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Estimation and Vanishing Results for Hodge numbers

Peter Petersen, Matthias Wink

Abstract

We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the Kähler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana's theorem for Kähler manifolds.

Estimation and Vanishing Results for Hodge numbers

Abstract

We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the Kähler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana's theorem for Kähler manifolds.

Paper Structure

This paper contains 8 sections, 19 theorems, 116 equations.

Table of Contents

  1. Preliminaries
  2. Tensors
  3. The Lie algebra $\mathfrak{u}(V)$
  4. Lichnerowicz Laplacians and holonomy
  5. $U(n)$-irreducible decomposition of $(p,q)$-forms
  6. Estimating the curvature term of the Lichnerowicz Laplacian
  7. The Lichnerowicz Laplacian on $(p,q)$-forms
  8. A Tachibana Theorem for Kähler manifolds

Key Result

Theorem A

Let $(M,g)$ be a compact connected Kähler manifold of complex dimension $n$. If then $(M,g)$ has the rational cohomology ring of $\mathbb{CP}^n.$

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary
  • Theorem D
  • Theorem E
  • Remark 1.1
  • Example 1.2
  • Proposition 1.3
  • proof
  • ...and 31 more