega _j'')$ satisfies

  $$\begin{aligned} \{t\zeta :\zeta \in {\hat{\omega }}_{j}''\}\subset {\hat{\omega }}_{j},\quad \forall j,\forall t\in [1-\delta ^*,1]. \end{aligned}$$

  (A.43)

Indeed, let $\phi _j(x_j)=(1,\dots , 1,{\tilde{x}}_j)$ with ${\tilde{x}}_j\in \mathbf{R}^{2n-m_j}$. We can take

$$\begin{aligned} {\hat{\omega }}_j''=(1-\delta ^*,1+\delta ^*)^{m_j}\times B^{\delta ''}_{2n-m_j}({\tilde{x}}_j) \end{aligned}$$

(A.44)

where $B^{\delta ''}_{2n-m_j}({\tilde{x}}_j)$ is the ball in $\mathbf{R}^{2n-m_j}$ centered at ${\tilde{x}}_j$ with a sufficiently small radius $\delta ''$. Note that

$$\begin{aligned} \phi _j(Q^\rho \cap \omega _j'')=(1-\delta ^*,\rho _1)\times \cdots \times (1-\delta ^*,\rho _{m_j})\times B^{\delta ''}_{2n-m_j}({\tilde{x}}_j). \end{aligned}$$

(A.45)

Define

$$\begin{aligned} M^*=\sup \{|f(z)|:z\in Q\setminus \cup _{j=1}^k\omega _j''\}. \end{aligned}$$

Then $M^*<1$. By the maximum principle, we have $|f|\le M^*$ on $Q\setminus \cup _{j=1}^k\omega _j''$. Fix $r_*$ so that

$$\begin{aligned} 1>r_*>\max \{1-\delta ^*, M^*, M_1,\dots , M_k\}. \end{aligned}$$

Suppose that $r_*\le \rho _i'\le \rho _i\le 1$ for $i=1,\dots , N$. Let $\Omega$ be a connected component of $Q^\rho _N$. Since $\Omega$ does not have a compact connected component, there exists $z^*\in \partial \Omega$ satisfying $|f_{i}(z^*)|=\rho _{i}$ for some i. Since $\rho _i > M^*$, then $z^*\in \omega _j''$ for some j. Let us assume that $z^*\in \omega _1''$, and $(\mu _{1,1},\dots ,\mu _{1,m_1})=(1,\dots , {m_1})$. Thus $\phi _1=(|f_1|,\dots , |f_{m_1}|,{\tilde{f}}_{m_1+1},\dots ,{\tilde{f}} _{2n})$. We now replace $z^*$ by some $z_*\in \Omega \cap \omega _1''$. We consider a path defined by

$$\begin{aligned} t\rightarrow \gamma (t):=\phi _{1}^{-1}(t\phi _{1}(z_*)), \quad 1-\delta ^*\le t\le 1. \end{aligned}$$

Note that by (A.43), $\gamma$ is well defined and is contained in $\omega _1$. We now have

$$\begin{aligned} |f_\ell (\gamma (t))|= t|f_\ell (z_*)|\le t\rho _\ell , \quad \ell \le m_1. \end{aligned}$$

(A.46)

Since $\gamma (t)\in \omega _1$, we also have

$$\begin{aligned} |f_\ell (\gamma (t))|\le M_1< r_*, \quad \ell >m_1. \end{aligned}$$

(A.47)

This shows that $\gamma (t)\in Q^{\rho }_N$. Since $\Omega$ is a connected component of $Q^\rho _N$ and $\gamma (1)=z_*\in \Omega$, we must have $\gamma (t)\in \Omega$. By the definition of $M_j$, at $t=1-\delta ^*$ we have $t\rho _\ell \le 1-\delta ^*<\rho _\ell '$. Combining with (A.46)–(A.47), we get $\gamma (1-\delta ^*)\in Q_N^{\rho '}$.

(b) Since $p_0,p_1$ are in the same ${\hat{\omega }}_j''$, the assertion also follows from the above construction of ${\hat{\omega }}_j''$ via (A.44)–(A.45) and the convexity of ${\hat{\omega }}_j''$. $\square$

In summary, by Proposition A.19 we cover C by generic analytic n-polyhedrons $U_i=\varphi _i^{-1}(\Delta _n)$ ($i=1,\dots , m$), which are in the general position. By Lemma A.16, each $U_i\cap U_j$, if non-empty, is a generic analytic polyhedron. Applying Proposition A.20 (a) to all non-empty $U_i\cap U_j$, we know that $\{U_i^r=\varphi _i^{-1}(\Delta _n^r):i=1,\dots , m\}$ for $r_*\le r\le 1$ is a family of nested coverings. Therefore, we can apply Theorem A.9 and Theorem A.12.