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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
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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
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