arXiv subjects

Alexandru I. Suciu

Publications and source records attributed to Alexandru I. Suciu.

3 recordsLinked to original sources

arXiv2024

Milnor fibrations of arrangements with trivial algebraic monodromy

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber $F$ can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for $b_1(F)$. We study here the "generic" case (in which $b_1(F)$ is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of $π_1(F)$. In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients.

math.AG↗

arXiv2017

On the topology of the Milnor fibration of a hyperplane arrangement

This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic varieties of the complement, and their tight relationship with orbifold fibrations and multinets on the underlying matroid. In favorable situations, this approach leads to a combinatorial formula for the first Betti number of the Milnor fiber and the algebraic monodromy. We also produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet are not homotopy equivalent: the difference is picked up by isolated torsion points in the higher-depth characteristic varieties.

math.AT↗

arXiv2009

Geometric and algebraic aspects of 1-formality

Formality is a topological property, defined in terms of Sullivan's model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting, the weaker 1-formality property allows one to reconstruct the rational pro-unipotent completion of the fundamental group, solely from the cup products of degree 1 cohomology classes. In this note, we survey various facets of formality, with emphasis on the geometric and algebraic implications of 1-formality, and its relations to the cohomology jump loci and the Bieri-Neumann-Strebel invariant. We also produce examples of 4-manifolds W such that, for every compact Kähler manifold M, the product M\times W has the rational homotopy type of a Kähler manifold, yet M\times W admits no Kähler metric.

math.AT↗