Cohomology algebra of plane curves, weak combinatorial type, and formality

Abstract:

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ of the complement to an algebraic curve $\mathcal {C}$ 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 $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ depends only on the following finite pieces of data: the number of irreducible components of $\mathcal {C}$ 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 $\mathcal {C}$. This finite set of data is referred to as the weak combinatorial type of $\mathcal {C}$. A further corollary is that the twisted cohomology jumping loci of $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ containing the trivial character also depend on the weak combinatorial type of $\mathcal {C}$. 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.