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Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces. (English) Zbl 0596.32016
The author computes the cohomology of holomorphic q-forms $$\Omega^ q$$ twisted by powers of a very ample line bundle L, $$\Omega^ q(k):=\Omega^ q\cdot L^ k$$, on Grassmann manifolds and non-singular quadric hypersurfaces, generalizing certain theorems of R. Bott [Ann. Math. 66, 203-248 (1957; Zbl 0094.357)] and J. Le Potier [Math. Ann. 226, 257-270 (1977; Zbl 0356.32018)). A complete description is given when X is a quadric hypersurface: All of the groups $$H^ p(X,\Omega^ q(k))$$ (with one obvious exception) are irreducible G- modules. If $$1\leq k\leq q$$, then $$H^ p(X,\Omega^ q(k))\neq 0$$ iff $$k=2q-n$$ and $$p=n-q,$$ in which case it is one dimensional. If $$k>q$$ then $$\Omega^ q(k)$$ is spanned and has a highest weight of the form $$(k-q- 1)\mu_ 1+\mu_ i.$$ When X is a Grassmann manifold, an explicit algorithm is given for computing which of the groups $$H^ p(X,\Omega^ q(k))$$ vanish. Several general statements are possible:
Let X be the Grassmann manifold of s-planes in $${\mathbb{C}}^{m+1}$$, $$s\leq m-s+1$$, and let $$n=\dim X$$. Then: $$H^ p(X,\Omega^ q(1))=0$$ if $$p+q>0$$; $$H^ p(X,\Omega^ q(2))=0$$ except for $$(p,q)=(a(a- 1)/2),a(a+1)/2),a\leq s;$$ $$H^ p(X,\Omega^ q(k))=0$$ if sp$$\geq (s-1)q$$ or $$p>n-q$$ or $$q>n-s$$; $$\Omega^ q(k)$$ is spanned if $$k>q$$ or $$k>m.$$
Similar statements for the other compact Hermitian symmetric spaces will be presented in a sequel to this paper.

##### MSC:
 32L20 Vanishing theorems 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 22E46 Semisimple Lie groups and their representations
##### Citations:
Zbl 0094.357; Zbl 0356.32018
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##### References:
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