Papers › On the cohomology groups of real Lagrangians in Calabi-Yau threefolds
On the cohomology groups of real Lagrangians in Calabi-Yau threefolds
Hülya Argüz, Thomas Prince
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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 L̆_ℝ, obtained as the fixed locus of an anti-symplectic involution in the mirror to X. We show that L̆_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 L̆_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 L̆_ℝ with certain Hodge numbers of X. Furthermore, we show that the integral cohomology groups Hʲ(L̆_ℝ,ℤ) of L̆_ℝ are 2-primary for j ≠0,3; we conjecture that this holds in much greater generality.
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