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Symmetric powers of symmetric bilinear forms, homogeneous orthogonal polynomials on the sphere and an application to compact hyperkähler manifolds. (English) Zbl 1374.14005

The Beauville-Bogomolov-Fujiki form \(q\) for a compact hyperkähler manifold \(X\) is a quadratic form on the integral cohomology \(H^2(X,\mathbb{Z})\) defined by an equation \(q(x)^k=I(x^{2k})\), \(I\) a linear form. The map \((f,g)\to I(fg)\) defines a symmetric bilinearform on \(\text{Sym}^kH^2(X, \mathbb{Z})\). A formula for the determinant of the Gram matrix of this bilinear form is given.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
15A63 Quadratic and bilinear forms, inner products
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
32Q15 Kähler manifolds

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References:

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