Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties

Gaiffi, Giovanni¹  ; Grassi, Michele¹

¹ Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo, 5 56125 Pisa - Italy

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 2, pp. 419-451.

Abstract (VO)

We prove that one can obtain natural bundles of Lie algebras on rank two $s$-Kähler manifolds, whose fibres are isomorphic respectively to $\mathbf{so}(s+1,s+1)$, $\mathbf{su}(s+1,s+1)$ and $\mathbf{sl}(2s+2,ℝ)$. These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of Kähler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of $\mathbf{su}(s+1,s+1)$ on (rational) Hodge classes of Abelian varieties with rational period matrix.

Résumé

Nous prouvons que on peut obtenir fibrés naturels des algèbres de Lie $\mathbf{so}(s+1,s+1)$, $\mathbf{su}(s+1,s+1)$ et $\mathbf{sl}(2s+2,ℝ)$ sur variétés $s$-Kähler de rang 2. Ces fibrés ont connexions naturelles dont les sections globales généralisent les opérateurs de Lefschetz de la géométrie de Kähler et agissent d’une façon naturelle sur la cohomologie. Pour première application nous construisons une représentation irréductible d’une forme rationnelle de $\mathbf{su}(s+1,s+1)$ sur les classes de Hodge (rationelles) de variétés abéliennes dont la matrice des periodes est rationelle.

Détail

DOI : 10.5802/afst.1249

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