地点：海纳苑2幢204室

时间：3月9日，16:00 - 17:20

报告人：江辰 Chen Jiang（复旦大学）

摘要:  

Pretalk:  Introduction to hyperkahler manifolds

Abstract: A hyperkahler manifold is a higher dimensional analogue of K3 surfaces. Such manifolds have many interesting geometric properties and are among one type of the building blocks of manifolds with trivial first Chern classes together with torus and Calabi-Yau manifolds. I will briefly recall basic definitions and properties of them.

Research talk: Positivity in hyperkahler manifolds via Rozansky-Witten theory

Abstract: For a hyperkahler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that $$\chi(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$ for any line bundle $L$ on $X$, where $q_X$ is the Beauville--Bogomolov--Fujiki quadratic form of $X$. Here the polynomial $\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann--Roch polynomial of $X$. In this talk, I will discuss the positivity of coefficients of the Riemann--Roch polynomial and also positivity of Todd classes. Such positivity results follows from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky—Witten theory.