On the cohomology ring of a simple hyperkähler manifold (on the results of Verbitsky)
type: Item
instance of: scholarly article
title: On the cohomology ring of a simple hyperkähler manifold (on the results of Verbitsky)
zbMATH Open document ID: 0862.53050
DOI: 10.1007/BF02247113
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-05-25T00:00:00Z
full work available at URL: https://eudml.org/doc/58240
review text: 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 \(A\).
reviewed by: M. A. Malakhaltsev
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 32G07
zbMATH DE Number: 934956
zbMATH Keywords: moduli space
zbMATH Keywords: cohomology
zbMATH Keywords: hyperkähler manifold
author: Fedor Bogomolov
MaRDI profile type: Publication
cites work: Variétés kähleriennes dont la première classe de Chern est nulle
cites work: Q3704110
cites work: Q5287201
cites work: Cohomology of compact hyperkähler manifolds and its applications
cites work: Tri-analytic subvarieties of hyperkaehler manifolds
OpenAlex ID: W2058023827
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description: scientific article; zbMATH DE number 934956