# Workshop on Geometry and Analysis of Discriminants

Period: February 7-8, 2011

Place: Balcony A , IPMU, Kashiwa Campus, The University of Tokyo

Organizer: K. Saito(IPMU), J. Sekiguchi(TUAT), K. Takeuchi(Tsukuba U)

### PROGRAM

7 February

13:30-14:30 S. Tajima:

μ-constant deformations and algebraic local cohomology.

14:45-15:45 K. Takeuchi:

Motivic Milnor fibers and Jordan normal forms of monodromies.

(joint work with Y. Matsui and A. Esterov)

16:00-17:00 J. Sekiguchi:

Saito free divisors in a four dimensional affine space.

8 February

10:00-11:00 T. Ishibe:

Monoids in the fundamental groups of the complement of logarithmic free divisors in C＾3.

(joint work with K. Saito)

11:15-12:15 S. Tanabé:

Hodge structure of period integrals revisited.

13:30-14:30 K. Ueda:

Mirror symmetry and Calabi-Yau hypersurfaces in weighted projective spaces.

14:45-15:45 M. Yoshinaga:

Minimal stratification for line arrangements and presentaions of fundamental groups.

16:00-17:00 M. Kato

Monodromy representations of uniformization equations of rank three.

### Title and Abstract of Talks

・Shin-ichi Tajima(Tsukuba Univ.)

Title: μ-constant deformations and algebraic local cohomology.

Abstract: We study μ-constant deformations for semi-quasihomogeneous hypersurface isolated singularities by using the Grothendieck local duality. We present a new method for computing Tjurina stratifications of μ-constant deformations. The key ingredient in this approach is the concept of parametric algebraic local cohomology.

・Kiyoshi Takeuchi(Tsukuba Univ.)

Title: Motivic Milnor fibers and Jordan normal forms of monodromies.(joint work with Y. Matsui nd A. Esterov)

Abstract: By computing the equivalent mixed Hodge numbers of motivic Milnor fibers introduced by Denef-Loeser etc., we obtain various formulae for the Jordan normal forms of the local and global monodromies of polynomials. For polynomials over affine complete intersection varieties the results will be described by the mixed volumes of the faces of their Newton polyhedrons.

・Jiro Sekiguchi(Tokyo Univ. of Agriculture and Technology)

Title: Saito free divisors in a four dimensional affine space.

Abstract: There is a relationship between Saito free divisors in C＾3 defined by weighted homogeneous polynomials and 1-parameter equisingular deformations of isolated curve curve singularities. This idea leads us to find several examples of Saito free divisors in C＾3 including discriminants of complex reflection groups of rank three. This kind of phenomena seem not occur in higher dimensional case. The speaker reports in this talk an idea to construct Saito free divisors in C＾4 and discuss a comparison of such divisors with discriminants of complex reflection gropus of rank four.

・Tadashi Ishibe(Hiroshima Univ.)

Title: Monoids in the fundamental groups of the complement of logarithmic free divisors in C＾3. (joint work with K. Saito)

Abstract: We study monoids generated by certain Zariski-van Kampen gererators in the 17 fundamental groups of the complement of logarithmic free divisors in C＾3 listed by Sekiguchi. They admit positive homogenerous presentations. Five of them are Artin monoids and eight of them are free abelian monoides. The remaining four monoids are not Gaussian and, hence, are neither Garside nor Artin. However, we introduce the concept of fundamental elements for positive homogenously presented monoids, and show that all 17 monoids possess fundamental elements. As an application of the study of monoids, we solve some decision problems for the fundamental groups except three cases.

・Susumu Tanabé(Galatasaray University, Istanbul)

Title: Hodge structure of period integrals revisited.

Abstract: Several years ago I proposed a method to calculate Mellin transform of period integrals defined for a certain class of affine complete intersection varieties. As a result of calculus, it turns out that the mixed Hodge structure of the variety reflects on the arrangement of poles of Mellin transform of period integrals. Almost at the same period, papers by Douai-Sabbah appeared. There the authors establish the existence of Kashiwara-Malgrange filtration for a D-module associated to the oscillatory integrals defined on an affine algebraic hypersurface. I would like to discuss the relation between these two approaches to the Hodge structure of integrals.

・Kazushi Ueda(Osaka Univ.)

Title: Mirror symmetry and Calabi-Yau hypersurfaces in weighted projective spaces.

Abstract: Based on joint works with Masahiro Futaki, Akira Ishii and Susumu Tanabe, I will discuss relations between

1. HMS(homological mirror symmetry) for weighted projective spaces,

2. HMS for Calabi-Yau Fermat hypersurfaces in weighted projective spaces,

3. HMS for Brieskorn-Pham singularities, and

4. monodromy of certain hypergeometric functions.

・Masahiko Yoshinaga(Kyoto Univ.)

Title: Minimal stratification for line arrangements and presentations of fundamental groups.

Abstract: The fundamental group of the complements to line arrangements has minimal presentation. This fact has been gereralized to "the minimality of the complement of hyperplane arrangements". Recently several descriptions of minimal CW complexes appear. In this talk, I would like to talk about "minimal stratification" which can be considered as a dual object to minimal CW complex and also discuss associated presentations of fundamental groups.

・Mitsuo Kato(Univ. of Ryukyu)

Title: Let *D *be a discriminant locus of a finite irreducible complex reflection group *G* of rank three. Then *D* is a Saito free divisor in a weighted homogeneous Affine space *X* of dimension three. Let *E* be a uniformization equation defined on *X* and singular along *D* whose monodromy group is a complex reflection group. We can construct a monodromy representation of *E*. In particular, we give three generators of *G *explicitely.

You can check the location of IPMU, Kashiwa campus, the University of Tokyo from

http://www.ipmu.jp/access/img/KashiwaCampusMap.png

The schedule of the seminar can be checked from