## 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)

### Keywords:

compact hermitian symmetric space; vanishing theorems

### Citations:

Zbl 0596.32016; Zbl 0605.32013
Full Text:

### References:

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