Speaker: Sergey Galkin

Title: Decompositions in Algebraic and Symplectic Geometry, Gamma Conjectures, Mirror Symmetry, and their applications

Date: 16:00-17:00, every Tuesday and Friday from Jan 14 to Feb 14, 2020.

Place: AG-77

Abstract:

In this series of lectures, assuming just basics of topology and algebraic geometry,
I will review some aspects of mirror symmetry for Fano manifolds and Dubrovin's 1998 conjectures. Both mirror symmetry and Dubrovin's conjectures relate symplectic topology and algebraic geometry, However, mirror symmetry relates two distinct dual geometries, while Dubrovin's conjectures relate algebraic and symplectic geometric invariants of the same Fano manifold,
under the assumptopn that quantum cohomology of a Fano manifold is semi-simple (symplectic property) and/or that the derived category of coherent sheaves has a full exceptional collection (algebraic property).

Gamma conjectures were formulated by myself jointly with Golyshev and Iritani. Gamma conjectures make last part of Dubrovin's conjectures precise and further generalize it. They can be thought of as a square root of the index theorem, and they express asymptotics of the solutions of the quantum differential equation of a Fano manifold in topological terms. The existence of asymptotic decompositions of the solutions space is suggested by the existence of the mirror dual Landau-Ginzburg potential. Moreover, it gives numerical candidates for some distinguished semi-orthogonal decompositions for the Fano manifold.

Finally, recently Katzarkov, Kontsevich and Pantev claimed that asymptotic decompositions exist for all compact symplectic manifolds (not necessarily monotone/Fano), and moreover they satisfy a blow-up relation. These combined would justify a well-definedness of a birational invariant that is sufficient to detect non-rationality of a very general cubic fourfold.