Sur la cohomologie à support des fibrés en droites sur les variétés symétriques complètes

Abstract

Let X be a complete symmetric variety, i.e., the wonderful compactification of a symmetric G-homogeneous space (where G is a simply connected semi-simple linear algebraic group). If L is a line bundle over X and if C is a Bialynicki-Birula cell of codimension c in X, then the Lie algebra \( \mathfrak{g} \) of G operates naturally on the cohomology group with support H C c (L). A necessary condition on C for the existence of a finite-dimensional simple subquotient of this \( \mathfrak{g} \)-module is given. As applications one calculates the Euler–Poincaré characteristic of L over X, estimates the higher cohomology group H d(X, L), d ≥ 0, and obtains the exact formulas in some cases including that of the complete conic variety.