Continuous \(q\)-convex exhaustion functions.

*(English)*Zbl 0599.32016Let \(X\) be a complex space. \(f: X\to {\mathbb{R}}\) is called a continuous q- convex function if given \(x\in X\) there are U a neighborhood of x and \(f_ 1,...,f_ s\in C^{\infty}\), \(q\)-convex (in the sense of Andreotti- Grauert) functions on \(U\) such that \(f/U=\max (f_ 1,...,f_ s)\). If in the definition of \(q\)-convex complex spaces (respectively \(q\)-complete complex spaces) \(C^{\infty}\) \(q\)-convex functions are replaced by continuous \(q\)-convex functions one obtains the \(q\)-convex with corners complex spaces (respectively \(q\)-complete with corners complex spaces). The importance of this class of functions (and the corresponding classes of complex spaces) has been put into light by recent results of H. Grauert [Kantenkohomologie, Compos. Math. 44, 79-101 (1981; Zbl 0512.32011)] and K. Diederich and J. E. Fornaess [’Smoothing q-convex functions and vanishing theorems’ (to appear)].

The paper investigates the convexity type in this large sense of some open subsets in complex spaces \(X\). The ambient complex spaces \(X\) are such that the complement of the diagonal \(\Delta_ X\subset X\times X\) (or rather a neighborhood of \(\Delta_ X)\) satisfies some convexity type conditions \((E_ r)\) (respectively \((E_ r'))\) which are slightly weaker than asking \((n+r)\)-convexity with corners. Here \(n=\inf \{\dim_ xX| x\in X\}\). It is proven that \({\mathbb{P}}_ n\) satisfies \((E_ 0)\) and the Hopf manifold satisfies \((E_ 1)\). We mention here two main results of the paper:

Theorem 4: Let \(X\) be a compact complex space such that \((E_ r)\) holds on \(X\) and \(M\subsetneqq X\) an open subset which is \(q\)-convex with corners. Then \(M\) is \((q+r)\)-complete with corners.

Theorem 6: Let \(X\) be a compact complex space such that \((E_ r')\) holds on \(X\) and \(A\subset X\) be an analytic set with \(\dim A_ i\geq n-q\) for every irreducible component \(A_ i\) of \(A\). Then \(X-A\) is \((q+r)\)-convex with corners. If \((E_ r)\) holds on \(X\), \(X-A\) is \((q+r)\)-complete with corners.

To construct continuous \(q\)-convex functions on open subsets of \(X\) out of the existing continuous \((n+r)\)-convex functions in the neighborhood of \(\Delta_ X\subset X\times X\) one relies on the following result: Let \(Y\) be a complex space, \(V\) an \(m\)-dimensional compact complex manifold and \(f\) a continuous r-convex function on \(Y\times V\). Suppose \(\sup f<\infty.\) Then \(s(y)=\sup \{f(y,v)| v\in V\}\) is locally approximable by continuous (\(r-m\))-convex functions.

The paper investigates the convexity type in this large sense of some open subsets in complex spaces \(X\). The ambient complex spaces \(X\) are such that the complement of the diagonal \(\Delta_ X\subset X\times X\) (or rather a neighborhood of \(\Delta_ X)\) satisfies some convexity type conditions \((E_ r)\) (respectively \((E_ r'))\) which are slightly weaker than asking \((n+r)\)-convexity with corners. Here \(n=\inf \{\dim_ xX| x\in X\}\). It is proven that \({\mathbb{P}}_ n\) satisfies \((E_ 0)\) and the Hopf manifold satisfies \((E_ 1)\). We mention here two main results of the paper:

Theorem 4: Let \(X\) be a compact complex space such that \((E_ r)\) holds on \(X\) and \(M\subsetneqq X\) an open subset which is \(q\)-convex with corners. Then \(M\) is \((q+r)\)-complete with corners.

Theorem 6: Let \(X\) be a compact complex space such that \((E_ r')\) holds on \(X\) and \(A\subset X\) be an analytic set with \(\dim A_ i\geq n-q\) for every irreducible component \(A_ i\) of \(A\). Then \(X-A\) is \((q+r)\)-convex with corners. If \((E_ r)\) holds on \(X\), \(X-A\) is \((q+r)\)-complete with corners.

To construct continuous \(q\)-convex functions on open subsets of \(X\) out of the existing continuous \((n+r)\)-convex functions in the neighborhood of \(\Delta_ X\subset X\times X\) one relies on the following result: Let \(Y\) be a complex space, \(V\) an \(m\)-dimensional compact complex manifold and \(f\) a continuous r-convex function on \(Y\times V\). Suppose \(\sup f<\infty.\) Then \(s(y)=\sup \{f(y,v)| v\in V\}\) is locally approximable by continuous (\(r-m\))-convex functions.

Reviewer: N.Mihalache

##### MSC:

32F10 | \(q\)-convexity, \(q\)-concavity |

##### Keywords:

approximation; exhaustion functions; continuous \(q\)-convex function; \(q\)- convex with corners complex spaces; \(q\)-complete##### References:

[1] | Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr.90, 193-259 (1962) · Zbl 0106.05501 |

[2] | Barth, W.: Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum. Math. Ann.187, 150-162 (1970) · Zbl 0189.36704 · doi:10.1007/BF01350179 |

[3] | Diederich, K., Fornaess, J.E.: Smoothingq-convex functions and vanishing theorems. (Preprint 1985) · Zbl 0586.32022 |

[4] | Diederich, K., Fornaess, J.E.: Smoothingq-convex functions in the singular case. (Preprint 1985) · Zbl 0586.32022 |

[5] | Grauert, H.: Kantenkohomologie. Comput. Math.44, 79-101 (1981) · Zbl 0512.32011 |

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