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Vanishing theorems on compact hermitian symmetric spaces. (English) Zbl 0631.32025
Let X be an irreducible compact hermitian symmetric space embedded minimally and equivariantly into projective space by an ample line bundle L and let \(\Omega ^ q(k)\) denote the sheaf of holomorphic q-forms on X tensored with the \(k^{th}\) power of L, \(\Omega ^ q(k)=\Omega ^ q\otimes L^ k\). This article determines the ranges of p, q and k for which \(H^ p(X,\Omega ^ q(k))\) vanishes and for which the sheaf \(\Omega ^ q(k)\) is spanned or ample when X is of type CI, DIII, EIII and EVII. For the exceptional spaces, EIII and EVII, the complete G- module structure of the cohomology of \(\Omega ^ q(k)\) is also worked out. Similar result for spaces of type AIII, BI and DI were obtained in a previous article by the same author, see Math. Ann. 276, 159-176 (1986; Zbl 0596.32016). In the light of these results, some general remarks are possible: (1) All of the cohomology of \(\Omega ^ q(1)\) vanishes if X is not of type BI or CI. (2) The non-singular values for the cohomological dimension of \(\Omega ^ q(k)\), \(cd(\Omega ^ q(k))\), are increasing in q and decreasing in k. (3) \(H^ p(X,\Omega ^ q(k))=0\) if \(k>0\) and \(p\geq (\dim X)/2\).

MSC:
32L20 Vanishing theorems
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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