Harvard-MIT Algebraic Geometry Seminar: Quantum periods, toric degenerations and intrinsic mirror symmetry

Mirror symmetry for Fano varieties predicts a relation between the enumerative geometry of a Fano variety Y and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition of Y into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor D. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce LG models associated to a pair (Y,D), assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In particular, we show that classical periods of this LG model recover the quantum periods of Y, and that these periods are equal to a certain naive curve count on Y. In the setting when Y\D is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and corresponding toric degenerations. As an example, we consider Y = Gr(n-k,n), D a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on Gr(k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams.