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Automorphisms and local rigidity of regular varieties. (English) Zbl 0910.14004

In this paper, the following vanishing theorem is proved: Let \(G\) be a connected affine algebraic group and \(X\) a complete regular \(G\)-variety. This implies that \(G\) acts with an open dense orbit \(\Omega\) which in turn is a spherical \(G\)-homogeneous space, i.e. there is a Borel subgroup of \(G\) having an open dense orbit in \(\Omega\). Let \({\mathcal S}_X\) be the “action sheaf”, i.e. the subsheaf of the tangent sheaf consisting of vector fields “tangent” to the boundary \(D:=X-\Omega\). Then under the assumption that \(D\) is a divisor with normal crossings and that \(\Omega\) is proper over an affine variety, the theorem states that the higher cohomology groups of \({\mathcal L}\otimes S^m ({\mathcal S}_X)\) vanish for all symmetric powers \(S^m ({\mathcal S}_X)\), \(m\geq 0\), and all invertible sheaves \({\mathcal L}\) on \(X\) that are generated by global sections.
The theorem has been generalized through a different approach by F. Knop [Ann. Math., II. Ser. 140, No. 2, 253-288 (1994; Zbl 0828.22017)]. His result avoids the extra hypothesis on \(\Omega\), but applies only to the untwisted case. The paper also contains applications of the vanishing theorem to automorphisms of \(X\) as above, and to the rigidity of spherical Fano varieties.

MSC:

14F17 Vanishing theorems in algebraic geometry
14M17 Homogeneous spaces and generalizations
14G20 Local ground fields in algebraic geometry

Citations:

Zbl 0828.22017
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References:

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