On the Cohomology Groups of Real Lagrangians in Calabi–Yau Threefolds

Abstract

The quintic threefold X is the most studied Calabi-Yau 3-fold in the mathematics literature. In this article, using Čech-to-derived spectral sequences, we investigate the mod 2 and integral cohomology groups of a real Lagrangian ${\stackrel{⌣}{L}}_{\mathbb{R}}$, obtained as the fixed locus of an anti-symplectic involution in the mirror to X. We show that ${\stackrel{⌣}{L}}_{\mathbb{R}}$ is the disjoint union of a 3-sphere and a rational homology sphere. Analyzing the mod 2 cohomology further, we deduce a correspondence between the mod 2 Betti numbers of ${\stackrel{⌣}{L}}_{\mathbb{R}}$ and certain counts of integral points on the base of a singular torus fibration on X. By work of Batyrev, this identifies the mod 2 Betti numbers of ${\stackrel{⌣}{L}}_{\mathbb{R}}$ with certain Hodge numbers of X. Furthermore, we show that the integral cohomology groups $H^j({\stackrel{⌣}{L}}_{\mathbb{R}},\mathbb{Z})$ of ${\stackrel{⌣}{L}}_{\mathbb{R}}$ are 2-primary for $j\ne 0,3$, we conjecture that this holds in much greater generality.

KEYWORDS:

• real Lagrangian
• Calabi-Yau threefold
• mirror symmetry
• Hodge numbers
• Heegard decomposition

Acknowledgments

We thank to Mark Gross and Tom Coates for many useful and inspiring conversations.

Additional information

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682603). H.A. was supported by Fondation Mathématique Jacques Hadamard. TP was partially supported by a Fellowship by Examination at Magdalen College, Oxford.