Numerical invariants of hyper-K\"ahler manifolds

TL;DR

This paper investigates the numerical invariants of hyper-K"ahler manifolds, focusing on the Beauville form and Huybrechts-Riemann-Roch polynomial, revealing new constraints and symmetries especially in dimension 6.

Contribution

It provides new constraints on key invariants of hyper-K"ahler manifolds and proves the polynomial's positivity and symmetry properties, especially in dimension 6.

Findings

Constraints on Beauville quadratic form in dimension 6

Huybrechts-Riemann-Roch polynomial expressed as a positive combination of explicit polynomials

Symmetry property of the Huybrechts-Riemann-Roch polynomial

Abstract

We study various constraints on the Beauville quadratic form and the Huybrechts-Riemann-Roch polynomial for hyper-K\"ahler manifolds, mostly in dimension 6 and in the presence of an isotropic class. In an appendix, Chen Jiang proves that in general, the Huybrechts-Riemann-Roch polynomial can always be written as a linear combination with nonnegative coefficients of certain explicit polynomials with positive coefficients. This implies that the Huybrechts-Riemann-Roch polynomial satisfies a curious symmetry property