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On strict Levi q-convexity and q-concavity on domains with piecewise smooth boundaries. (English) Zbl 0628.32021
Let X be an n-dimensional complex manifold. An open subset D of X is said to have a generic k-edge at \(x_ 0\in \partial D\) if we can find real valued, smooth functions \(\rho _ 1,...,\rho _ k\), defined on an open neighborhood U of \(x_ 0\) in X, such that \(D\cap U=\cap _{j=1,...,k}\{x\in U| \rho _ j(x)<0\}\) and \(\partial \rho _ 1(x_ 0)\wedge...\wedge \partial \rho _ k(x_ 0)\neq 0.\) It is said to be strictly Levi q-convex at \(x_ 0\) in the codirection \(\xi ^ 0=d\rho _ 1(x_ 0)+...d\rho _ k(x_ 0)\) if the Levi form \(-i<\partial {\bar \partial}(\rho _ 1+...+\rho _ k)(x_ 0),\eta \wedge J\eta >\) has q negative and \(n-q-k\) positive eigenvalues on the complex tangent space \(H_{x_ 0}=\cap _{j=1,...,k}\{\eta \in T_{x_ 0}X| <\partial \rho _ j(x_ 0),\eta >=0\}\). Under this assumption, it is proved that the local cohomology group \(\underset \sim H^ q_{x_ 0}(D)=\lim _{V\;open\;\ni x_ 0}H^ q(D\cap U,{\mathcal O})\) is infinite dimensional. This is an extension of a theorem proved for domains with a smooth boundary by A. Andreotti and F. Norguet [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 20, 197-241 (1966; Zbl 0154.335)]. The result extends to domains with singularities at the boundary only controlled by a generic edge and for general locally free coherent sheaves. A furthr extension is given for the local cohomology for the Dolbeault complex on functions that are smooth up to the boundary. The existence of infinitely many global q-cohomology classes on D, having linearly independent restriction to \(\widetilde H^ q_{x_ 0}(D)\), is proved under the additional assumption that X be pseudoconvex and that D be contained in a q-pseudoconvex open set \(\Omega\) with a smooth boundary, such that \(x_ 0\in \partial \Omega\) and \(\xi ^ 0\) is the exterior conormal to \(\Omega\) at \(x_ 0.\)
The open set D is said to have a generic k-identation at \(x_ 0\in \partial D\) if, in an open neighborhood U of \(x_ 0\) one can find real valued, smooth functions \(\rho _ 1,...,\rho _ k\), such that \(D\cap U=\cup _{j=1,...,k}\{x\in U| \rho _ j(x)<0\}\) and \(\partial \rho _ 1(x_ 0)\wedge...\wedge \partial \rho _ k(x_ 0)\neq 0.\) Such a domain is said to be strictly Levi q-concave at \(x_ 0\) in the codirection \(\xi ^ 0=d\rho _ 1(x_ 0)+..+d\rho _ k(x_ 0)\) if the Levi form of \(\rho _ 1(x)+...+\rho _ k(x)\) at \(x_ 0\) has q negative and \(n-k-q\) positive eigenvalues on \(H_{x_ 0}\) (defined as above). Under this condition the same results on local cohomology groups are obtained as in the strict Levi q-convexity case.
The proof of these theorems rely on the violation of a ”radiation principle” that, roughly speaking, states that there should be no \({\bar \partial}\) closed forms that are rapidly fading in D away from \(x_ 0.\)
In the last part of the paper, the local cohomology groups of the tangential Cauchy-Riemann complex at boundary points of an open submanifold \(\Omega\) of a real k-codimensional generic submanifold S of X are considered. Let S be described on the neighborhood U of its point \(x_ 0\) as the set of common zeros of k real valued, smooth functions \(\rho _ 1,...\rho _ k\), having linearly independent holomorphic differentials at \(x_ 0\) and let \(V\cap \Omega =\{x\in S\cap U| \phi (x)<0\}\) for a real valued, smooth function \(\phi\) on U with \(\phi (x_ 0)=0\) and \(\partial \phi (x_ 0)\wedge \partial \rho _ 1(x_ 0)\wedge...\wedge \partial \rho _ k(x_ 0)\neq 0\). One considers the Levi form \(-i<\partial {\bar \partial}(\phi +\lambda ^ 1\rho _ 1+...+\lambda ^ k\rho _ k)(x_ 0),\eta \wedge J\eta >\) restricted to the analytic tangent space \(H_{x_ 0}\cap \{\eta \in T_{x_ 0}X| <\partial \phi (x_ 0),\eta >=0\}\). Then \(\Omega\) is said to be d-convex (resp. q-concave) at \(x_ 0\) if this form has at least \(n-k-d\) positive (resp. q negative) eigenvalues for all \(\lambda ^ 1,...,\lambda ^ k\in R\). If for a choice of \(\lambda _ 1,...,\lambda ^ k\) in R it has exactly q negative and \(n-k-q-1\) positive eigenvalues, then \(\Omega\) is said to be strictly Levi q-concave at \(x_ 0\). Denoting by \(H^ j_{x_ 0}(Q_ S^{p,*}({\bar \Omega}),{\bar \partial}_{S^ *})=\lim _{V open\;\ni x_ 0}H^ j(Q_ S^{p,*}({\bar \Omega}\cap V),{\bar \partial}_{S^ *})\) the local cohomology groups (with regularity up to the boundary) of the tangential Cauchy Riemann complex on \(\Omega\), one proves that they vanish for \(j>d\) and \(1\leq j<q\) if \(\Omega\) is d-convex and q-concave at \(x_ 0\), while are infinite dimensional for \(j=q\) if \(\Omega\) is strictly q-concave at \(x_ 0\).

MSC:
32F10 \(q\)-convexity, \(q\)-concavity
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32V40 Real submanifolds in complex manifolds
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