# Pseudo-convex and pseudo-concave

Properties of domains in complex spaces, as well as of complex spaces and functions on them, analogous to convexity and concavity properties of domains and functions in the space $\mathbf R ^ {n}$. A real-valued function $\phi$ of class $C ^ {2}$ on an open set $U \subset \mathbf C ^ {n}$ is called $p$- pseudo-convex (or $p$- convex) if the Hermitian form

$$H ( \phi ) = \sum _ { j,k } \frac{\partial ^ {2} \phi }{\partial z _ {j} \partial \overline{z}\; _ {k} } u _ {j} \overline{u}\; _ {k}$$

has, at each point of $U$, at least $n - p + 1$ non-negative eigen values. If $H ( \phi )$ has at least $n - p + 1$ positive eigen values, then one says that $\phi$ is strictly (or strongly) $p$- pseudo-convex. In particular, a (strictly) $1$- pseudo-convex function is a (strictly) plurisubharmonic function of class $C ^ {2}$. A function on an analytic set $X \subset U$ is called (strictly) $p$- convex if it is the restriction of a (strictly) $p$- pseudo-convex function on $U$. Finally, a (strictly) $p$- convex function on an arbitrary complex space $X$ is a continuous function on $X$ that is, in a neighbourhood of each point, a (strictly) convex function on the corresponding model (cf. Analytic space).

A complex space $X$ is called $( p, q)$- convex-concave if there is a continuous function $\phi : X \rightarrow \mathbf R$ and two numbers $d _ {0} , c _ {0}$, $- \infty \leq d _ {0} \leq c _ {0} \leq \infty$, such that for any $c \geq c _ {0}$ and $d \leq d _ {0}$ the set

$$X _ {c,d} = \ \{ {x \in X } : {d < \phi ( x) < c } \}$$

is relatively compact in $X$, while $\phi$ is strictly $p$- convex on $X _ {\infty , c _ {0} }$ and strictly $q$- concave on $X _ {d _ {0} , \infty }$. If $d _ {0} = - \infty$ or $c _ {0} = \infty$, then $X$ is called strongly $p$- pseudo-convex or strongly $q$- pseudo-concave, respectively. If $d _ {0} = c _ {0} = - \infty$, then $X$ is called $p$- complete.

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### Examples.

1) An open set $X$ with smooth boundary $\partial X$ in a complex manifold $M$ is called strongly $p$- pseudo-convex (strongly $p$- pseudo-concave) if every point $x _ {0} \in \partial X$ has a neighbourhood $U$ in which there is a strictly $p$- pseudo-convex (strictly $p$- pseudo-concave) function $\phi$ such that $X \cap U = \{ {x \in U } : {\phi ( x) < 0 } \}$( respectively, $X \cap U = \{ {x \in U } : {\phi ( x) > 0 } \}$). Every strongly $p$- pseudo-convex (strongly $p$- pseudo-concave) relatively-compact open set is a strongly $p$- convex (strongly $p$- concave) manifold. If certain components of the boundary $\partial X$ satisfy the $p$- pseudo-convexity condition, while the remaining satisfy the $q$- pseudo-concavity condition, then one obtains an example of a $( p , q )$- convex-concave manifold.

2) Compact complex spaces are naturally considered to be $0$- convex.

3) The class of $1$- complete spaces coincides with the class of Stein spaces (cf. Stein space).

4) The class of strongly $1$- convex spaces coincides with the class of spaces obtained from Stein spaces by proper modifications at a finite set of points.

5) Let $X$ be a compact complex manifold of dimension $n$, and let $S$ be a closed submanifold of it all components of which have dimension $q$. Then $X \setminus S$ is strongly $( q + 1 )$- concave, and if the normal bundle over $S$ is positive, then $X \setminus S$ is a strongly $( n - q )$- convex space.

6) If $S$ is a closed submanifold of codimension $p$ in a Stein manifold $X$, then $X \setminus S$ is $p$- complete.

7) A holomorphic vector bundle $E$ of rank $r$ over a manifold $X$ is called $p$- positive ( $q$- negative) if there is a fibre-wise Hermitian metric $h$ on $E$ such that $\chi ( \nu ) = - h ( \nu , \nu )$ is a strictly $( p + r )$- convex (respectively, $- \chi$ is a strictly $q$- convex) function on $E$ outside the zero section (if $p = q = 1$ one obtains the concepts of a positive vector bundle and a negative vector bundle). If $X$ is compact, then the space of a $p$- positive vector bundle $E$ is strongly $( p + r )$- concave, while the space of a $q$- negative vector bundle is strongly $q$- convex. The space of a holomorphic vector bundle over a $p$- complete space is always $p$- complete.

For $( p , q )$- convex-concave spaces one has proved theorems on the finite dimensionality and separability of certain cohomology spaces with values in coherent analytic sheaves (cf. Finiteness theorems in the theory of analytic spaces). Analogous finiteness theorems have been proved for strictly $( p , q )$- convex-concave mappings (cf. , ). A space $X$ is strongly $1$- convex if and only if $\mathop{\rm dim} H ^ {r} ( X , F ) < \infty$ for all $r$ and any coherent analytic sheaf $F$ on $X$. If $X$ is $p$- complete, then $H ^ {r} ( X , F ) = 0$ for all $r \geq p$ and any coherent analytic sheaf $F$ on $X$.

The homology groups of $p$- convex and $p$- complete spaces have the following properties. If $X$ is an $n$- dimensional reduced strongly $p$- convex ( $p$- complete) complex space, then $\mathop{\rm dim} H _ {r} ( X , \mathbf C ) < \infty$( respectively, $H _ {r} ( X , \mathbf C ) = 0$) for $r \geq n + p$. For strongly $1$- convex spaces it is also known that $H _ {r} ( X , \mathbf Z )$ is a finitely-generated group for $r \geq n + 1$, while for $p$- complete manifolds it is known that $H _ {r} ( X , \mathbf Z ) = 0$ for $r \geq n + p$ and that $H _ {n+} p- 1 ( X , \mathbf Z )$ is free.

A complex space $X$ is called pseudo-concave if there is a relatively-compact open set $U$ in $X$, intersecting each non-degenerate component of $X$ and satisfying the following condition: Any point $x _ {0} \in \partial U$ has a neighbourhood $V$ in $X$ such that for any $x \in V$ sufficiently close to $x _ {0}$,

$$| f ( x) | \leq \sup _ {y \in V \cap U } | f ( y) |$$

for all holomorphic functions $f$ in $V$. If $X$ is an $n$- dimensional manifold, $n \geq 2$, then it is sufficient that $U$ is a strongly $( n - 1 )$- pseudo-concave set in $X$. Any compact space is pseudo-concave. For pseudo-concave spaces $X$ the following finiteness theorems have been proved: The space of holomorphic sections of any holomorphic vector bundle over $X$ is finite-dimensional; if $X$ is connected, then all holomorphic functions on $X$ are constant; the field of meromorphic functions on $X$ is an algebraic function field whose transcendence degree does not exceed $\mathop{\rm dim} X$. The latter theorem has important applications to automorphic functions (cf. Automorphic function), based on the fact that the space $D / \Gamma$, where $\Gamma$ is a properly-discontinuous group of automorphisms of a bounded domain $D \subseteq \mathbf C ^ {n}$, turns out to be pseudo-concave in many cases (one says in this case that $\Gamma$ is a pseudo-concave group). E.g., arithmetic subgroups of automorphism groups of bounded symmetric domains are pseudo-concave.

How to Cite This Entry:
Pseudo-convex and pseudo-concave. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-convex_and_pseudo-concave&oldid=48344
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article