Title:
Homological Mirror Symmetry for singularities from SYZ
Abstract:
A recent preprint of Abouzaid, Auroux, and Katzarkov constructs SYZ mirrors
for certain "generalized conic bundles" over toric varieties. In this talk,
I will begin by explaining the background, recalling the important elements
of the SYZ program and the construction by AAK. We then examine their
construction from the point of view of Homological Mirror Symmetry. In the
second part of the talk, we will examine conic bundles over $\mathbb{C}^n$,
focusing on the case of smoothings of the $A_{n,m}$ singularity. Using the
SYZ philosophy, we then construct mirrors to the $A_{n,m}$ chain of
Lagrangian spheres and we prove that these objects generate a certain
subcategory of compactly supported matrix factorizations. In the two
dimensional case, we also construct mirrors to non-compact Lagrangians
which generate the wrapped Fukaya category and prove that they generate the
entire category of matrix factorizations on the mirror as well. If time
remains, I will describe joint work in progress with Kazushi Ueda and
Kwokwai Chan where we compute wrapped Floer homology of Lagrangian sections
of the SYZ fibration for generalized conic bundles over $\mathbb{C}^*$ and
identify it with endomorphisms of certain line bundles on the mirror
Calabi-Yau.