Resumen de Cohomology Algebra of plane curves, weak combinatorial type, and formality

José Ignacio Cogolludo Agustín , Daniel Matei

• We determine an explicit presentation by generators and relations of the cohomology algebra H(P2 \ C,C) of the complement to an algebraic curve C in the complex projective plane P2 via the study of log-resolution logarithmic forms on P2. As a first consequence, we derive that H(P2 \ C,C) depends only on the following data: the number of irreducible components of C together with their degrees and genera, the pairwise intersection numbers of distinct local branches at the singular points of C, and, at each singular point, the incidence relations between the local branches and the global irreducible components passing through it. A further corollary is that the twisted cohomology jumping loci of H(P2 \ C,C) containing the trivial character also depend on the same data. We relate, in this context, the geometric and combinatorial properties of the curve C through the notion of combinatorial pencil. This notion will be used to derive a combinatorial version of the Max-Noether Theorem. Finally, we find that the relations in the cohomology algebra are in fact satisfied, for appropriate choices of representatives, at the level of differential forms, concluding that H(P2 \ C,C) is a formal space.