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

摘要

五次三重X是数学文献中研究最多的 Calabi-Yau 三重 X。在这篇文章中，我们使用 Çech-to-derived 谱序列，研究了实拉格朗日量的 mod 2 和积分上同调群 ${\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}}$，作为X 的镜像中反辛对合的固定轨迹获得。我们表明 ${\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}}$ 是 3-球和有理同调球的不相交并集。进一步分析 mod 2 上同调，我们推导出 mod 2 Betti 数之间的对应关系 ${\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}}$ 以及基于X上的奇异环面纤维化的某些积分点数。通过 Batyrev 的工作，这确定了 mod 2 Betti 数字 ${\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}}$ 具有X的某些霍奇数。此外，我们证明积分上同调群 $H^j({\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}},\mathbb{Z})$ 的 ${\stackrel{⌣}{\mathrm{大号}}}_{\mathbb{R}}$ 是 2-primary 的 $j\ne 0,\mathrm{3个}$，我们推测这具有更大的普遍性。