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The twistor space of a compact hypercomplex manifold is never Moishezon

Yulia Gorginyan

Abstract

Let (X,I,J,K) be a compact hypercomplex manifold, i.e. a smooth manifold X with an action of the quaternion algebra (Id,I,J,K) on the tangent bundle TX, inducing integrable almost complex structures. For any $(a, b, c) \in S^2$, the linear combination $L := aI + bJ + cK$ defines another complex structure on X. This results in a $C P^1$-family of complex structures called the twistor family. Its total space is called the twistor space. We show that the twistor space of a compact hypercomplex manifold is never Moishezon and, moreover, it is never Fujiki class C (in particular, never Kahler and never projective).

The twistor space of a compact hypercomplex manifold is never Moishezon

Abstract

Let (X,I,J,K) be a compact hypercomplex manifold, i.e. a smooth manifold X with an action of the quaternion algebra (Id,I,J,K) on the tangent bundle TX, inducing integrable almost complex structures. For any , the linear combination defines another complex structure on X. This results in a -family of complex structures called the twistor family. Its total space is called the twistor space. We show that the twistor space of a compact hypercomplex manifold is never Moishezon and, moreover, it is never Fujiki class C (in particular, never Kahler and never projective).

Paper Structure

This paper contains 13 sections, 1 theorem, 25 equations.

Table of Contents

  1. Introduction
  2. 4-dimensional world
  3. Hypercomplex world
  4. Preliminaries
  5. Hypercomplex manifolds
  6. Hodge structures
  7. Variation of Hodge structures
  8. The Gauss-Manin connection and monodromy
  9. Cohomology of the blow-up
  10. Fujiki class $\mathcal{C}$ manifolds
  11. Variation of Hodge structures on Moishezon manifolds
  12. Hypercomplex Hopf manifolds and their twistor spaces
  13. The twistor space of a compact hypercomplex manifold

Key Result

theorem Theorem 2.3

(Newlander--Nirenberg) If $I$ is an integrable almost complex structure on $M$, then $M$ admits the structure of a complex manifold compatible with $I$.

Theorems & Definitions (1)

  • theorem Theorem 2.3