Symmetric Powers of Symmetric Bilinear Forms, Homogeneous Orthogonal Polynomials on the Sphere and an Application to Compact Hyperk\"ahler Manifolds

TL;DR

This paper explores symmetric powers of bilinear forms on hyperk"ahler manifolds, relates them to orthogonal polynomials on spheres, and constructs orthogonal polynomial bases with applications to geometry.

Contribution

It generalizes the Beauville-Fujiki relation to symmetric bilinear forms on symmetric powers and connects this to orthogonal polynomials on spheres.

Findings

Properties of symmetric powers of bilinear forms are characterized.

A basis of orthogonal homogeneous polynomials on spheres is constructed.

The construction links geometric forms to classical orthogonal polynomial theory.

Abstract

The Beauville-Fujiki relation for a compact Hyperk\"ahler manifold $X$ of dimension $2k$ allows to equip the symmetric power $\text{Sym}^kH^2(X)$ with a symmetric bilinear form induced by the Beauville-Bogomolov form. We study some of its properties and compare it to the form given by the Poincar\'e pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere $\mathbb{S}^d$, or equivalently, over $\mathbb{R}^{d+1}$ with a Gaussian kernel.