Positivity of Riemann–Roch polynomials and Todd classes of hyperkähler manifolds

For a hyperkähler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0$, $a_2$, …, $a_{2n}$ such that \begin{equation*} \chi (L) =\sum _{i=0}^n\frac {a_{2i}}{(2i)!}q_X(c_1(L))^{i} \end{equation*} 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 paper, we show that all coefficients of the Riemann–Roch polynomial of $X$ are positive. This confirms a conjecture proposed by Cao and the author, which implies Kawamata’s effective non-vanishing conjecture for projective hyperkähler manifolds. It also confirms a question of Riess on strict monotonicity of Riemann–Roch polynomials.

In order to estimate the coefficients of the Riemann–Roch polynomial, we produce a Lefschetz-type decomposition of $\mathrm {td}^{1/2}(X)$, the root of the Todd genus of $X$, via the Rozansky–Witten theory following the ideas of Hitchin and Sawon, and of Nieper-Wißkirchen.