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

摘要

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