Seminars

The Gauss-Bonnet-Chern theorem for singular spaces and Donaldson-Thomas invariants

Author:    Publicsh date:2018-06-21    Clicks:
题目:  The Gauss-Bonnet-Chern theorem for singular spaces and Donaldson-Thomas invariants报告人:蒋云峰 (University of Kansas)时间:2018年6月22日 14:00 - 15:00 地点:创新研究院813 摘要: The Gauss-Bonnet-Chern theorem states that for a smooth compact complex manifold X, the integration of the top Chern class over X is the topological Euler characteristic of X. In order to study Chern classes f...

题目:  The Gauss-Bonnet-Chern theorem for singular spaces and Donaldson-Thomas invariants
报告人:蒋云峰 (University of Kansas)

时间:2018年6月22日 14:00 - 15:00 

地点:创新研究院813 
摘要:

The Gauss-Bonnet-Chern theorem states that for a smooth compact complex manifold X, the integration of the top Chern class over X is the topological Euler characteristic of X. In order to study Chern classes for singular varieties or schemes, R. MacPherson introduced the notion of local Euler obstruction of singular varieties. The local Euler obstruction is an integer value constructible function on X, and the constant function 1_X can be written down as a linear combination of local Euler obstructions.  A characteristic class for a local Euler obstruction was defined by using Nash blow-ups, and is called its Chern-Mather class or Chern-Schwartz-MacPherson class. The Chern-Schwartz-MacPherson class of the constant function 1_X is defined as the Chern class for X.

Inspired by gauge theory in higher dimension and string theory, the curve counting theory via stable coherent sheaves was constructed by Donaldson-Thomas on projective 3-folds, which is now called the Donaldson-Thomas theory. In the case of Calabi-Yau threefolds, the Donaldson-Thomas invariants are proved by Behrend to be "weighted Euler characteristics" of the moduli space X, where the weights come from the local Euler obstruction of the moduli space X. In this talk I will survey some results of Donaldson-Thomas invariants along this line and talk about one case that how the Behrend weighted Euler characteristic is related to Y. Kiem and J. Li's cosection localization invariants.  If time permits, I will also survey the formal and non-Archimedean version of the moduli space (stack) of stable coherent sheaves and its relation to motivic Donaldson-Thomas invariants.