# Differential and combinatorial aspects of singularities

T.U. Kaiserslautern, August 3-7, 2015.

## Titles and Abstracts:

Takuro Abe: Divisionally free arrangements of hyperplanes (slides)

We show the division theorem for free arrangements, which is a generalization of Terao's addition-deletion theorems. The division theorem allows us to define a new category of free arrangements, called divisionally free arrangements which is purely combinatorial. This new category contains, and is strictly larger than that of inductively free arrangements. In particular, Terao's conjecture is true among divisionally free arrangements.

Ragnar-Olaf Buchweitz: A McKay Correspondence for Reflection Groups

Let $$G$$ be a finite subgroup of $$GL(n,K)$$ for a field $$K$$ whose characteristic does not divide the order of $$G$$. The group $$G$$ then acts linearly on the polynomial ring $$S$$ in $$n$$ variables over $$K$$ and one may form the corresponding twisted or skew group algebra $$A = S*G$$. With $$e$$ in $$A$$ the idempotent corresponding to the trivial representation, consider the algebra $$A/AeA$$. If $$G$$ is a finite subgroup of $$SL(2,K)$$, then it is known that $$A$$ is Morita-equivalent to the preprojective algebra of an extended Dynkin diagram and $$A/AeA$$ to the preprojective algebra of the Dynkin diagram itself. This can be seen as a formulation of the McKay correspondence for the Kleinian singularities.

We want to establish an analogous result when $$G$$ is a group generated by reflections. With $$D$$ the coordinate ring of the discriminant of the group action on $$S$$, we show that $$A/AeA$$ is maximal Cohen-Macaulay as a module over $$D$$ and that it is of finite global dimension as a ring.

The ring $$A/AeA$$ is the endomorphism ring of a maximal Cohen-Macaulay module over the ring of the discriminant, namely of the direct image of the coordinate ring of the associated hyperplane arrangement.

In this way one obtains a noncommutative resolution of singularities of that discriminant, a hypersurface that is a free divisor, thus, singular in codimension one.

Nero Budur: Local systems of rank one (slides)

We give a survey of recent results on the cohomology jump loci of local systems of rank one, showing their connections with classical singularity theory, Hodge theory, Bernstein-Sato polynomials, and algebraic statistics. Joint work with Botong Wang.

Alex Dimca: Nearly free divisors and rational cuspidal curves (slides)

We introduce a class of complex projective plane curves, called nearly free curves, having properties similar to those of free curves. Conjecturally any rational cuspidal plane curve is either free or nearly free. This conjecture is proved in a number of cases by using a recent result due to Uli Walther.

Clément Dupont: Bi-arrangements of hyperplanes and Orlik-Solomon bi-complexes (slides)

Motivated by the study of certain periods such as the values at the Riemann zeta function at integer points, we introduce the notion of a bi-arrangement of hyperplanes, which generalizes that of an arrangement of hyperplanes. We study the combinatorial and cohomological properties of bi-arrangements by using algebraic tools which generalize the Orlik-Solomon algebra.

Eleonore Faber: Desingularizing free divisors: discriminants

In this talk we will comment on desingularizations of free divisors, especially on existence and constructions of non-commutative resolutions of singularities. In particular we will focus on discriminants of finite reflection groups $$G$$, for which we can give a quite natural construction of a non-commutative desingularization as a quotient of the twisted group algebra of $$G$$. This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.

Michel Granger: Logarithmic differential forms and questions of residues (slides)

In this talk I will recall first Kyoji Saito's theory of logarithmic differential forms and vector fields paying a particular attention to residues along a divisor. These divisor may be free in some situation but I will not limit myself to this case. I will explain a result of Mathias Schulze and myself about the characterisation of divisors with normal crossings in codimension by a minimality property of the module of residues of 1-forms. I will then discuss the natural question of describing more explicitly the modules of residues of 1-forms for more general divisors. I will explain recent results of Delphine Pol which concern the set of valuations of residues along reduced plane curves as well the relationship of this object with the classification of plane curves due to Hefez and Hernandez in the case of branches.

Herwig Hauser: Lie algebras of logarithmic vector fields (slides)

For germs $$X$$ of analytic subvarieties in $$({\mathbb C}^n,0)$$ the collection of holomorphic vector fields on $$({\mathbb C}^n,0)$$ which are logarithmic (or tangent) to $$X$$ forms a Lie-algebra $$D_X$$. It is known that this Lie-algebra (equipped with a natural topology) carries quite a lot of information about the geometry of $$X$$. Among others, it determines for $$n\geq 3$$ the analytic type of $$X$$.

Actually, the variety $$X$$ can be interpreted as an integral variety of $$D_X$$, similar to Malgrange's singular Frobenius theorem.

We will recall in the lecture some of the geometric features of $$D_X$$ and will then discuss to what extent vector fields could be used to build up a category which mimics Lie-algebra theoretically the passage from varieties to the theory of affine schemes.

Xia Liao: Hirzebruch classes of line arrangements

The motivic Hirzebruch class theory is a characteristic class theory which unifies the Chern-Schwarz-MacPherson class theory, singular Todd class theory and Thom-Milnor’s L-class theory. In this talk, I will discuss a conjectured formula concerning the computation of the Hirzebruch class of a free divisor $$D$$ in a nonsingular variety $$X$$ by its sheaf of logarithmic differentials $$\Omega_X (\log D)$$. While the conjectured formula is still difficult to understand, I will report some of my recent progress focusing on the simplest case $$X=\mathbb{P}^2$$ and $$D$$ is a line arrangement.

Luis Narvaez: Symmetries of the roots of b-functions: a survey (notes)

The symmetry of the roots (different from -1) of b-functions of quasi-homogeneous isolated singularities has been known for long time and has been related with the symmetry of the spectrum. More recently, Granger and Schulze (2010) proved that the b-function of any reductive prehomogeneous determinant (these are the non-reduced version of linear free divisors) or of any regular special linear free divisor satisfies the equality $$b(−s−2)=±b(s)$$. Motivated by this result and by the behaviour under duality of the b-function of some examples of logarithmic integrable connections with respect to quasi-homogeneous plane curves, the author has proven that the same symmetry property occurs for free divisors for which the module $$D[s]f^s$$ admits a logarithmic Spencer resolution. This applies in particular to free divisors of linear Jacobian type (e.g. locally quasi-homogeneous free divisors, or free hyperplane arrangements). Results of Granger-Schulze and the author overlap, but there is still missing a common generalization. In this talk I will survey the use of duality theory for proving the above symmetry property. I will also present some examples suggesting extensions of this “duality principle” and other families of examples where symmetry with other intermediate shiftings occurs.

Claudiu Raicu: Characters of equivariant $$D$$-modules on spaces of matrices

I will describe the characters of the simple $$GL$$-equivariant $$D$$-modules on a complex vector space of matrices (general, symmetric, or skew-symmetric) and explain how this information can be used to compute local cohomology modules, as well as to (dis)prove some cases of a conjecture of Levasseur.

Thomas Reichelt: Hodge theory of GKZ systems

I will explain a close relationship between GKZ hypergeometric systems after Gelfand, Kapranov and Zelevinsky, and Gauss-Manin systems of families of Laurent polynomials. This induces a mixed Hodge module structure on these GKZ systems whose Hodge filtration is explicitly computable. As an application, I will show how to construct non-affine Landau-Ginzburg models which are mirror partners for complete intersections in toric varieties.

Claude Sabbah: On the Kontsevich logarithmic complex

To any function $$f$$ on a smooth quasi-projective variety $$U$$, Kontsevich associates a logarithmic de Rham complex defined on a suitable compactification, which computes the cohomology of the de Rham complex of $$U$$ with twisted differential $$d + df$$. This complex allows one to define a filtration on this cohomology, called the irregular Hodge filtration, which satisfies the degeneration at $$E_1$$ property, like the standard Hodge filtration. The talk will survey recent work of Esnault-Sabbah-Yu and Katzarkov-Kontsevich-Pantev.

Jörg Schürmann: Singular Todd classes of tautological sheaves on Hilbert schemes of points on a smooth surface

Let $$X$$ be a quasi-projective smooth complex algebraic surface, with $$X^{[n]}$$ the Hilbert scheme of $$n$$ points on $$X$$, so that the (rational) cohomology of all these Hilbert schemes together can be generated by the cohomology of $$X$$ in terms of Nakajima creation operators. Given an algebraic vector bundle $$V$$ on $$X$$, there exist universal formulae for the characteristic classes of the associated tautological vector bundles $$V^{[n]}$$ on $$X^{[n]}$$ in terms of the Nakajima creation operators and the corresponding characteristic classes of $$V$$. But in general the corresponding coefficients are not known. Based on the derived equivalence of Bridgeland-King-Reid and work of Haiman and Scala, we give an explicit formula in case of the singular Todd classes, but in terms of Nakajima creation operators of the delocalized equivariant cohomology of all $$X^n$$ with its natural combinatorial $$S_n$$-action.

Hiro Terao: On Parabolic Subarrangements and Restrictions of Weyl arrangements (slides)

Recently Takuro Abe introduced a new combinatorial criterion, called the DF (divisionally freeness), for the freeness of an arrangement of hyperplanes.　In this talk, we apply the DF to the restrictions of Weyl arrangements. We also make use of the height-free theorem proved by T. Abe, M. Barakat, M. Cuntz, T. Hoge andH. Terao (to appear in J. Euro. Math. Soc.) (This is a joint work with T. Abe.)

Michele Torielli: Homotopy type, Orlik-Solomon algebra and Milnor fiber of supersolvable arrangements

In this talk I will give a very natural description of the bijections between the minimal CW-complex homotopy equivalent to the complement of a supersolvable arrangement $$\mathcal A$$, the nbc-basis of the Orlik-Solomon algebra associated to $$\mathcal A$$ and the set of chambers of $$\mathcal A$$. I will use these bijections to get results on the first (co)homology group of the Milnor fiber of $$\mathcal A$$.

Wim Veys: Bounds for p-adic exponential sums and log-canonical thresholds (paper)

In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo $$p^m$$ lying $$p$$-adically close to $$y$$, and the proposed bounds are uniform in $$p$$, $$y$$, and $$m$$. We give evidence for the conjecture, by showing uniform bounds in $$p$$, $$y$$, and in some values for $$m$$. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.

Uli Walther: Annihilators of powers of arrangements (slides)

We consider the differential operators that annihilate $$f^s$$ where s is a new unknown and f the defining equation of an arrangement. An interesting question is: when is this D-ideal generated by derivations? The issue is related to tameness and maz be equivalent. We discuss the connection, and some possible applications.

Sergey Yuzvinsky: Supersolvable arrangements: higher topological complexity of the complements, randomness

In this talk we continue the calculation of higher topological complexity of hyperplane arrangement complements. This time we summarize and apply the technique from certain previous papers to the class of supersolvable arrangements. This allows us to prove that, for the complement of every irreducible supersolvable arrangement, $$TC_s=sr-1$$, where $$r$$ is the rank of the arrangement.

Also we discuss some initial results about random supersolvable graphic arrangements.