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Formulae for Line Bundle Cohomology on Calabi‐Yau Threefolds
Fortschritte der Physik ( IF 3.9 ) Pub Date : 2019-11-10 , DOI: 10.1002/prop.201900084
Andrei Constantin 1 , Andre Lukas 1
Affiliation  

We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi‐Yau threefolds realised as complete intersections in products of projective spaces. The formulae have been obtained by systematising and extrapolating concrete calculations and they have been checked computationally. Although the intermediate calculations often involve laborious computations of ranks of Leray maps in the Koszul spectral sequence, the final results for cohomology follow a simple pattern. The space of line bundles can be divided into several different regions, and in each such region the ranks of all cohomology groups can be expressed as polynomials in the line bundle integers of degree at most three. The number of regions increases and case distinctions become more complicated for manifolds with a larger Picard number. We also find explicit cohomology formulae for several non‐simply connected Calabi‐Yau threefolds realised as quotients by freely acting discrete symmetries. More cases may be systematically handled by machine learning algorithms.

中文翻译:

Calabi-Yau三元组上线束同调的公式

我们以在投影空间积中完全交集的几个Calabi-Yau三重性上的全纯线束的所有同调组的秩的形式,给出封闭形式的表达式。这些公式是通过对具体计算进行系统化和外推而获得的,并且已经过计算检查。尽管中间计算通常涉及对Koszul光谱序列中Leray映射等级的费力计算,但同​​调的最终结果遵循简单的模式。线束的空间可以划分为几个不同的区域,并且在每个这样的区域中,所有同调群的秩可以表示为度数最大为3的线束整数中的多项式。对于具有较大皮卡德数的歧管,区域数量增加并且区分大小写变得更加复杂。我们还发现了通过自由作用离散对称性而实现为商的几个非简单连接的Calabi-Yau三折式的显式同调公式。机器学习算法可以系统地处理更多情况。
更新日期:2019-11-10
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