Positivity in hyperkahler manifolds via Rozansky-Witten theory

Positivity in hyperkahler manifolds via Rozansky-Witten theory

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Chen Jiang, Fudan University

Zoom link:  https://princeton.zoom.us/j/91248028438

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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 recent progress on the positivity of coefficients of the Riemann-Roch polynomial and also positivity of Todd classes. Such positivity results follow from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky-Witten theory, following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.