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Hodge-theoretic variants of the Hopf and Singer Conjectures

Donu Arapura, Laurentiu Maxim, Botong Wang

Abstract

We propose several Hodge theoretic analogues of the conjectures of Hopf and Singer, and prove them in some special cases.

Hodge-theoretic variants of the Hopf and Singer Conjectures

Abstract

We propose several Hodge theoretic analogues of the conjectures of Hopf and Singer, and prove them in some special cases.
Paper Structure (9 sections, 7 theorems, 28 equations)

This paper contains 9 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Singer-Hopf conjecture.
  3. Hodge-theoretic variants of the Singer-Hopf conjecture.
  4. Hopf conjecture for positive sectional curvature and generalizations.
  5. Proofs of results
  6. On Conjecture \ref{['aw1']}
  7. On Conjecture \ref{['aw1b']}
  8. On Conjecture \ref{['ch']}
  9. Acknowledgments

Key Result

Theorem 1.5

If $X$ is a compact Kähler (resp., complex projective) manifold with a nef (resp., ample) tangent bundle $TX$ (e.g., $X$ has a non-negative (resp., positive) sectional curvature), then $\chi(X) \geq 0$ (resp., $>0$).

Theorems & Definitions (26)

  • Conjecture 1.1: Singer-Hopf
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4: Hopf
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7: Campana-Peternell
  • Conjecture 1.8
  • Conjecture 1.9
  • Conjecture 1.10
  • ...and 16 more