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

Gaiffi, Giovanni; Grassi, Michele

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

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.

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.

[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 $\mathbf{sl}(6,ℂ)$, 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 $\mathrm{SO}\left(5\right)$ 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