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Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties
Giovanni Gaiffi; Michele Grassi
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 2, p. 419-451

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(2s+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.

Nous prouvons que on peut obtenir fibrés naturels des algèbres de Lie so(s+1,s+1), su(s+1,s+1) et sl(2s+2,) 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 su(s+1,s+1) sur les classes de Hodge (rationelles) de variétés abéliennes dont la matrice des periodes est rationelle.

Received : 2009-04-26
Accepted : 2010-01-12
Published online : 2010-09-01
DOI : https://doi.org/10.5802/afst.1249
@article{AFST_2010_6_19_2_419_0,
     author = {Giovanni Gaiffi and Michele Grassi},
     title = {Lie Algebra bundles on s-K\"ahler manifolds, with applications to Abelian varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 19},
     number = {2},
     year = {2010},
     pages = {419-451},
     doi = {10.5802/afst.1249},
     mrnumber = {2674769},
     zbl = {1206.53031},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2010_6_19_2_419_0}
}
Gaiffi, Giovanni; Grassi, Michele. Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 2, pp. 419-451. doi : 10.5802/afst.1249. afst.centre-mersenne.org/item/AFST_2010_6_19_2_419_0/

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