SYZ mirror symmetry in the complement of a divisor and regular functions on the mirror
Hot Topics: Cluster algebras and wallcrossing March 28, 2016  April 01, 2016
Location: SLMath: Eisenbud Auditorium
symplectic mirror symmetry
homological mirror symmetry
Fukaya category
Lagrangian Floer homology
sheaf cohomology
51F20  Congruence and orthogonality in metric geometry [See also 20H05]
51F25  Orthogonal and unitary groups in metric geometry [See also 20H05]
51F30  Lipschitz and coarse geometry of metric spaces [See also 53C23]
51E20  Combinatorial structures in finite projective spaces [See also 05Bxx]
14476
We will give an overview of the StromingerYauZaslow (SYZ) approach to mirror symmetry in the setting of noncompact CalabiYau varieties given by the complement U of an anticanonical divisor D in a projective variety X. Namely, U is expected to carry a Lagrangian torus fibration, and a mirror CalabiYau variety U' can then be
constructed as a (suitably corrected) moduli space of Lagrangian torus fibers equipped with local systems. (Partial) compactifications of U deform the symplectic geometry of these Lagrangian tori by introducing holomorphic discs; counting these discs yields distinguished regular functions on the mirror U'. The goal of the talk will be to illustrate these concepts on simple examples, such as the complement of a conic in C^2.
If time permits we will also try to explain the relation of this story to the symplectic cohomology of U and its product structure
Auroux Notes

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