Homological mirror symmetry for elliptic Hopf surfaces

Abstract

We show homological mirror symmetry results relating coherent analytic sheaves on some complex elliptic surfaces and objects of certain Fukaya categories. We first define the notion of a non-algebraic Landau-Ginzburg model on $\mathbb{R}\times S^1$ and its associated Fukaya category, and show that non-Kähler sur-faces obtained by performing two logarithmic transformations to the product of the projective plane and an elliptic curve have non-algebraic Landau-Ginzburg models as their mirror spaces; this class of surface includes the classical Hopf surface $S^1\times S^3$ and other elliptic primary and secondary Hopf surfaces. We also define localization maps from the Fukaya categories associated to the Landau-Ginzburg models to partially wrapped and fully wrapped categories. We show mirror symmetry results that relate the partially wrapped and fully wrapped categories to spaces of coherent analytic sheaves on open submanifolds of the compact complex surfaces in question, and we use these results to sketch a proof of a full HMS result.

arXiv comment69 pages, 3 figures, comments welcome

References in corpus (6)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Covariantly functorial wrapped Floer theory on Liouville sectors
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Family Floer cohomology and mirror symmetry
• Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions
• Orlov and Viterbo functors in partially wrapped Fukaya categories