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

Giovanni Gaiffi; Michele Grassi

doi:10.5802/afst.1249

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, Serie 6, Volume 19 (2010) no. 2, pp. 419-451.

Abstract
Abstract

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.

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

EuDML MR Zbl

DOI: 10.5802/afst.1249

Author's affiliations:

Giovanni Gaiffi ¹; Michele Grassi ¹

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

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     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},
     pages = {419--451},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {2},
     year = {2010},
     doi = {10.5802/afst.1249},
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Giovanni Gaiffi; Michele Grassi. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1249/

References
Cited by

[BMP] U. Bruzzo, G. Marelli, F. Pioli A Fourier transform for sheaves on real tori Part II. Relative theory J. of Geometry and Phy. 41 (2002) 312-329 | MR | Zbl

[CDGP] P. Candelas, X.C. De la Ossa, P.S. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B359 (1991), p 21-74 | MR | Zbl

[DOPW] R. Donagi, B.A. Ovrut, T. Pantev, D. Waldram Standard-model bundles, Adv. Ther. Math. Phys. 5 (2001) n.3, p. 563-615 | MR | Zbl

[GG1] G. Gaiffi, M. Grassi A geometric realization of $sl (6, ℂ)$ , arXiv:0704.0104v1, to appear on Rend. Sem. Mat. Univ. Padova. | Numdam | MR

[GG2] G. Gaiffi, M. Grassi A natural Lie superalgebra bundle on rank three WSD manifolds, J. Geom. Phys. 59 (2009), p. 207-220 | MR | Zbl

[G1] M. Grassi, Polysymplectic spaces, $s$ -Kähler manifolds and lagrangian fibrations, math.DG/0006154 (2000)

[G2] M. Grassi, Mirror symmetry and self-dual manifolds, math.DG/0202016 (2002)

[G3] M. Grassi, Self-dual manifolds and mirror symmetry for the quintic threefold, Asian J. Math 9 (2005) 79-102 | MR | Zbl

[GH] P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York (1978) | MR | Zbl

[Gr] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser P.M. 152, Boston 1999 | MR | Zbl

[Gu] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian $𝕋^{n}$ -spaces, Birkhäuser P.M. 122 (1994) | MR | Zbl

[HT] T. Hausel, M. Thaddeus Mirror symmetry, Langlands duality, and the Hitchin system Invent. Math. 153 (2003), n.1 197-229 | MR | Zbl

[LM] H.B. Lawson, M-L. Michelsohn, Spin Geometry, Princeton M.S. 38 (1989) | Zbl

[M] A. McInroy, Orbifold mirror symmetry for complex tori, preprint | MR

[KS] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, math.SG/0011041 (2001) | MR | Zbl

[SYZ] A. Strominger, S.T. Yau, E. Zaslow, Mirror Symmetry is T-Duality, Nucl. Phys. B479 (1996) 243-259; hep-th/9606040 | MR | Zbl

[V1] M.S. Verbitsky, Action of the Lie algebra of $SO (5)$ on the cohomology of a hyper-Kähler manifold, (Russian) Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 70–71; translation in Funct. Anal. Appl. 24 (1990), no. 3, 229–230 (1991) | MR | Zbl

[V2] M.S. Verbitsky, Hyperholomorphic bundles over a hyper-Kähler manifold J. Algebraic Geom. 5 (1996), no. 4, 633–669 | MR | Zbl

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