## 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.