Speaker: Sergey Galkin

Title: The conifold point, conjecture O, and related problems

Date: June 9, 2015, 16:30

Place: Max Planck Institute for Mathematics, Bonn

Conjecture O describes the geometry in the complex line of the eigenvalues u_i of the operator of quantum multiplication by the first Chern class acting on the cohomology of a Fano manifold. In particular, it says that eigenvalues with maximal absolute value have multiplicity one and one of them is real and positive number T. Fano manifolds tend to have mirror dual Ginzburg-Landau potentials f, which tend to have a distinguished non-degenerate critical point which we name the conifold point. Explicitly the conifold point is the unique critical point P on real positive locus, and the respective critical value T_{con} = f(P) is the global minimum on the real positive locus. In this case it is conjectured that T_{con} coincides with T, that is any eigenvalue u_i has absolute value at most T_{con}, and that the conifold point is the unique critical point with value T_{con}. In most cases the existence of the conifold point is clear and the conjectures can be checked to be true, however we do not know how to prove them even for abstract toric Fano manifolds, or complete intersections therein. These conjectures are basic for formulation of Gamma conjectures about the appearance of Gamma function in the symplectic topology of Fano manifolds. Two references are arXiv:1404.7388 and my joint work with Golyshev and Iritani arXiv:1404:6407.