Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties
نویسنده
چکیده
Bott-Samelson varieties were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. In particular, the cohomology of some line bundles on Bott-Samelson varieties were used to prove that Schubert varieties are normal, Cohen-Macaulay and with rational singularities (see for example [BK05]). In this paper, we will be interested in the cohomology of all line bundles of Bott-Samelson varieties. We consider a Bott-Samelson variety Z(w̃) over an algebraically closed field k associated to an expression w̃ = sβ1 . . . sβN of an element w in the Weyl group of a Kac-Moody group G over k (see Definition 1.1 (i)). In the case where G is semi-simple, N. Lauritzen and J.F. Thomsen proved, using Frobenius splitting, the vanishing of the cohomology in positive degree of line bundles on Z(w̃) of the form L(−D) where L is any globally generated line bundle on Z(w̃) and D a subdivisor of the boundary of Z(w̃) corresponding to a reduced expression of w [LT04, Th7.4]. The aim of this paper is to give the vanishing in some degrees of the cohomology of any line bundles on Z(w̃).
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