Cohomology of compact hyperkaehler manifolds

1995/01/03 by Misha Verbitsky, Verbitsky, Misha
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Business #Cohomology #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #alg-geom #dg-ga #math.AG #math.DG

paper · pdf · doi:10.48550/arxiv.alg-geom/9501001

87 pages LaTeX 2.09

openalex publication_date 1995/01/03 · arxiv created 1995/05/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, dim H²(M)=n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie algebra so(n-3,3) acts by automorphisms on the cohomology ring H^*(M). Under this action, the space H²(M) is isomorphic to the fundamental representation of so(n-3,3). Let A^r be the subring of H^*(M) generated by H²(M). We construct an action of the Lie algebra so(n-2,4) on the space A, which preserves A^r. The space A^r is an irreducible representation of so(n-2,4). This makes it possible to compute the ring A^r explicitely.

Citations

• Principles of Algebraic Geometry
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I

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