Symmetries of Lagrangian fibrations

2009/08/07 by Ricardo Castaño-Bernard, Diego Matessi, Jake P. Solomon · 20 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Bundle #Fiber bundle #Fibration #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Holomorphic function #Homogeneous space #Homotopy #K3 surface #Lagrangian #Line bundle #Mathematics #Mirror symmetry #Pure mathematics #Symplectic geometry #Symplectic manifold #Symplectomorphism #Torus #math.AG #math.SG #msc:14F05 #msc:53D12 #msc:53D40

paper · pdf · doi:10.1016/j.aim.2010.04.001

published in Advances in Mathematics 225(3), 1341-1386 (Elsevier BV) · 45 pages

arxiv created 2009/08/07 · openalex publication_date 2010/05/12 · arxiv updated 2012/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 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.

Citations

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Disk enumeration on the quintic 3-fold
• Mirror symmetry is T-duality
• On global action‐angle coordinates
• Hypergeometric functions and mirror symmetry in toric varieties
• Proceedings of the International Congress of Mathematicians
• Almost toric symplectic four-manifolds
• Microlocal branes are constructible sheaves
• Calibrated geometries