Takuro Abe

The Euler multiplicity and addition-deletion theorems for multiarrangements

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the e-multiplicity, of a restricted multiarrangement. We compute the e-multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements.

The freeness of ideal subarrangements of Weyl arrangements

AbstractA Weyl arrangement is the arrangement defined by the root sys-tem of a finite Weyl group. When a set of positive roots is an ideal inthe root poset, we call the corresponding arrangement an ideal sub-arrangement. Our main theorem asserts that any ideal subarrange-ment is a free arrangement and that its exponents are given by thedual partition of the height distribution, which was conjectured bySommers-Tymoczko. In particular, when an ideal subarrangement isequal to the entire Weyl arrangement, our main theorem yields thecelebrated formula by Shapiro, Steinberg, Kostant, and Macdonald.Our proof of the main theorem heavily depends on the theory of freearrangements and thus greatly differs from the earlier proofs of theformula. 1 Introduction Let Φ be an irreducible root system of rank l and fix a simple system (orbasis) ∆ = {α 1 ,...,α l }. Let Φ + be the set of positive roots. Define thepartial order ≥ on Φ + such that α ≥ β if α − β ∈ Z ≥0 α 1 + ···+ Z ≥0 α l forα,β ∈ Φ + . A subset I of Φ

Splitting criterion for reflexive sheaves

The purpose of this paper is to study the structure of reflexive sheaves over projective spaces through hyperplane sections. We give a criterion for a reflexive sheaf to split into a direct sum of line bundles. An application to the theory of free hyperplane arrangements is also given.

Signed-eliminable graphs and free multiplicities on the braid arrangement

We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signed-eliminable graphs as a generalization of Stanley�s classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs.

Exponents of 2-multiarrangements and multiplicity lattices

We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.

Simple-root bases for Shi arrangements

Abstract In his affirmative answer to the Edelman–Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, we know very little about the bases themselves except their existence. In this article, we prove the unique existence of two distinguished bases which we call the simple-root basis plus (SRB+) and the simple-root basis minus (SRB−). They are characterized by nice divisibility properties relative to the simple roots.

The freeness of Shi-Catalan arrangements

Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H ? A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H . We say that a deformation is W -equivariant if the number of parallel hyperplanes of each hyperplane H ? A depends only on the W -orbit of H . We prove that the conings of the W -equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinagas theorem conjectured by Edelman-Reiner.

Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements

In this article, we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual partition formula. Then, it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each subarrangement of the filtration satisfies the dual partition formula. This generalizes the main result in Abe et al. (J. Eur. Math. Soc., to appear) which affirmatively settled a conjecture by Sommers and Tymoczko (Trans. Am. Math. Soc. 358:3493–3509, 2006).

The stability of the family of B2-type arrangements

We introduce the family of B 2-type arrangements as a generalization of the classical Coxeter arrangement of type B 2 and consider the stability and the freeness of it. We show the freeness and (semi)stability are determined by the combinatorics. Moreover, we give a partial answer to the 4-shift problem, which is a conjecture on the combinatorics and geometry induced from the B 2-type arrangements.

Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements

Abstract We define primitive derivations for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the primitive filtrations of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration was studied in Abe and Terao (2010) [2] , which generalizes the Hodge filtration introduced by K. Saito (e.g., Saito, 2003 [6] ).