Hyperplane Arrangements and Characteristic classes

11 (Mon) November, 2013.
One-day workshop on hyperplane arrangements, at Rm 3-127, Dept. of Math. Kyoto Univ.

12(Tue)--15(Fri) November, 2013.
RIMS, Kyoto University (Room 111).

Invited Speakers:

Schedule
11 (Mon) Nov. (Rm 3-127, Department of Mathematics, Kyoto University)
13:15-14:15 Schenck, Logartithmic vector fields and quasihomogeneous curve configurations.
14:30-15:30, Huh, Likelihood Geometry.
15:45-16:45, Tsukamoto, Symmetry and realizations of oriented matroids.

12 (Tue) Nov. (Rm-111, RIMS)
9:30-10:30, Aluffi, Segre classes and hyperplane arrangements.
10:45-11:45, Liao, Chern classes of logarithmic derivations for free divisors.
13:30-14:30, Enomoto, A representation theoretic approach to the Johnson cokernels for the mapping class group of surfaces.
14:45-15:45, Okuma, Good ideals for normal surface singularities.
16:00-17:00, Terao, The freeness of ideal subarrangements of Weyl arrangements.

13 (Wed) Nov. (Rm-111, RIMS)
9:15-10:15, Suciu, Topology and combinatorics of Milnor fibrations of hyperplane arrangements.
10:30-11:30, Torielli, On the admissibility of certain local systems.
11:45-12:45, Nakashima, Canonical systems of basic invariants for reflection groups.
13:00- , free discussion.

14 (Thu) Nov. (Rm-111, RIMS)
9:30-10:30, Maxim, Characteristic classes of singular toric varieties.
10:45-11:45, Wakefield, Counting arrangements and the multivariable Tutte polynomial
13:30-14:30, Settepanella, Braid groups in complex spaces and grassmannians.
14:45-15:45, Denham, Intersection-theoretic characteristic polynomial formulas.
16:00-17:00, Libgober, Periodicity of motives of abelian and cyclic covers associated with arrangements.

15 (Fri) Nov. (Rm-111, RIMS)
9:30-10:30, Matei, Topology of arrangements of surfaces
10:45-11:45, Kohno, Discriminantal arrangements and fusion rules in WZW model.
12:00-15:00, free discussion.

Abstract: 11 (Mon) Nov.
13:15-14:15 Schenck,
Title: Logartithmic vector fields and quasihomogeneous curve configurations.
Abstract: Let A be the union U(C_i) of a finite number of smooth plane curves C_i in the projective plane, such that the singular points of A are quasihomogeneous. We prove that if C is a smooth curve such that the singularities of A U C are quasihomogeneous, then there is a short exact sequence relating the bundle of logarithmic derivations on A to the bundle of logarithmic derivations on A U C. This yields an inductive tool for studying the splitting of these bundles in terms of the geometry of the divisor A|_C on C. (joint with H. Terao and M. Yoshinaga, Hokkaido)

14:30-15:30, Huh,
Title: Likelihood Geometry.
Abstract: Likelihood geometry is a study of embedded geometry of very affine varieties (closed subvarieties of an algebraic torus). Many of our favorite questions on arrangement complements, especially those concerning characteristic classes, can be asked more generally for very affine varieties in the likelihood geometric setting. This is a joint work with Bernd Sturmfels.

15:45-16:45, Tsukamoto,
Title: Symmetry and realizations of oriented matroids.
Abstract: An oriented matroid is a combinatorial type of a(pseudo)hyperplane arrangement, and the hyperplane arrangement is called a realization of the oriented matroid. Some oriented matroids which have symmetry and realizations do not have any symmetric realizations. This property often causes the disconnectedness of their realization spaces. In this talk, I construct some examples of symmetric oriented matroids, and talk about symmetry of their realizations.

12 (Tue) Nov.
9:30-10:30, Aluffi,
Title: Segre classes and hyperplane arrangements.
Abstract: Segre classes are important invariants in intersection theory, with applications to enumerative geometry and singularity theory. This talk will focus on recent progress in methods for the computations of these classes, and on the relation between the Segre class of the singularity subscheme of a hyperplane arrangement and other invariants of the arrangement, particularly its characteristic polynomial.

10:45-11:45, Liao,
Title: Chern classes of logarithmic derivations for free divisors.
Abstract: The sheaf of logarithmic derivations is an important object to study. In this presentation, I will focus on the question of comparing the Chern class of the logarithmic derivations of a free divisor and the Chern-Schwartz-MacPherson class of the complement of the divisor. When the Jacobian ideal of the free divisor is of linear type, the two classes mentioned above are the same. I will also explain how this result is related to the Logarithmic Comparison Theorem (LCT).

13:30-14:30, Enomoto,
Title: A representation theoretic approach to the Johnson cokernels for the mapping class group of surfaces.
Abstract: Let $\Sigma_{g,1}$ be a compact oriented surface of genus g with one boundary component and $M_{g,1}$ its mapping class group. The Johnson homomorphism gives an approximation of the mapping class group $M_{g,1}$ and its Torelli subgroup using a certain graded Lie algebras and the derivation algebra of the free Lie algebras generated by the first homology group of $\Sigma_{g,1}$ . In this talk, we will introduce a representation theoretic approach to study the structure of the Johnson homomorphism and their cokernels. We will give a new class and detect some explicit series of Sp-irreducible components in the Johnson cokernels, for example the “anti-Morita obstractions” Finally, I will discuss some relationships between our new class and other recent developments for the Johnson cokernels for the mapping class group of surfaces.

14:45-15:45, Okuma,
Title: Good ideals for normal surface singularities. Abstract: S. Goto, S. Iai, and K. Watanabe introduced the notion of good ideals in local rings as good ones next to the parameter ideals and characterized good ideals for rational Gorenstein surface singularities in terms of desingularization. However the existence of good ideals for surface singularities is still an open question. I will discuss the existence of good ideals for Gorenstein singularities and rational singularities and the structure of the set of good ideals. This is a joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.

16:00-17:00, Terao,
Title: The freeness of ideal subarrangements of Weyl arrangements.
Abstract: The exponents of an irreducible root system (or of a Weyl arrangement), which are the most important integers arising from the root system, control the corresponding Lie group, the reflection group, and the arrangement of hyperplanes. The celebrated SSKM formula, due to A. Shapiro, R. Steinbrg, B. Kostant and I. G. Macdonald, connects the exponents to the height distribution of positive roots by the concept of dual partitions. The SSKM formula was first proved (without using the classification) by Kostant by studying 3-dimensional Lie subgroups. On the other hand, E. Sommers - J. Tymoczko (2006) conjectured that any ideal subarrangement of the Weyl arrangement is a free arrangement and that the exponents and the height distribution are dual partitions to each other. In this talk, we will prove that the Sommers-Tymoczko conjecture holds true. Our proof, even in the case of the entire Weyl arrangement, gives a new proof of the SSKM formula. (jointly with Takuro Abe，Mohamed Barakat, Michael Cuntz, Torsten Hoge)

13 (Wed) Nov.
9:15-10:15, Suciu,
Title: Topology and combinatorics of Milnor fibrations of hyperplane arrangements.
Abstract: I will discuss recent progress in understanding the homology of the Milnor fiber and the algebraic monodromy of the Milnor fibration of a hyperplane arrangement, and the way these homological invariants are related to the combinatorics of the underlying matroid.

10:30-11:30, Torielli,
Title: On the admissibility of certain local systems.
Abstract: In this talk we will recall the notion of admissible local systems and we will describe how these systems behave. Moreover, we will describe which points of the characteristic variety of a given line arrangements are admissible. This is a joint work with S. Nazir and M. Yoshinaga.

11:45-12:45, Nakashima,
Title: Canonical systems of basic invariants for reflection groups.
Abstract: A system of basic invariant is said to be canonical if the system satsfy certain differential equations. Explicit formulas of canonical systems play an important part in the study for mean value problems related with polytopes. It has shown by Flatto and Wiener that there exist canonical systems for all finite real reflection groups. We give a proof of the existance of canonical systems for finite unitary reflection groups, and a construction for explicit formulas of canonical systems.

14 (Thu) Nov.
9:30-10:30, Maxim,
Title: Characteristic classes of singular toric varieties.
Abstract: We discuss the computation of the homology Hirzebruch characteristic classes of (possibly singular) toric varieties. We present two different perspectives for the computation of these characteristic classes. First, we take advantage of the torus-orbit decomposition and the motivic properties of the homology Hirzebruch classes to express the latter in terms of the (dual) Todd classes of closures of orbits. The obtained formula is then applied to weighted lattice point counting in lattice polytopes. Secondly, in the case of simplicial toric varieties, we make use of the Lefschetz-Riemann-Roch theorem in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. This is joint work with Joerg Schuermann.

10:45-11:45, Wakefield,
Title: Counting arrangements and the multivariable Tutte polynomial
Abstract: The degree of the realization space M of a matroid can be interpreted as the number of arrangements that contain dim M general position points. In the case of generic arrangements and cones of generic arrangements we use some basic intersection theory and Schubert calculus to compute this degree. We will also discuss a relation to the multivariable Tutte polynomial.

13:30-14:30, Settepanella,
Title: Braid groups in complex spaces and grassmannians.
Abstract: The ordered and unordered configuration spaces of $k$ distinct points in a manifold M and their fundamental groups have been widely studied. In a recent paper Berceanu and Parveen introduced new configuration spaces. They stratify the classical ordered and unordered configuration spaces of $k$ distinct points in the complex projective space with submanifolds defined as the ordered (resp. unordered) configuration spaces of all $k$ points generating a projective subspace of dimension $i$. Then they compute their fundamental groups. Recently Manfredini, Parveen and Settepanella, using similar techniques, computed the fundamental groups of ordered and unordered configuration spaces of $k$ distinct points in the affine space $\mathbb{C}^n$ (resp. in the grassmannian manifold parametrizing $k$-dimensional subspaces of $\mathbb{C}^n$) generating a subspace of fixed dimension.

14:45-15:45, Denham,
Title: Intersection-theoretic characteristic polynomial formulas.
Abstract: The critical points of a product of powers of polynomial functions are parameterized by a "maximum likelihood variety." In the special case of linear polynomials, one can recover the characteristic polynomial of a complex hyperplane arrangement from the homology cycle of this variety in its natural embedding. Some specializations of this result include a formula counting critical points due to Varchenko, Orlik and Terao, and a Hilbert series formula due to Solomon and Terao. It turns out that an analogous formula also holds if one replaces the complex hyperplane arrangement complement by the reciprocal plane: i.e., the image of the complement in the complex torus under the inverse map. Using recent results of June Huh, the problem amounts to computing the Chern-Schwartz-MacPherson class of the reciprocal plane, which can be done by a deletion-restriction argument. In particular, we obtain some (apparently) new inequalities that the characteristic polynomial of a complex arrangement must satisfy. This is joint work with June Huh.

16:00-17:00, Libgober,
Title: Periodicity of motives of abelian and cyclic covers associated with arrangements.
Abstract: I will discuss structure of isogeny components of Albanese varieties associated with abelian and cyclic covers ramified over arrangements of hyperplanes and relation to open problems about topology of the complements.

15 (Fri) Nov.
9:30-10:30, Matei,
Title: Topology of arrangements of surfaces
Abstract:

10:45-11:45, Kohno,
Title: Discriminantal arrangements and fusion rules in WZW model.
Abstract: In WZW model, the space of conformal blocks in the space where the partition functions of the WZW action live. We construct a period map from the homology of local systems on the complements of discriminantal arrangements to the dual of the space of conformal blocks. It turns out that this period map is surjective. We show that the kernel of the period map coincides with the one for the natural map from the homology to the homology with locally finite chains.

Contact
Takuro Abe(abe.takuro.4c (at) kyoto-u.ac.jp)
Toru Ohmoto(ohmoto (at) math.sci.hokudai.ac.jp)
Masahiko Yoshinaga(yoshinaga (at) math.sci.hokudai.ac.jp)