The local Steiness problem with singularities

TL;DR

This paper proves that certain unbranched Riemann domains over Stein spaces are cohomologically q-complete under specific conditions, providing a positive answer to the local Steinness problem even with singularities.

Contribution

It generalizes previous results by establishing cohomological q-completeness for unbranched Riemann domains with singularities, extending the local Steinness problem to broader cases.

Findings

Proves cohomological q-completeness under specified conditions.

Extends known results to spaces with singularities.

Provides a positive solution to the local Steinness problem.

Abstract

In this article, we prove that if $\Pi: X\rightarrow \Omega$ is an unbranched Riemann domain with $\Omega$ Stein of dimension $n$ and $\Pi$ a locally $q$-complete morphism, then $X$ is cohomologically $q$-complete if $n\geq 3$ and $1\leq q\leq n-2$ or if $\Omega$ has dimension $2$ and $1\leq q\leq 2$. This generalizes a well-known result which is obtained in ~\cite{ref3} for $q=1$ when $X$ and $\Omega$ have isolated singularities and, gives in particular a positive answer to the local Steiness problem, namely if $X$ is a Stein space and $\Omega$ a locally Stein open subset of $X$, then $\Omega$ is Stein.

Equations

$$K\subset \{x\in \Omega :\varphi (x)<0\}\subset \subset D$$

$$H^p(\Omega ,\mathcal{F})⟶H^p(D,\mathcal{F})$$

$$H_{S_k}^p(V,\mathcal{F})⟸E_2^{p,q}=H^p(V,\underline{H_{S_k}^p}(\mathcal{F}))$$

$$H_S^p(X,\mathcal{F})⟸E_2^{p,q}=H^p(X,\underline{H_S^p}(\mathcal{F}))$$

$$\cdots \to H_S^p(X,\mathcal{F})\to H^p(X,\mathcal{F})\to H^p(X_c^{\prime },\mathcal{F})\to H_S^{p+1}(X,\mathcal{F})\cdots$$

$$\bigcup _{i\le j}{U}_i\cap U_{j+1}=\{x\in U_{j+1}:{\varphi }_{j+1}(x)<0\}$$

$$H^p(X_{j+1},\mathcal{F})\cong H^p(Y_{j+1},\mathcal{F})\cong H^p(Y_j,\mathcal{F})\oplus H^p(Y_{j+1}^{\prime },\mathcal{F})forallp\le r$$

$$0\to \mathcal{I}(Z^{\prime })\to {\mathcal{O}}_X\to {\mathcal{O}}_{Z^{\prime }}\to 0$$

Full text

1. Introduction

A holomorphic map $\Pi:X\rightarrow Y$ of complex spaces is said to be a locally $r$-complete morphism if for every $x\in Y$, there exists an open neighborhood $U$ of $x$ such that $\Pi^{-1}(U)$ is $r$-complete. When $r=1$, $\Pi$ is called locally $1$-complete or locally Stein morphism.

   In  [3] Coltoiu and Diederich proved the following

Theorem 1**.**

Let $X$ and $Y$ be complex spaces with isolated singularities and $\Pi:X\rightarrow Y$ an unbranched Riemann domain such that $Y$ is Stein and $\Pi$ is a locally Stein morphism. Then $X$ is Stein.

In this article, we prove that the same result follows if we assume only that $Y$ is an arbitrary Stein space.

   An immediate consequence of this result is the

Corollary 1**.**

Let $X$ be a Stein space, and let $\Omega\subset X$ be an open subset which is locally Stein in the sense that every point $x\in\partial{\Omega}$ has an open neighborhood $U$ in $X$ such that $U\cap\Omega$ is Stein. Then $\Omega$ is itself Stein.

2. Preliminaries

We start by recalling some definitions which are important for our purposes.

   Let $\Omega$ be an open set in $\mathbb{C}^{n}$ with complex coordinates $z_{1},\cdots,z_{n}$. Then it is known that a function $\phi\in C^{\infty}(\Omega)$ is $q$-convex if for every point $z\in\Omega$, there exists a complex vector subspace $E(z)$ of $\mathbb{C}^{n}$ of dimension at least $n-q+1$ such that the levi form $L_{z}(\phi,\xi)=\displaystyle\sum_{i,j}\frac{\partial^{2}\phi(z)}{\partial{z_{i}}\partial{\overline{z}_{j}}}\xi_{i}\overline{\xi}_{j}$ is positive definite at each point $\xi\in E(z)$.

   A smooth real valued function $\phi$ on a complex space $X$ is called $q$-convex if every point $x\in X$ has an open neighborhood $U$ isomorphic to a closed analytic set in a domain $D\subset\mathbb{C}^{n}$ such that the restriction $\phi|_{U}$ has an extension $\tilde{\phi}\in C^{\infty}(D)$ which is $q$-convex on $D$.

   We say that $X$ is $q$-complete if there exists a $q$-convex function $\phi\in C^{\infty}(X,\mathbb{R})$ which is exhaustive on $X$ i.e. $\{x\in X:\phi(x)<c\}$ is relatively compact for any $c\in\mathbb{R}.$

   The space $X$ is said to be cohomologically $q$-complete if for every coherent analytic sheaf ${\mathcal{F}}$ on $X$ the cohomology groups $H^{p}(X,{\mathcal{F}})$ vanish for all $p\geq q$.

   An open subset $D$ of $\Omega$ is called $q$-Runge if for every compact set $K\subset D$, there is a $q$-convex exhaustion function $\phi\in C^{\infty}(\Omega)$ such that

[TABLE]

It is shown in $[2]$ that if $D$ is $q$-Runge in $\Omega$, then for every ${\mathcal{F}}\in coh(\Omega)$ the cohomology groups $H^{p}(D,\mathcal{F})$ vanish for $p\geq q$ and, the restriction map

[TABLE]

has dense image for all $p\geq q-1$.

3. Main result

Lemma 1**.**

*Let $X$ and $Y$ be complex spaces and $\Pi:X\rightarrow Y$ an unbranched Riemann domain. Assume that there exists a smooth $q$-convex function $\phi$ on $Y$. Then, for any real number $c$ and every coherent analytic sheaf ${\mathcal{F}}$ on $X$ if $dih({\mathcal{F}})>q$ and $X^{\prime}_{c}=\{x\in X:\phi o\Pi(x)>c\},$ the restriction map $H^{p}(X,{\mathcal{F}})\rightarrow H^{p}(X^{\prime}_{c},{\mathcal{F}})$ is an isomorphism for $p\leq dih({\mathcal{F}})-q-1.$

Here $dih({\mathcal{F}})$ denotes the homological dimension of ${\mathcal{F}}$.*

Let $V$ be a closed analytic set in a domain $D\subset\mathbb{C}^{n}$ and $\phi\in C^{\infty}(V)$ a $q$-convex function. Let $\xi\in V$ and suppose we can find a $q$-convex function $\hat{\phi}\in C^{\infty}(D)$ with $\hat{\phi}|_{V}=\phi$ and that $n$ is equal to the dimension of the Zariski tangent space at $\xi$.

   Then in order to prove lemma $1$ we shall need the following result due to Andreotti-Grauert  [2].

Theorem 2**.**

For any coherent analytic sheaf ${\mathcal{F}}$ on $V$ with $dih({\mathcal{F}})>q,$ there exists a fundamental system of Stein neighborhoods $U\subset D$ of $\xi$ such that if $Y=\{z\in V:\phi(z)>0\}$, then $H^{p}(Y\cap U,{\mathcal{F}})=0$ for $0<p<dih_{\xi}({\mathcal{F}})-q$ and $H^{0}(U\cap V,{\mathcal{F}})\rightarrow H^{p}(U\cap Y,{\mathcal{F}})$ is an isomorphism.

Proof.

Let $\xi\in X$ such that $\phi o\Pi(\xi)=c,$ and let $V\subset\subset X$ be a hyperconvex open neighborhood of $\xi,$ biholomorphic by $\Pi$ to the open subset $W=\Pi(V)\subset Y$. We may take $V$ so that $W$ is biholomorphic to a closed analytic subset of a domain $D$ in $\mathbb{C}^{n}$ of minimal dimension and $\phi|_{W}$ extends to a smooth $q$-convex function in a neighborhood $W_{1}\subset D$ of $W$. Let $\psi:V\rightarrow]-\infty,0[$ be a continuous strictly plurisubharmonic function. Then it is clear that $\psi_{k}=\frac{1}{k}\psi+\phi o\Pi,$ $k\geq 1,$ is an increasing sequence of $q$-convex functions on $V.$ If we put $V_{k}=\{x\in V:\psi_{k}(x)>c\},$ then $\displaystyle\bigcup_{k\geq 1}V_{k}=V\cap X^{\prime}_{c}.$ Moreover, for any $x\in V,$ there exists, by theorem $2,$ a fundamental system of connected Stein neighborhoods $U\subset V$ such that $H^{r}(U\cap V_{k},{\mathcal{F}})=0$ for $1\leq r<dih({\mathcal{F}})-q$ and $H^{o}(U,{\mathcal{F}})\rightarrow H^{o}(U\cap V_{k},{\mathcal{F}})$ is an isomorphism, or equivalently (See  [4] or  [1]) $H^{r}_{S_{k}}({\mathcal{F}})=0$ for $r\leq dih({\mathcal{F}})-q,$ where $\underline{H^{r}_{S_{k}}}({\mathcal{F}})$ is the cohomology sheaf with support in $S_{k}=\{x\in V:\psi_{k}(x)\leq c\}$ and coefficients in ${\mathcal{F}}.$ Furthermore, there exists a spectral sequence

[TABLE]

Since $\underline{H^{p}_{S_{k}}}({\mathcal{F}})=0$ for $p\leq dih({\mathcal{F}})-q,$ then for any $p\leq dih({\mathcal{F}})-q$ the cohomology groups $H^{p}_{S_{k}}(V,{\mathcal{F}})=$ vanish and, the exact sequence of local cohomology

$\cdots\rightarrow H^{p}_{S_{k}}(V,{\mathcal{F}})\rightarrow H^{p}(V,{\mathcal{F}})\rightarrow H^{p}(V_{k},{\mathcal{F}})\rightarrow H^{p+1}_{S_{k}}(V,{\mathcal{F}})\rightarrow\cdots$

implies that $H^{p}(V_{k},{\mathcal{F}})\cong H^{p}(V,{\mathcal{F}})$ for all $p\leq dih({\mathcal{F}})-q-1.$ Hence

$H^{p}(V_{k},{\mathcal{F}})=0$ for $1\leq p\leq dih({\mathcal{F}})-q-1$ and $H^{o}(V_{k},{\mathcal{F}})\cong H^{o}(V,{\mathcal{F}})$ for every integer $k$. Since $V\cap X^{\prime}_{c}$ is an increasing union of $V_{k}$, $k\in\mathbb{N}$, then, by ( [2], lemma, p.$250$), we deduce that $H^{p}(V\cap X^{\prime}_{c},{\mathcal{F}})=0$ for $1\leq p\leq n-q-1$ and $H^{o}(V,{\mathcal{F}})\rightarrow H^{o}(V\cap X^{\prime}_{c},{\mathcal{F}})$ is an isomorphism. Since each point of $X$ has a fundamental system of hyperconvex neighborhoods, then, if

$S=\{x\in X:\phi o\Pi(x)\leq c\},$ the cohomology sheaf $H^{p}_{S}({\mathcal{F}})$ vanishes for all $p\leq dih({\mathcal{F}})-q.$ Therefore the spectral sequence

[TABLE]

shows that $H^{p}_{S}(X,{\mathcal{F}})=0$ for any $p\leq dih({\mathcal{F}})-q,$ and from the exact sequence

[TABLE]

we see that $H^{r}(X,{\mathcal{F}})\cong H^{r}(X^{\prime}_{c},{\mathcal{F}})$ for any $c\in\mathbb{R}$ and all $r\leq n-q-1.$ ∎

Lemma 2**.**

Let $Y$ be a Stein space and $\Pi:X\rightarrow Y$ an unbranched Riemann domain and locally $r$-complete morphism. Then for any coherent analytic sheaf ${\mathcal{F}}$ on $X$ with $dih({\mathcal{F}})\geq r+2$, the cohomology group $H^{p}(X,{\mathcal{F}})=0$ for all $p\geq r.$

Proof.

Since $\Pi:X\rightarrow Y$ is locally $r$-complete, it follows from  [7] that $H^{p}(X,{\mathcal{F}})=0$ for all $p\geq r+1.$ It is therefore enough to prove that $H^{r}(X,{\mathcal{F}})=0.$

   We consider a covering ${\mathcal{V}}=(V_{i})_{i\in\mathbb{N}}$ of $Y$ by open sets $V_{i}\subset\Omega$ such that $\Pi^{-1}(V_{i})$ is $r$-complete for all $i\in\mathbb{N}$. By the Stein covering lemma of Sthelé  [6], there exists a locally finite covering ${\mathcal{U}}=(U_{i})_{i\in\mathbb{N}}$ of $Y$ by Stein open subsets $U_{i}\subset\subset Y$ such that ${\mathcal{U}})$ is a refinement of ${\mathcal{V}}$, $\displaystyle\bigcup_{i\leq j}U_{i}$ is Stein for all is Stein for all j. Moreover, there exists for all $j\in\mathbb{N}$ a continuous strictly plurisubharmonic function $\phi_{j+1}$ on $\displaystyle\bigcup_{i\leq j+1}U_{i}$ such that

[TABLE]

Note also that $\Pi^{-1}(U_{i})$ is $r$-complete for all $i\in\mathbb{N}$ and, if $X_{j}=\Pi^{-1}(\displaystyle\bigcup_{i\leq j}U_{i}$ and $X^{\prime}_{j+1}=\Pi^{-1}(U_{j+1})$, then $X_{j}\cap X^{\prime}_{j+1}=\{x\in X^{\prime}_{j+1}:\phi_{j+1}o\Pi(x)<0\}$ is clearly $r$-Runge in $X^{\prime}_{j+1}$.

   We shall first prove by induction on $j$ that $H^{r}(X_{j},{\mathcal{F}})=0.$ For $j=0$, this is clear, since $\Pi^{-1}(U_{o})$ is $r$-complete. Assume that $j\geq 1$, $H^{r}(X_{j},{\mathcal{F}})=0$ and put $Y_{j}=\{x\in X_{j}:\phi_{j+1}o\Pi(x)>0\}$ and $Y^{\prime}_{j+1}=\{x\in X^{\prime}_{j+1}:\phi_{j+1}o\Pi(x)>0\}$. Then, by lemma $1$, $H^{p}(Y_{j},{\mathcal{F}})\cong H^{p}(X_{j},{\mathcal{F}})$ and $H^{p}(Y^{\prime}_{j+1},{\mathcal{F}})\cong H^{p}(X^{\prime}_{j+1},{\mathcal{F}})$ for $p\leq r.$ Since $Y_{j+1}=\{x\in X_{j+1}:\phi_{j+1}o\Pi(x)>0\}=Y_{j}\cup Y^{\prime}_{j+1}$ and $Y_{j}\cap Y^{\prime}_{j+1}=\emptyset$, then we have

[TABLE]

This proves in particular that $H^{r}(X_{j},{\mathcal{F}})=0$ for all $j\in\mathbb{N}.$

   Moreover, since $X$ is an increasing union of $(X_{j})_{j\geq 0}$ and $H^{r-1}(X_{j+1},{\mathcal{F}})\cong H^{r-1}(X_{j},{\mathcal{F}})\oplus H^{r-1}(X^{\prime}_{j+1},{\mathcal{F}})$, then, by [$2$, lemma, p. $250$], the restriction map $H^{r}(X,{\mathcal{F}})\rightarrow H^{r}(X_{0},{\mathcal{F}})$ is an isomorphism, which implies that $H^{r}(X,{\mathcal{F}})=0.$ ∎

Theorem 3**.**

Let $\Pi:X\rightarrow Y$ be an unbranched Riemann domain with $Y$ a Stein space of dimension $n$ and $\Pi$ a locally Stein morphism. Then $X$ is Stein.

Proof.

The proof is by induction on the dimension of $Y$.

   In order to prove theorem $3$ we have only to verify that $H^{1}(X,{\mathcal{O}}_{X})=0.$ (See  [5]).

   Suppose that $n=2,$ and let $\xi:\tilde{Y}\rightarrow Y$ be a normalization of $Y$. If $\tilde{X}$ denotes the fiber product of $\Pi:X\rightarrow Y$ and the normalization $\xi:\tilde{Y}\rightarrow Y$, then $\tilde{X}=\{(x,\tilde{y})\in X\times\tilde{Y}:\Pi(x)=\xi(\tilde{y})\}$ and, it is clear that the projection $\Pi_{2}:\tilde{X}\rightarrow\tilde{Y}$ is an unbranched Riemann domain over the $2$-dimensional Stein normal space $\tilde{Y}.$ Moreover, since $\Pi_{2}$ is obviously a locally Stein morphism, it follows from  [3] that $\tilde{X}$ is Stein. On the other hand, it is easy to verify that the projection $\Pi_{1}:\tilde{X}\rightarrow X$ is a finite holomorphic surjection, which implies that $X$ is Stein. (See e.g.  [8]).

   We now suppose that $n\geq 3$ and that the theorem has already proved if $dim(Y)\leq n-1.$

   Since a complex space $X$ is Stein if and only if each irreducible component $X_{i}$ of $X$ is Stein, then we may assume that $Y$ is irreducible.

   Let $f$ be a holomorphic function on $Y$, $f\neq 0$, but $Z=\{f=0\}\neq\emptyset.$ Then $\Pi|_{Z^{\prime}}:Z^{\prime}=\Pi^{-1}(Z)\rightarrow Z$ is an unbranched Riemann domain and locally Stein. Therefore $Z^{\prime}$ is Stein by the induction hypothesis. Furthermore, if ${\mathcal{I}}(Z^{\prime})$ denotes the ideal sheaf of $Z^{\prime}$, it follows from $[2]$ that $dih({\mathcal{I}}(Z^{\prime})=dih({\mathcal{O_{Z^{\prime}}}})+1\geq 3$ and, by lemma $2$, we obtain $H^{1}(X,{\mathcal{I}}(Z^{\prime}))=0.$ Consider now the exact sequence of sheaves

[TABLE]

Since $H^{1}(X,{\mathcal{I}}(Z^{\prime}))=0$ and $Z^{\prime}$ is Stein, we deduce from the long exact sequence of cohomology that $H^{1}(X,{\mathcal{O}_{X}})=0.$ ∎