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On real Calabi-Yau threefolds twisted by a section

Diego Matessi

Abstract

We study the mod $2$ cohomology of real Calabi-Yau threefolds given by real structures which preserve the torus fibrations constructed by Gross. We extend the results of Castaño-Bernard-Matessi and Arguz-Prince to the case of real structures twisted by a Lagrangian section. In particular we find exact sequences linking the cohomology of the real Calabi-Yau with the cohomology of the complex one. Applying SYZ mirror symmetry, we show that the connecting homomorphism is determined by a ``twisted squaring of divisors'' in the mirror Calabi-Yau, i.e. by $D \mapsto D^2 + DL$ where $D$ is a divisor in the mirror and $L$ is the divisor mirror to the twisting section. We use this to find an example of a connected $(M-2)$-real quintic threefold.

On real Calabi-Yau threefolds twisted by a section

Abstract

We study the mod cohomology of real Calabi-Yau threefolds given by real structures which preserve the torus fibrations constructed by Gross. We extend the results of Castaño-Bernard-Matessi and Arguz-Prince to the case of real structures twisted by a Lagrangian section. In particular we find exact sequences linking the cohomology of the real Calabi-Yau with the cohomology of the complex one. Applying SYZ mirror symmetry, we show that the connecting homomorphism is determined by a ``twisted squaring of divisors'' in the mirror Calabi-Yau, i.e. by where is a divisor in the mirror and is the divisor mirror to the twisting section. We use this to find an example of a connected -real quintic threefold.
Paper Structure (40 sections, 14 theorems, 140 equations, 10 figures)

This paper contains 40 sections, 14 theorems, 140 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Lagrangian fibrations with real structures
  3. The Leray spectral sequence and mirror symmetry
  4. Main results
  5. Structure of the paper
  6. Lagrangian fibrations and mirror symmetry
  7. Dimension two: focus-focus points.
  8. The affine quartic
  9. Dimension three: positive and negative vertices
  10. The affine quintic
  11. Singular fibres
  12. Topological mirror symmetry
  13. The Leray spectral sequence
  14. Real structures
  15. The standard real structure
  16. ...and 25 more sections

Key Result

Theorem 1

Let $\tau$ be a Lagrangian section of $f: X \rightarrow B$ and $\iota_{\tau}$ the associated real structure. There exist sheaves $\mathcal{L}^{1}_{\tau}$ and $\mathcal{L}^{2}_{\tau}$ over $B$ and a short exact sequence such that $\mathcal{L}^{1}_{\tau}$ and $\mathcal{L}^{2}_{\tau}$ are related to the topology of $X$ by the following short exact sequences

Figures (10)

  • Figure 1: Charts around a focus-focus point
  • Figure 2: Charts near a negative vertex
  • Figure 3: Discriminant of an affine quintic
  • Figure 4: Triangulation of $2$-dimensional faces
  • Figure 5: Triple intersection graph
  • ...and 5 more figures

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Theorem 4.1
  • Theorem 5.1
  • ...and 10 more