On the cohomology groups of real Lagrangians in Calabi–Yau threefolds

Journal article

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 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 $j \neq 0,3$; we conjecture that this holds in much greater generality.

Authors: Arguz, HA; Prince, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Taylor & Francis
• Journal: Experimental Mathematics
• Volume: 32
• Issue: 1
• Pages: 169-190
• Publication date: 2021-06-26
• Acceptance date: 2021-06-26
• DOI: 10.1080/10586458.2021.1926006
• EISSN: 1944-950X
• ISSN: 1058-6458
• Language: English
• Keywords: Heegard decomposition; real Lagrangian; Calabi-Yau threefold; mirror symmetry; Hodge numbers