12:30 pm Tuesday, March 31, 2015
GADGET : Homological mirror symmetry for GIT quotients - a case study by Gabe Kerr (KSU) in 9th floor
The homological mirror symmetry conjecture for toric varieties has been thoroughly studied and, in many cases, proven. However, few of these proofs are natural from the perspective of GIT where a toric variety X is viewed as the geometric quotient of a quasi-affine variety U by a complex torus T. This is due to the absence of an honest homological mirror to quasi-affine varieties. In this talk, I will recall a construction of the mirror skeleton Z for a general toric variety X (affine, quasi-affine and projective) introduced by Bondal and studied by Fang, Liu, Treumann and Zaslow. Using this skeleton, I will describe a partially wrapped Fukaya category F(Z), defined on a neighbourhood of Z, which can be taken to be the categorical mirror of X. I will sketch the proof of the categorical equivalence of F(Z) with D^b (X) and detail the natural GIT correspondence. If time permits, I will discuss semi-orthogonal decompositions of the mirror categories from crossing walls in the GIT, or Mori, fan. This talk is based on joint work in progress with Ludmil Katzarkov and Maxim Kontsevich.
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