21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

Abe, Takuro : The 8th COE Lecture Series

Hyperplane arrangements and geometry of logarithmic vector fields

A hyperplane arrangement is a finite collection of affine hyperplanes in a fixed vector space. This is a very simple mathematical object, but there are a lot of interesting studies on it using several mathematics, e.g., combinatorics, topology, algebra, complex and algebraic geometry.

In this talk, we begin with the elementary definitions of hyperplane arrangements and show some theorems, especially those related to combinatorics, algebra and algebraic geometry. Moreover, we focus attention on the logarithmic vector fields, and want to explain how these geometric objects show the combinatorial aspects and structures of hyperplane arrangements.

The organization of this talk is as follows:

In the first talk, we introduce the elementary and combinatorial theory of arrangements and prove Zaslavsky's theorem, which asserts that, in a real vector space, the cardinality of chambers of the complement space of hyperplanes in an arrangement is determined by the combinatoriacs of arrangements.

In the second talk, we introduce a module of logarithmic vector fields with respect to an arrangement. Introducing some related topics (e.g., multiarrangements or free arrangements), we prove Terao's factorization theorem, which is the formula on the combinatorics and logarithmic vector fields.

In the third talk, we consider the sheaf of logarithmic vector fields. This (reflexive) sheaf is intensively studied in these years by Schenck or Yoshinaga. We show some recent results by them, and prove the theorem which asserts the freeness and (Mumford-Takemoto's) stability of some arrangements induced from Coxeter groups (type A_2 and B_2) are determined by some combinatorics.

21st Century COE Program : Mathematics of Nonlinear Structure via Singularity