2010/06/30 by Kwokwai Chan, Siu-Cheong Lau, Naichung Conan Leung · 28 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Conjecture #Fibration #Geometric and Algebraic Topology #Geometry and complex manifolds #Manifold (fluid mechanics) #Mirror symmetry #Symmetry (geometry) #Toric variety #Torus #math.AG #math.SG
paper · pdf · doi:10.4310/jdg/1335230845
published in Journal of Differential Geometry 90(2) (Lehigh University) · v3: final version, published in JDG 90 (2012), no. 2, 177-250. 71 pages, 14 figures; substantially revised and expanded
openalex publication_date 2012/02/01 · arxiv created 2012/03/14 · openalex created_date 2016/06/24 · arxiv updated 2017/05/19 · openalex updated_date 2026/08/05
We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold X, we construct a complex manifold \checkX using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold \checkX, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the Kähler parameters of X have integral coefficients. Applying the results in "A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry," to appear in Pacific J. Math., and "A relation for Gromov-Witten invariants of local Calabi-Yau threefolds," to appear in Math. Res. Lett., we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including Kℙ2 and Kℙ1.