Niveau: Supérieur, Doctorat, Bac+8
VANISHING THEOREMS FOR DOLBEAULT COHOMOLOGY OF LOG HOMOGENEOUS VARIETIES Michel Brion Abstract We consider a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections. We obtain an optimal vanishing theorem for logarithmic Dolbeault cohomology of nef line bundles in that setting. This implies a vanishing theorem for ordinary Dolbeault cohomology which generalizes results of Broer for flag varieties, and of Mavlyutov for toric varieties. Introduction The main motivation for this work comes from the well-developed theory of complete intersections in algebraic tori (C?)n and in their equivariant compactifications, toric vari- eties. In particular, the Hodge numbers of these complete intersections were determined by Danilov and Khovanskii, and their Hodge structure, by Batyrev, Cox and others (see [11, 2, 26]). This is made possible by the special features of toric geometry; two key ingredients are the triviality of the logarithmic tangent bundle TX(? logD), where X is a complete nonsingular toric variety with boundary D, and the Bott–Danilov–Steenbrink vanishing theorem for Dolbeault cohomology: H i(X,L ? ?jX) = 0 for any ample line bundle L on X and any i ≥ 1, j ≥ 0. A natural problem is to generalise this theory to complete intersections in algebraic homogeneous spaces and their equivariant compactifications.
- associated logarithmic tangent
- logarithmic tangent
- tangent bundle
- rx ?
- abelic varieties satisfy
- ym has normal
- varieties
- broer's vanishing