## 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
Full Text:

### References:

 [1] Bott, R.: Homogeneous vector bundles. Ann Math.66, 203-248 (1957) · Zbl 0094.35701 [2] Demazure, M.: A very simple proof of Bott’s theorem. Invent math.33, 271-272 (1976) · Zbl 0383.14017 [3] Helgason, S.: Differential geometry and symmetric spaces. New York, San Francisco, London: Academic Press 1978 · Zbl 0451.53038 [4] Humphreys, J.: Introduction to lie algebras and representation theory. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0254.17004 [5] Humphreys, J.: Linear algebraic groups. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0325.20039 [6] Kostant, B.: Lie algebra cohomology and the generalized Borel Weil Theorem. Ann. Math.74, 329-387 (1961) · Zbl 0134.03501 [7] Le Potier, J.: Annulation de la cohomologie ? valeurs dans un fibr? vectoriel holomorphe positif de rang quelconque. Math. Ann.218, 35-53 (1975) · Zbl 0313.32037 [8] Le Potier, J.: Cohomologie de la Grassmannienne ? valeurs dans les puissances ext?rieures et symmetriques du fibr? universel. Math. Ann.266, 257-270 (1977) · Zbl 0356.32018 [9] Serre, J.P.: Repr?sentations lin?ares et espaces homog?nes K?hlerians des groupes de Lie compacts (d’apr?s Borel et Weil). Sem. Bourbaki, expos? 100, May 1954. New York: Benjamin 1966 [10] Shiffman, B., Sommese, A.: Vanishing theorems on complex manifolds. Boston, Basel, Stuttgart: Birkh?user 1985 · Zbl 0578.32055 [11] Snow, D.: On the ampleness of homogeneous vector bundles. Trans. Am. Math. Soc.294, 585-594 (1986) · Zbl 0588.32038 [12] Snow, D.: Vanishing theorems on compact hermitian symmetric spaces. To appear · Zbl 0631.32025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.