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Calabi-Yau threefolds with boundary

Simon Donaldson, Fabian Lehmann

Abstract

We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the 5-dimensional boundary. In the case of strongly pseudoconvex boundary, we obtain an analogue of Hitchin's local Torelli Theorem for compact manifolds, modulo a finite dimensional obstruction space, which we show is zero in many cases of interest.

Calabi-Yau threefolds with boundary

Abstract

We develop the deformation theory of Calabi-Yau threefolds, by which we mean 3-dimensional complex manifolds with a nowhere-vanishing holomorphic 3-form, on manifolds with boundary. The boundary data is a closed, real 3-form on the 5-dimensional boundary. In the case of strongly pseudoconvex boundary, we obtain an analogue of Hitchin's local Torelli Theorem for compact manifolds, modulo a finite dimensional obstruction space, which we show is zero in many cases of interest.
Paper Structure (16 sections, 44 theorems, 250 equations)

This paper contains 16 sections, 44 theorems, 250 equations.

Table of Contents

  1. Background
  2. Complex Geometry
  3. Some elliptic boundary value problems
  4. Review of the $\bar{\partial}$-Neumann problem
  5. Geometry on the boundary
  6. Gauge fixing
  7. Gauge fixing for the domain of the torsion-free $\mathrm{SL}(3,{\mathbb{C}})$-equation
  8. Gauge fixing for target space of the torsion-free $\mathrm{SL}(3,{\mathbb{C}})$-equation
  9. Proof of Theorem \ref{['MainThm']}
  10. Existence under stronger hypothesis
  11. Proof under general hypothesis
  12. The deformation space
  13. The space of infinitesimal deformations
  14. Comparison with CR deformation theory and extension to higher dimensions
  15. An example: $T^{3}\times B^{3}$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Psi_0$ be the real part of a Calabi-Yau structure on $Z$ with enhanced boundary value $\hat{\psi}_0$. If $\Psi_0$ is rigid then for any enhanced boundary value close to $\hat{\psi}_0$ there is a Calabi-Yau structure with real part $\Psi$ close to $\Psi_0$ with that enhanced boundary value and

Theorems & Definitions (75)

  • Theorem 1
  • Lemma 1.2: see DonaldsonBdryRemarks
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • Proposition 1.9
  • ...and 65 more