On the Cohomology Groups of Real Lagrangians in Calabi–Yau Threefolds. Issue 1 (2nd January 2023)

• Record Type: Journal Article
• Authors: Argüz, Hülya; Prince, Thomas
• Is Part Of: Experimental mathematics. Volume 32:Issue 1(2023)
• Page Start: 169
• Page End: 190
• Publication Date: 2023-01-02
• Subjects: real Lagrangian -- Calabi-Yau threefold -- mirror symmetry -- Hodge numbers -- Heegard decomposition; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2021.1926006
• Languages: English
• ISSNs: 1058-6458

Abstract

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 L ⌣ R, obtained as the fixed locus of an anti-symplectic involution in the mirror to X . We show that L ⌣ 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 L ⌣ 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 L ⌣ R with certain Hodge numbers of X . Furthermore, we show that the integral cohomology groups H j ( L ⌣ R, Z ) of L ⌣ R are 2-primary for j ≠ 0, 3, we conjecture that this holds in much greater generality.