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Algebraic approximations of compact Kähler threefolds

Hsueh-Yung Lin

Abstract

We prove that every compact Kähler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

Algebraic approximations of compact Kähler threefolds

Abstract

We prove that every compact Kähler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

Paper Structure

This paper contains 41 sections, 68 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The Kodaira problem in dimension 3
  3. A general approach proving the existence of algebraic approximations
  4. Bimeromorphic models of compact Kähler threefolds
  5. Algebraic approximations of non-uniruled compact Kähler threefolds of algebraic dimension $a \le 1$
  6. Algebraic approximations of uniruled threefolds
  7. Organization of the article and remark on the dependence on HYLbimkod1HYLkodfibellip
  8. Preliminaries and general results
  9. Basic notions and terminologies
  10. $G$-equivariant locally trivial deformations
  11. General approaches proving the existence of algebraic approximations
  12. Campana's criterion
  13. Good Stein open covers
  14. Smooth torus fibrations and tautological families
  15. Smooth torus fibrations as torsors under the Jacobian fibration
  16. ...and 26 more sections

Key Result

Theorem 1.1

Every compact Kähler manifold of dimension 3 has an algebraic approximation.

Theorems & Definitions (144)

  • Theorem 1.1
  • Definition 1.2: Algebraic approximation
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture 1.5: Peternell
  • Corollary 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Theorem 1.10: HYLkodfibellip
  • ...and 134 more