Topological invariants of the complement to arrangements of rational plane curves
About this Title
José Ignacio Cogolludo-Agustín
Publication: Memoirs of the American Mathematical Society
Publication Year 2002: Volume 159, Number 756
ISBNs: 978-0-8218-2942-4 (print); 978-1-4704-0349-2 (online)
MathSciNet review: 1921584
MSC (2000): Primary 32S22; Secondary 14H30, 52C30, 55N25
In the present work we analyze two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Our main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2-forms and its structure depends on a finite number of invariants of the curve (its combinatorial type). This theorem generalizes similar results for arrangements of lines by Brieskorn and Orlik-Solomon. We also define a family of complexes (Generalized Aomoto Complexes) that play an important role in determining the characteristic varieties. They are equivalent to purely combinatorial complexes that were already defined for hyperplane arrangements (Aomoto Complexes). The resonance varieties of such complexes allow one to study the cohomology support loci of rank one local systems on the complement of a curve. In particular, we prove that the irreducible subgroups of the characteristic varieties of a rational arrangement are fully determined by its combinatorial data.
Graduate students and research mathematicians interested in several complex variables, analytic spaces, algebraic geometry, and algebraic topology.
Table of Contents
- 1. Preliminaries
- 2. Ring structure of
- 3. Generalized Aomoto complexes
- 4. Characteristic varieties, local systems and rational arrangements
- 5. Examples