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

Giovanni GaiffiMichele Grassi — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that one can obtain natural bundles of Lie algebras on rank two s -Kähler manifolds, whose fibres are isomorphic respectively to so ( s + 1 , s + 1 ) , su ( s + 1 , s + 1 ) and sl ( 2 s + 2 , ) . 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 su ( s + 1 , s + 1 ) on (rational) Hodge classes of Abelian varieties with rational period matrix.

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