We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) , which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.
Bibliographical noteFunding Information:
We would like to thank P. Seidel for introducing us to Conjecture 1.3 and for many helpful comments. We would like to thank R. Bezrukavnikov for his patient help with the proof of Corollary 8.2. We would also like to thank K. Fukaya, D. Kazhdan, and G. Tian for their helpful comments. R. Castaño-Bernard was partially supported by I.H.E.S. and Kansas State FDA and USRG awards. Matessi was partially supported by MIUR (“Riemannian metrics and differentiable manifolds”, PRIN 05). J. Solomon was partially supported by NSF grant DMS-0703722. This project was partially supported by NSF award DMS-0854989: “FRG: Mirror Symmetry & Tropical Geometry”.
- Calabi-Yau manifolds
- Homological mirror symmetry
- Lagrangian fibrations
- Symplectic manifolds