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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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