Positivity of Riemann-Roch polynomials and Todd classes of hyperk\"{a}hler manifolds

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 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 hyperk"{a}hler 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 ${\text{td}}^{1\mathrm{/}2}(X)$, the root of the Todd genus of $X$, via the Rozansky-Witten theory following the ideas of Hitchin, Sawon, and Nieper-Wi{\ss}kirchen.