Rational points and Coxeter group actions on the cohomology of toric varieties

Lehrer, Gustav I.

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 671-688.

On donne une formule simple pour l’action d’un groupe de Coxeter fini crystallographique sur la cohomologie de la variété torique complexe associée. La méthode utilise la structure de Hodge sur la cohomologie pour relier le nombre des points rationnels sur un corps fini à cette action. On utilise la formule pour quelques applications, telles que la détermination de la multiplicité graduée de la représentation par réflexions dans la cohomologie.

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

Classification : 14M25, 14F40, 14G05, 20G40, 14L30
Keywords: Toric varieties, cohomology, Hodge theory, rational points
Mot clés : variétés toriques, cohomologie, théorie de Hodge, points rationnels

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