Kapfer, Simon Symmetric powers of symmetric bilinear forms, homogeneous orthogonal polynomials on the sphere and an application to compact hyperkähler manifolds. (English) Zbl 1374.14005 Commun. Contemp. Math. 19, No. 2, Article ID 1650007, 19 p. (2017). 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. Reviewer: Gerhard Pfister (Kaiserslautern) 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 Keywords:symmetric bilinear forms on symmetric powers; orthogonal polynomials in several variables; homogeneous orthogonal polynomials; Gegenbauer polynomials; ultraspherical polynomials; Hermite polynomials; spherical harmonics; Hankel matrices; hyperkähler manifolds; irreducible holomorphically symplectic manifolds; Beauville-Bogomolov form; Beauville-Fujiki relation Software:HilbK3 PDFBibTeX XMLCite \textit{S. Kapfer}, Commun. Contemp. Math. 19, No. 2, Article ID 1650007, 19 p. (2017; Zbl 1374.14005) Full Text: DOI arXiv References: [1] Boissière, S., Nieper-Wißkirchen, M. and Sarti, A., Smith theory and irreducible holomorphic symplectic manifolds, J. Topology6(2) (2013) 361-390. · Zbl 1295.14023 [2] Chihara, T. S., An Introduction to Orthogonal Polynomials, , Vol. 13 (Gordon and Breach Science Publishers, 1978). · Zbl 0389.33008 [3] F. Dai and Y. Xu, Spherical harmonics, preprint (2013); arXiv:1304.2585. [4] Dolgachev, I. V., Classical Algebraic Geometry (Cambridge University Press, 2012). · Zbl 1252.14001 [5] Dunkl, C. F. and Xu, Y., Orthogonal Polynomials of Several Variables, , Vol. 81 (Cambridge University Press, 2001). · Zbl 0964.33001 [6] Folland, G. B., How to integrate a polynomial over a sphere, Amer. Math. Monthly108(5) (2001) 446-448. · Zbl 1046.26503 [7] Gross, M., Huybrechts, D. and Joyce, D., Calabi-Yau Manifolds and Related Geometries, (Springer, 2003). · Zbl 1001.00028 [8] S. Kapfer, Computing cup-products in integral cohomology of Hilbert schemes of points on K3 surfaces, to appear in LMS J. Comput. Math. · Zbl 1347.14005 [9] McGarraghy, S., Symmetric powers of symmetric bilinear forms, Algebra Colloq.12(1) (2005) 41-57. · Zbl 1070.11013 [10] K. G. O’Grady, Compact Hyperkähler manifolds: General theory, Lecture notes (2014); www.mimuw.edu.pl/ gael/Document/hk-theory.pdf. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.