On the cohomology ring of a simple hyperkähler manifold (on the results of Verbitsky)
Let \(M\) be a simple compact hyperkähler manifold, i.e., a compact Kähler manifold supplied with a nondegenerate holomorphic \((2,0)\)-form \(w\) such that \(\dim H^{(2,0)} (M)=1\) and \(\dim H^{(1,0)} (M)=0\). \textit{M. Verbitsky} proved a structure theorem for the \(H^2(M,\mathbb{C})\)-generated part of \(H^* (M,\mathbb{C})\) [Geom. Funct. Anal. 6, No. 4, 601-611 (1996; Zbl 0861.53069)]. His technique used the existence of a special (hyperkähler) metric on \(M\). The goal of the present paper is to show that ``some of Verbitsky's results can be obtained via another approach based on the nonobstructedness of the deformations of the complex structure of \(M\)''. Let \(M\) be a simple compact hyperkähler manifold, \(\dim_\mathbb{C} M= 2n>2\), and \(X \to \text{Mod}\) a locally universal deformation of \(M\). Mod can be embedded in the projectivization \(P= P(H^2(M, \mathbb{C}))\), and the author proves that Mod is contained in a smooth quadric \(Q\): \(q=0\). The second order Laplacian differential operator \(\Delta\) corresponding to \(q\) acts on the space \(S^nH^2 (M,\mathbb{C})\) considered as the space of homogeneous functions on \((H^2 (M, \mathbb{C}))^*\). Let \(I_n\) be the subspace of harmonic elements \((\Delta x= 0)\) in \(S^nH^2 (M, \mathbb{C})\). It is proved: Theorem. The subalgebra \(A\) of \(H^*(M,C)\) generated by \(H^2(M, \mathbb{C})\) is the quotient of the symmetric algebra \(S^*H^2 (M,\mathbb{C})\) by the homogeneous ideal generated by \(I_{n+1}\). Corollary. The action of the orthogonal group \(O(q)\) on \(H^2(M, \mathbb{C})\) extends into the automorphisms of the algebra \(A\).
scientific article; zbMATH DE number 934956
scientific article