Applications of quantum cohomology to birational geometry

主讲人 Speaker:Maxim Kontsevich
时间 Time:16:30-17:30, 2024-03-21/27
地点 Venue:Jian Hua Building 建华楼LG1-21报告厅, Tsinghua University; Zoom Meeting ID: 4552601552 Passcode: YMSC


2024-03-21 West Lecture Hall(西阶), Tsinghua University; Zoom Meeting ID: 4552601552 Passcode: YMSC

2024-03-27  Jian Hua Building 建华楼LG1-21报告厅; Zoom Meeting ID: 4552601552 Passcode: YMSC


Theory of quantum multiplication and Gromov-Witten invariants of a smooth projective algebraic variety X deals with enumerative questions concerning curves in X. A long time ago physicists E.Witten, R.Dijkgraaf, E.Verlinde and H.Verlinde discovered a remarkable system of non-linear differential equations on the generating series for Gromov-Witten invariants. In particular, one can calculate the number of rational curves of degree N in projective plane passing through generic 3N-1 points via a certain solution of Painleve VI equation.


The topic of my lectures is an application of theory of quantum multiplication to birational geometry (joint work with L.Katzarkov, T.Pantev and T.Yu), based on the recent blowup formula by H.Iritani.

In particular, we solved a long-standing conjecture on the non-rationality of a generic cubic hypersurface in 5-dimensional projective space.

In my first lecture I'll give a brief introduction to the theory of quantum cohomology and introduce a notion of atoms which play the central role in rationality questions.  In the second lecture I'll propose a series of conjectures relating atoms and derived category of coherent sheaves.