Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Mirror symmetry of abelian varieties and multi-theta functions

Author: Kenji Fukaya
Journal: J. Algebraic Geom. 11 (2002), 393-512
Published electronically: February 27, 2002
MathSciNet review: 1894935
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Abstract: We study homological mirror symmetry conjecture of symplectic and complex torus. We will associate a mirror torus $(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$ to each symplectic torus $(T^{2n},\omega )$ together with a closed 2 form $B$ which we call a $B$-field. We will associate a coherent sheaf ${\mathcal E}(L,{\mathcal L})$ on $(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$ to each pair $(L,{\mathcal L})$ of affine Lagrangian submanifolds $L$ and a flat complex line bundle ${\mathcal L}$ on $L$. In the case of affine Lagrangian submanifolds, we show that the Floer homology of Langrangian submanifolds is isomorphic to the extension of the mirror sheaf ${\mathcal E}(L,{\mathcal L})$. We construct a canonical isomorphism in the case when a certain transversality condition is satisfied. Our isomorphism then is functorial.

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Additional Information

Kenji Fukaya
Affiliation: Department of Mathematics, Faculty of Sciences, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto, 602-8502 Japan
MR Author ID: 217346

Received by editor(s): July 29, 1998
Published electronically: February 27, 2002
Additional Notes: Partially supported by Grant-in-Aid for Scientific Research 13852001