Toric GIT quotients and some homological mirror symmetry
Michela Barbieri, University College London
Anything about mirror symmetry refers to some mysterious relationships
between complex and symplectic geometry. Roughly speaking, homological
mirror symmetry says that if you have some complex geometry X, there
is a mirror symplectic geometry Y such that D^b(Coh X) is equivalent
to the Fukaya category of Y, denoted Fuk(Y). In fact, starting from a
complex geometry (think an algebraic variety) there isn't just one mirror.
There's a family of mirrors living over a parameter space, which is
sometimes referred to as the Stringy Kähler Moduli Space (SKMS). The
fundamental group of the SKMS acts naturally on Fuk(Y) via monodromy,
and by mirror symmetry, we expect to see this action carry over. The way
it carries over is believed to be through spherical functors.
My goal will be to explain some details of this story in the context of
Calabi-Yau toric geometric invariant theory. I'll start off explaining some things
about GIT, mainly focusing on examples, and see where we get to!