# Topological Invariants of the Complement to Arrangements of Rational Plane Curves

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*José Ignacio Cogolludo-Agustín*

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.

#### Table of Contents

# Table of Contents

## Topological Invariants of the Complement to Arrangements of Rational Plane Curves

- Contents vii8 free
- Introduction ix10 free
- Chapter 1. Preliminaries 116 free
- Chapter 2. Ring structure of H[sup(•)](P[sup(2)] \ R;C) 1530
- 2.1. The spaces H[sup(k)](P[sup(2)] \ D;C) and the residue maps 1530
- 2.2. Logarithmic forms on non-normal crossing divisors 1833
- 2.3. A basis for H[sup(1)](P[sup(2)] \ D;C) 1934
- 2.4. The problem with the 2-forms 2136
- 2.5. Logarithmic ideals 2338
- 2.6. Construction of logarithmic trees 3045
- 2.7. A holomorphic presentation for H[sup(2)](P[sup(2)] \ R;C) 3853
- 2.8. The product in H[sup(•)](P[sup(2)] \ R;C) 4459

- Chapter 3. Generalized Aomoto Complexes 4762
- Chapter 4. Characteristic varieties, local systems and rational arrangements 5772
- Chapter 5. Examples 6378
- Bibliography 7489