On the cohomology groups of real Lagrangians in Calabi-Yau threefolds.

The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using \v{C}ech-to-derived spectral sequences, we investigate the 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 of $\breve{L}_\mathbb{R}$ are $2$-primary; we conjecture that this holds in much greater generality.