| Résume | L'exposé sera diffusé (396 751 8661, demander le mot de passe à Oliver B. ou D. ou Frederic H.)
Abstract: For a hyperk\"{a}hler 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 follows 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. |
