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

Abstract

For a hyperkähler manifold X X of dimension 2 n 2n , Huybrechts showed that there are constants a 0 a_0 , a 2 a_2 , …, a 2 n a_{2n} such that χ ( L ) = ∑ i = 0 n a 2 i ( 2 i ) ! q X ( c 1 ( L ) ) i \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 L on X X , where q X q_X is the Beauville–Bogomolov–Fujiki quadratic form of X X . Here the polynomial ∑ i = 0 n a 2 i ( 2 i ) ! q i \sum _{i=0}^n\frac {a_{2i}}{(2i)!}q^{i} is called the Riemann–Roch polynomial of X X .

In this paper, we show that all coefficients of the Riemann–Roch polynomial of X 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 t d 1 / 2 ( X ) \mathrm {td}^{1/2}(X) , the root of the Todd genus of X X , via the Rozansky–Witten theory following the ideas of Hitchin and Sawon, and of Nieper-Wißkirchen.