Differential Geometry Seminar | Holomorphic discs and extended deformations in mirror symmetry

Abstract：

Given a Lagrangian torus fibration on the complement of an anticanonical divisor in a Kahler manifold, one usually constructs a mirror space by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations determined by counts of Maslov index 0 holomorphic discs; this mirror also comes equipped with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. However, holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations; the geometric features of the resulting mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon (and show that it actually occurs) on an explicit example (a 4-fold obtained by blowing up a Calabi-Yau toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting.

About the speaker：

Denis Auroux is a professor at Department of Mathematics，Harvard University. His research interests are Symplectic geometry, low-dimensional topology, mirror symmetry.

Personal Homepage：

https://people.math.harvard.edu/~auroux/