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摘要翻译：
通过对$\mathbbp^2$上的对数分辨对数形式的研究，我们确定了复射影平面$\mathbbp^2$上补的上同调代数$h^*(\mathbbp^2\set-C,\mathbbc)$与代数曲线$C$的关系。作为第一个结果，我们推导出$H^*(\mathbb p^2\set-c，\mathbb C)$只依赖于以下有限的数据：$C$的不可约分量及其度和属的数目，每个分量在每个奇点上的局部分支的数目，以及在$C$的每个奇点上每两个不同的局部分支的交数。这个有限的数据集被称为弱组合类型$C$。进一步的推论是包含平凡特征的扭曲上同调跳跃轨迹$h^*(\mathbb p^2\setminus C,\mathbb C)$也依赖于弱组合类型$C$。最后，通过生成元和关系的显式构造，证明了平面射影曲线的补是Sullivan意义下的形式空间。
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英文标题：
《Cohomology algebra of plane curves, weak combinatorial type, and
formality》
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作者：
J.I.Cogolludo-Agustin and D.Matei
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最新提交年份：
2010
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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英文摘要：
We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the study of log-resolution logarithmic forms on $\mathbb P^2$. As a first consequence, we derive that $H^*(\mathbb P^2\setminus C,\mathbb C)$ depends only on the following finite pieces of data: the number of irreducible components of $C$ together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of $C$. This finite set of data is referred to as the weak combinatorial type of $C$. A further corollary is that the twisted cohomology jumping loci of $H^*(\mathbb P^2\setminus C,\mathbb C)$ containing the trivial character also depend on the weak combinatorial type of $C$. Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan.
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PDF链接：
https://arxiv.org/pdf/0711.1951
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