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 paper, using Čech-to-derived spectral sequences, we investigate the mod $2$ and integral cohomology groups of a real Lagrangian ${\breve{L}}_{\mathbb{R}}$, obtained as the fixed locus of an anti-symplectic involution in the mirror to $X$. We show that ${\breve{L}}_{\mathbb{R}}$ is the disjoint union of a $3$-sphere and a rational homology sphere. Analysing the mod $2$ cohomology further, we deduce a correspondence between the mod $2$ Betti numbers of ${\breve{L}}_{\mathbb{R}}$ 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 ${\breve{L}}_{\mathbb{R}}$ with certain Hodge numbers of $X$. Furthermore, we show that the integral cohomology groups $H^j({\breve{L}}_{\mathbb{R}},\mathbb{Z})$ of ${\breve{L}}_{\mathbb{R}}$ are $2$-primary for $j\ne 0,3$; we conjecture that this holds in much greater generality.

36 pages, 7 figures. Improved exposition: several explanations on the main computations are added, some minor shuffling and corrections are done