Hiroaki Terao

Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

In this paper we strenghten a theorem by Esnault-Schechtman-Viehweg, [3], which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain ”Aomoto non-resonance conditions” for monodromies are fulfilled at some ”edges” (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges, see Theorem 4.1. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations, cf. [8]), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras, see Theorem 7.1.

Multiderivations of Coxeter arrangements

Abstract.Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose αH∈V* such that H=ker(αH). For each nonnegative integer m, define the derivation module D(m)(?)={θ∈DerS|θ(αH)∈SαmH}. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D(m)(?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+mi(1≤i≤ℓ) (when m is odd). Here m1≤···≤mℓ are the exponents of G and h=mℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.

Modular elements of lattices and topological fibration

An l-arrangement is a finite collection of (I 1 )-dimensional vector subspaces of an I-dimensional vector space V over a field K. We restrict our attention to the case that K =C for a while. The complement M of the hyperplanes is an open manifold. In [3], Falk and Randell studied a libration of M: M is called strictly linearly jibered if after a suitable linear change of coordinates the restriction of the projection to the first (I 1) coordinates is a fiber bundle projection with base N the complement of an arrangement in C? ’ and fiber a complex plane C with finitely many points removed. The arrangement is called fiber-type when it allows a successive strictly linearly fibrations finally reaching C* as base. In this case they investigated several topological properties of M. Here we are interested in a combinatorial characterization of a strictly linear fibration in lattice theoretic terms. For this, the concept of modular elements defined by Stanley [S] is crucial: For example, concerning a geometric lattice arising from an essential (= the intersection of all the elements is the origin) arrangement defined over C, the existence of l-dimensional modular element is equivalent to a strictly linear libration of the complement M (2.15). This is a corollary of Theorem (2.9). As another corollary we have a characterization (2.17) that the arrangement is fiber-type if and only if it gives rise to an SS (= supersolvable) lattice, which is defined also by Stanley [9] to be a geometric lattice, which is defined also by Stanley [9] to be a geometric lattice with an ascending maximal chain consisting of modular elements. These are done in Section 2. In Section 3, we state Theorem (3.8), asserting that the algebra A associated with a finite geometric lattice studied in [S, 63 factors as a tensor product of graded subspaces by a modular element, which is Theorem (3.8). This has a topological interpretation (3.9) when the lattice originates from an arrangement defined over C: the factorization of the cohomological ring of M into that of the base and that of the fiber. Since 135 OoOl-8708/86

Free arrangements of hyperplanes and supersolvable lattices

7.50

Commutative algebras for arrangements

A finite family of hyperplanes through the origin in C’+’ is called an Iarrangement of hyperplanes, or simply an l-arrangement. An l-arrangement is said to be free if the module of logarithmic vector fields (=holomorphic vector fields tangent to all hyperplanes in the arrangement) is a free module (the accurate definition will be given in (2.1)). Let A be a free I-arrangement. Then a set {do,..., d,} of I + 1 nonnegative integers can be defined (2.3). These (d, ,..., d,) are called the exponents of A. On the other hand, one can associate a geometric lattice L(A) with any arrangement A (3.1). The characteristic polynomial XLca,(t) E Z[t] is defined by using the Mobius function (3.4). The second author [7] proved that

The number of critical points of a product of powers of linear functions

Let V be a vector space of dimension l over some field K. A hyperplane H is a vector subspace of codimension one. An arrangement is a finite collection of hyperplanes in V . We use [7] as a general reference.

A free resolution of the module of logarithmic forms of a generic arrangement

be the complement of .A. Given a complex n-vector 2 = (21,...,2n) E Cn, consider the multivalued holomorphic function defined on M by Studying Bethe vectors in statistical mechanics, A. Varchenko [18] conjectured that for generic 2 all critical points of qS;~ are nondegenerate and the number of critical points is equal to the absolute value of the Euler characteristic of M,)~(M). He proved the conjecture for complexified real arrangements in [18]. A similar result was known to K. Aomoto [1, Example 1] who stated without proof that for positive 2 the number of critical points of ~b;. equals the number of bounded components of the complement of the real arrangement. See [2, Theorem 4.4.1.1] for details. In this paper we prove Varchenkos conjecture for all arrangements. More generally we give a formula for the number of critical points of the multivalued holomorphic function

The Euler multiplicity and addition-deletion theorems for multiarrangements

1.1. The Setup of This Paper Let V be an I-dimensional vector space over a field K. We assume that the characteristic of K is zero. (For the positive characteristic case, see 4.6.) Let d be a generic arrangement of II hyperplanes: d is a finite family of one-codimensional subspaces of V satisfying 1. n= #d>1>3, 2. Every 2 hyperplanes of d intersect only at the origin. Let S denote the symmetric algebra S( V*) of the dual space V* of V. Then 5’ can be considered as the K-algebra of all polynomial functions on V. Let 0 6 q < 1. Let 524 = ~2; denote the S-module of all regular q-forms on V. Then each Qq is a free S-module of rank (i). For each HE d choose CAKE V* such that ker(a,) = H. Let

The freeness of ideal subarrangements of Weyl arrangements

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.

Free arrangements and relation spaces

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