Applications of quantum cohomology to birational geometry I & II

主讲人 / Speaker

马克西姆 · 孔采维奇

Maxim Kontsevich

法国高等科学研究所(IHES)常任教授

菲尔兹奖得主(1998)

马克西姆·孔采维奇, 法国俄裔数学物理学家，1998年菲尔兹奖得主，法国科学院院士。主要研究领域包括扭结理论，量子化和镜像对称。他的主要贡献有：对任意泊松流形有效的形变量子化，拓扑场论中的稳定映像的模空间，利用一种类似费曼路径积分的复杂积分构造的扭结不变量。

除了1998年菲尔兹奖，孔采维奇教授还在1997年获庞加莱奖、2008年获克拉福德奖、2012年获邵逸夫奖和基础物理学奖、2014年获数学突破奖。瑞典皇家科学院曾这样评价他与合作者：孔采维奇和威滕使用物理学方法论开创了一门用于研究不同类型集合物体的新数学学科，具有“突破性意义”。

Abstract

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.