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.


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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