论文标题
在卡拉比Yau三倍的皇家拉格朗日人的同时组中
On the cohomology groups of real Lagrangians in Calabi-Yau threefolds
论文作者
论文摘要
五倍的$ x $是研究最多的卡拉比（Calabi-Yau）$ 3 $折 - 数学文献中。在本文中，使用čech到衍生的频谱序列，我们研究了一个真正的拉格朗日$ \ breve {l} _ {\ mathbb {r}} $的mod $ 2 $和整体共同体学组，作为反镜像到$ x $的反效果的固定位点所获得的。我们表明$ \ breve {l} _ {\ mathbb {r}} $是$ 3 $ - sphere和一个理性同源性领域的不相交联合。进一步分析mod $ 2 $共同体，我们推断出mod $ 2 $ 2 $ betti的$ \ breve {l} _ {\ mathbb {r}} $之间的对应关系与$ x $ nistular fiblation的基础上的积分点的某些积分点。通过BatyRev的工作，这将标识$ \ breve {l} _ {\ mathbb {r}} $的mod $ 2 $ betti号码，其中某些hodge号码为$ x $。此外，我们表明积分共同体组$ h^j（\ breve {l} _ {\ Mathbb {r}}，\ breve {l} _ {l} _ {\ mathbb {\ mathb {r}} $ 2 $ $ j $ j $ j $ j $ $ j n n n n neq 0.我们猜想这具有更大的一般性。
The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using Čech-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.
