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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Author: Chen Jiang
Journal: J. Algebraic Geom. 32 (2023), 239-269
Published electronically: August 3, 2022
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Abstract | References | Additional Information


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.

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Additional Information

Chen Jiang
Affiliation: Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, 2005 Songhu Road, Shanghai 200438, People’s Republic of China
ORCID: 0000-0003-0562-6240

Received by editor(s): December 28, 2020
Received by editor(s) in revised form: November 21, 2021
Published electronically: August 3, 2022
Additional Notes: The author was supported by National Key Research and Development Program of China (Grant No. 2020YFA0713200)
Article copyright: © Copyright 2022 University Press, Inc.