Hirzebruch-Riemann-Roch formulae on irreducible symplectic Kähler manifolds

In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible symplectic Kähler manifold $X$ of dimension $2n$, there are numbers $a_0, a_2, \dots , a_{2n}$ such that \begin{equation*} \chi (L) = \sum _{k = 0}^n \frac {a_{2k}}{(2k)!} q_X(c_1(L))^k \end{equation*} for the Euler characteristic of a line bundle $L$, where $q_X: \mathrm {H}^2(X, \mathbb {C}) \to \mathbb {C}$ is the Beauville-Bogomolov quadratic form of $X$. Using Rozansky-Witten classes similarly to Hitchin and Sawon, we obtain a closed formula expressing the $a_{2k}$ in terms of Chern numbers of $X$.