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

Chen Jiang

Abstract

For a hyperkähler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that $$χ(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$ 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 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 $\text{td}^{1/2}(X)$, the root of the Todd genus of $X$, via the Rozansky-Witten theory following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.

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

Abstract

For a hyperkähler manifold of dimension , Huybrechts showed that there are constants such that for any line bundle on , where is the Beauville-Bogomolov-Fujiki quadratic form of . Here the polynomial is called the Riemann-Roch polynomial of . In this paper, we show that all coefficients of the Riemann-Roch polynomial of 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 , the root of the Todd genus of , via the Rozansky-Witten theory following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.

Paper Structure

This paper contains 21 sections, 29 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Positivity of Riemann--Roch polynomials
  3. A Lefschetz-type decomposition of ${\rm{td}}^{1/2}(X)$ via the Rozansky--Witten theory
  4. Preliminaries
  5. Complex structures
  6. Beauville--Bogomolov--Fujiki form and Riemann--Roch polynomial
  7. Characteristic values
  8. Todd genus and Todd classes
  9. Some linear algebra on symplectic forms
  10. An $\mathfrak{sl}_2$-action on the cohomology of a hyperkähler manifold
  11. The Rozansky--Witten theory
  12. The graph homology space
  13. Rozansky--Witten classes in general setting
  14. Rozansky--Witten classes of hyperkähler manifolds
  15. Conclusions
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be a hyperkähler manifold. Then all coefficients of the Riemann--Roch polynomial ${\rm{RR}}_X(q)$ are positive.

Figures (2)

  • Figure 1: A labelling of ${\bf w}_2\cup \ell$
  • Figure 3: A labelling of ${\bf w}_{2k}$

Theorems & Definitions (71)

  • Theorem 1.1
  • Example 1.2
  • Conjecture 1.3
  • Theorem 1.4: =Proposition \ref{['prop primitive']}+Theorem \ref{['td=sum tp']}
  • Corollary 1.5: =Corollary \ref{['lambda td']}
  • Corollary 1.6: =Corollary \ref{['upper td1/2']}
  • Theorem 2.1: fujiki, gross
  • Definition 2.2: nieper-jag
  • Proposition 2.3: cf. nieper-jag
  • Lemma 2.4: cf. huybrechts
  • ...and 61 more