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

Giovanni Gaiffi[1]; Michele Grassi[1]

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

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 2, page 419-451
  • ISSN: 0240-2963

Abstract

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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 , 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 su ( s + 1 , s + 1 ) on (rational) Hodge classes of Abelian varieties with rational period matrix.

How to cite

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Gaiffi, Giovanni, and Grassi, Michele. "Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties." Annales de la faculté des sciences de Toulouse Mathématiques 19.2 (2010): 419-451. <http://eudml.org/doc/115887>.

@article{Gaiffi2010,
abstract = {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 $\mathbf\{sl\}(2s + 2,\mathbb\{R\})$. 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.},
affiliation = {Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo, 5 56125 Pisa - Italy; Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo, 5 56125 Pisa - Italy},
author = {Gaiffi, Giovanni, Grassi, Michele},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Lie algebra; Kähler manifold; complex structure; Kähler forms},
language = {eng},
month = {4},
number = {2},
pages = {419-451},
publisher = {Université Paul Sabatier, Toulouse},
title = {Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties},
url = {http://eudml.org/doc/115887},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Gaiffi, Giovanni
AU - Grassi, Michele
TI - Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 2
SP - 419
EP - 451
AB - 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 $\mathbf{sl}(2s + 2,\mathbb{R})$. 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.
LA - eng
KW - Lie algebra; Kähler manifold; complex structure; Kähler forms
UR - http://eudml.org/doc/115887
ER -

References

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