Speaker: Sergey Galkin

Title: On positive aspects of mirror symmetry

Date: Jan 10, 2014, 10:10 - 11:10

Place: National Taiwan University

Together with Coates, Corti, Golyshev and Kasprzyk we're trying to reshape the classification of Fano manifolds by use of mirror symmetry, and to understand better the case of mirror symmetry in presence of natural positive structures. The key point of our programme is that it is much easier to construct a suitable mirror dual to a Fano manifold than to invent geometrically a new Fano manifold. We redone quite easily the classification of Fano threefolds (which was originally obtained by Fano, Iskovskikh and Mori-Mukai as a result of very involved geometric parsing), and we're in progress of preparing the list of the perspective mirrors for perspective Fano fourfolds. I'll sketch the programme and address the technical requirements for its final realization. To a Fano manifold X we associate a collection of Laurent polynomials m_0(L) for various (special, monotone, weakly unobstructed, SYZ) Lagrangian tori on X. Those potentials differ by cluster transformations, and as a whole they provide a structure of a cluster variety U on the universal family of the mirror-dual Calabi-Yau manifolds over their moduli space. The space U has a distinguished positive structure, and its totally positive locus (points with positive coordinates) form a distinguished Lagrangian, whose Floer homology coincides with the anti-canonical ring of X, so X can be constructed as a projective spectrum of a Floer homology algebra of one particular explicit (non-compact) Lagrangian.