### International Conference on Singularity Theory 2019

2019-12-09 ~ 2019-12-13 3446

Dates: 9-13 December 2019

This conference intends to provide a good exchange platform on exciting new results on singularities and promote further the development of algebraic and geometric aspects of singularity theory.

Organizers

Name University
Stanislaw Janeczko Banach Institute
Gerhard Pfister University of Kaiserslautern
Stephen Yau Tsinghua University
Huaiqing Zuo Tsinghua University

### Participants

 NO. English Name Chinese Name Employer's Name in English Employer's Name in Chinese 1 Igor Burban University of Paderborn 2 Stanisaw Janeczko Banach Center and Warsaw   University of Technology 3 Gerhard Pfister University of Kaiserslautern 4 Stephen Shing-Toung Yau Tsinghua   University 5 Huaiqing Zuo 左怀青 Tsinghua University 6 Anne Frühbis Krüger Leibniz Universität Hannover 7 Duco van Straten Hohannes Gutenberg   Universität Mainz 美因茨大学 8 Mutsuo Oka Tokyo University of Science 东京理科大学 9 Viktor Kulikov Steklov Mathematical Instit 10 Ngo Viet Trung Vietnam Academy of Science   and Technology 越南科学院 11 Shihoko Ishii TokyoWoman's　Chiristian　University 12 Antonio Campillo University of Valladolid,   Spain 西班牙瓦拉多利德大学 13 Claus Hertling University of Mannheim 缅因大学 14 Wolfram Decker Technische Universität   Kaiserslautern 15 Victor Goryunov University of Livepool 利物浦大学 16 Sabir Guzein-Zade Moscow State University 莫斯科国立大学 17 Kyoji Saito Kavli IPMU, University of   Tokyo 京都大学数理解析研究所 18 Morihiko Saito Kyoto University 19 Lê Dũng Tráng The Abdus Salam,   International Centre for Theoretical Physics 20 Mihai Tibar University of Lille, FRANCE 法国里尔大学 21 Wolfgang Ebeling University of Hannover 汉诺威大学 22 Eleonore Faber the University of Leeds 利兹大学 23 Thomas Reichelt Universität Heidelberg 24 Thomas Markwig the University of Tübinge 图宾根大学 25 Nguyen Duc Hong BCAM – Basque Center for   Applied Mathematics Mazarredo, 14, 48009 Bilbao, Basque Country – Spain 26 Pham Thuy Huong Quy Nhon University 昆恩大学 27 Xiaotao Sun 孙笑涛 AMSS 中国科学院数学与系统科学研究院   国家数学与交叉科学中心 28 Guangfeng Jiang 姜广峰 Beijing University of   Chemical Technology 北京化工大学 29 Fanning Meng 孟凡宁 Guangzhou University 广州大学 30 Chuangqiang Hu 胡创强 Tsinghua University 清华大学 31 Xiankui Meng 孟宪奎 Tsinghua University 清华大学 32 Naveed Hussian 哈那维 Huashang College Guangdong   University of Fiance and Economics 广东财经大学 33 Shuanghe  Fan Tsinghua University 清华大学 34 Klaus Altmann Freie Universität Berlin,   Fachbereich Mathematik und Informatik, Mathematisches Institut 柏林自由大学 35 Gert-Martin Wilhelm Greuel technische universitt   kaiserslautern 德国凯撒斯劳滕工业大学 36 Hua Zheng 华诤 University of Hong Kong 香港大学 37 Alexander Elashvili 38 Guorui Ma 马国瑞 Tsinghua University 清华大学 39 Xiping Zhang 40 Qiwei Zhu 朱其蔚 Tsinghua University 清华大学

### Schedule

===============================  Monday  ===================================

9:00-10:00

Claus Hertling

Marked singularities, their moduli spaces, distinguished bases and Stokes regions

10:00-10:30 Coffee break

10:30-11:30

Zheng Hua

Noncommutative Mather-Yau theorem and its applications

11:30-14:30  Lunch break

14:30-15:30

Thomas Reichelt

Hodge theory of GKZ systems

15:30-16:00 Coffee break

16:00-16:40

Mutsuo Oka

On the Milnor fibration of $f(z)\bar g(z)$

16:50-17:30

Wolfgang Ebeling

Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic

===============================  Tuesday  ===================================

9:00-10:00

Classification of Reflexive Modules on Gorenstein Surface Singularities and a conjecture of Drodz, Greuel and Kashuba

10:00-10:30 Coffee break

10:30-11:30

Eleonore Faber

Countable Cohen-Macaulay type and the infinity-gon

11:30-14:00 Lunch break

14:00-14:40

Mihai-Marius Tibar

Polar degree conjectures

14:50-15:30

Ngyuen Duc Hong

Cohomology of contact loci

15:30-16:00 Coffee break

16:00-16:40

Goryunov, Victor

Vanishing cycles of matrix singularities

16:50-17:30

Victor Kulikov

On rigid germs of finite morphisms of smooth surfaces

===============================  Wednesday =================================

9:00-10:00

Kyoji Saito

Primitive forms without metric structure and integrable hierarchy

10:00-10:30 Coffee break

10:30-11:30

Gert-Martin Greuel

On Semicontinuity of Singularity Invariants in Families of Formal Power Series

11:30-14:00 Lunch break

===============================  Thursday ===================================

9:00-10:00

Shihoko Ishii

A new bridge between positive characteristic and characteristic 0

10:00-10:30 Coffee break

10:30-11:30

Klaus Altmann

Universal extensions of semigroups

11:30-14:00 Lunch break

14:00-14:40

Stanislaw Janeczko

Geometric and algebraic restrictions of differential forms

14:50-15:30

Guangfeng Jiang

Free subarrangements of SHI and ISH arrangements

15:30-16:00 Coffee break

16:00-16:40

Ngo Viet Trung

Depth functions of homogeneous ideals

16:50-17:30

Antonio Campillo

Poincare series of matroids

===============================  Friday  ====================================

9:00-10:00

Xiaotao Sun

A Miyaoka-Yau type inequality of surfaces in characteristic $p>0$

10:00-10:30 Coffee break

10:30- 11:30

Poincaré polynomials of filtrations and algebraic links in the Poincaré sphere and in the links of simple surface singularities.

11:30-14:00 Lunch break

### Titles and Abstracts

==========================Monday  ========================================

Claus Hertling

Marked singularities, their moduli spaces, distinguished bases and Stokes regions

Abstract: One part of the talk is on a global study of μ-constant

families of holomorphic function germs with isolated singularities.

Some new data are defined and discussed, μ-constant monodromy groups,

marked singularities, their global moduli spaces, and a global Torelli

type conjecture for their Brieskorn lattices. Another part of the talk

is on universal unfoldings, Brieskorn lattices at semisimple points,

their Stokes data and distinguished bases, and a global

Lyashko-Looijenga map. For the simple singularities (by work of

Looijenga and Deligne 73/74) and the simple elliptic singularities

(joint work with C. Roucairol 18) this leads to an understanding of a

certain global base space as an atlas of Stokes data.

===================================================================================

Zheng Hua

Noncommutative Mather-Yau theorem and its applications

Abstract: I will survey some recent progress in noncommutative differential calculus of formal and analytic potentials and their applications to algebraic geometry and cluster algebra. This is based on joint work with Guisong Zhou and with Bernhard Keller.

===================================================================================

Thomas Reichelt

Hodge theory of GKZ systems

Abstract: GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss

hypergeometric differential equation.  It can be shown that for certain parameters the GKZ-systems carry the  structure of

a mixed Hodge module in the sense of Morihiko Saito. We will discuss the Hodge  and weight filtration of these D-modules.

===================================================================================

Mutsuo Oka

On the Milnor fibration of $f(z)\bar g(z)$

Abstract: We consider a mixed function of type $H(\mathbf z,\bar{\mathbf z})=f(\mathbf z)\bar g({\mathbf z})$ where $f$ and $g$ are which  are non-degenerate  with respect to the Newton boundary.We assume  also that  the variety $f=g=0$ is  a non-degenerate  complete intersection variety.Under the convenience assumption of the Newton boundary and the multiplicity condition,we will show that $H$ has both  a tubular and  a spherical Milnor fibrations. For non-convenience case, we assume the local tame non-degeneracy, we can also prove the existence of Milnor fibration.

===================================================================================

Wolfgang Ebeling

Dual invertible polynomials with permutation symmetries and the orbifold Euler characteristic

Abstract:

P.Berglund, T.H\"ubsch, and M.Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial  and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A.Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. It turns out that in order to get certain mirror symmetric properties in this case, one needs a special condition on the permutation group called parity condition (PC). We prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign. This is joint work with Sabir Gusein-Zade.

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======================== Tuesday==========================================

Classification of Reflexive Modules on Gorenstein Surface Singularities and a conjecture of Drodz, Greuel and Kashuba

Abstract: In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems we prove several isomorphisms between different deformation functors that we expect that will be useful in further work.

====================================================================================

Eleonore Faber

Countable Cohen-Macaulay type and the infinity-gon

Abstract: By a result of Buchweitz-Greuel-Schreyer, a hypersurface is of countable Cohen-Macaulay representation type, if and only if it is isomorphic to a singularity of type $A_{\infty}$ or $D_{\infty}$. In this talk, we show how the category CM(R) of maximal Cohen-Macaulay modules over the coordinate ring $R$ for the $A_{\infty}$-curve gives a categorical model for arcs in an $\infty$-gon''. This allows us to construct triangulations of the $\infty$-gon, making use of the language of cluster categories.

This is joint work with J. August, M. Cheung, S. Gratz, and S. Schroll.

====================================================================================

Mihai-Marius Tibar

Polar degree conjectures

My talk is based on joint work with Dirk Siersma and Joseph Steenbrink.

Dolgachev (Michigan Math J, 2000) has initiated the study of Cremona polar transformations i.e. birational maps gradf:ℙn⇢ℙn defined by the gradient map of a homogeneous polynomial. He conjectured that the topological degree of gradf depends only on the projective zero locus V of f,so that it can be called polar degree of V, denoted pol(V). The hypersurfaces with pol(V)=1 are called homaloidal; Dolgachev classified the homaloidal plane curves. I'll discuss the proof of Dolgachev's conjecture found by Dimca and Papadima (Annals of Math 2003),their conjecture on the classification of homaloidal hypersurfaces with isolated singularities proved by Huh (Duke Math J, 2014), and Huh's conjecture on the classification of hypersurfaces with isolated singularities and pol(V)=2 with its recent proof.

====================================================================================

Nguyen Duc

Cohomology of contact loci

Abstract: We construct a  spectral sequence converging to the

cohomology with compact support of the $m$-th contact locus of a

complex polynomial.  The first page is explicitly described in terms

of a log resolution and coincides with the first page of McLean's

spectral sequence converging to the Floer cohomology of the $m$-th

iterate of the monodromy, when the polynomial has an isolated

singularity. Inspired by this connection, we conjecture that if two

germs of holomorphic functions are embedded topologically equivalent,

then the Milnor fibers of the their tangent cones are homotopy

equivalent. (Joint work with J. Fern\'andez de Bobadilla, N. Budur,Q.T. Le).

====================================================================================

Goryunov, Victor

Vanishing cycles of matrix singularities

The talk is about holomorphic map germs M : (Cs,0) → Matn, where the target is the space of either square, or symmetric, or skew-symmetric n × n matrices. The target contains the set ∆ of all degenerate matrices, and our main object will be the vanishing topology of M−1(∆). Our attention is on the singular Milnor fibre of M, that is, the local inverse image V of ∆ under a generic small perturbation of M. The variety V is highly singular, but, according to Lˆe Du ̃ng Tr ́ang’s theorem, it is homotopic to a wedge of (s − 1)-dimensional spheres.

The talk will start with introduction of local models for the spheres vanishing in the matrix context.

We will then prove the μ = τ conjecture formulated by Damon for corank 1 map-germs M with a generic linear part, and a generalisation of this conjecture to the matrix version of boundary function singularities.

Bifurcation diagrams of matrix singularities will also be discussed, and a rather unexpected appearance of the discriminants of certain Shephard-Todd groups as such diagrams will be highlighted.

If time permits, possible approaches to the study of the monodromy will be mentioned.

====================================================================================

Victor Kulikov

On rigid germs of finite morphisms of smooth surfaces

Abstract: In the talk, questions related to deformations of germs of finite

morphisms of smooth surfaces and a correspondence between the set of

rigid germs of finite morphisms of smooth surfaces and the set of Belyi

rational functions  will be discussed.

====================================================================================

=======================  Wednesday =========================================

Kyoji Saito

Primitive forms without metric structure and integrable hierarchy

Abstract: We introduce primitive forms without higher residue structure.

Just as a classical primitive form induces a flat structure (Frobenius

manifold structure), they induces flat structure without metric. They,

further, induce cerain integrable hierarchies which generalize Gelfand

Dikii Hierarchy.

====================================================================================

Gert-Martin Greuel

On Semicontinuity of Singularity Invariants in Families of Formal Power Series

Abstract: The problem we are considering  came up in connection with the classification of singularities in positive characteristic.

Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal power series parametrized by some affine algebraic variety. In contrast to the case of analytic

or algebraic families, where such a bound is well known, the problem is rather subtle, since the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the fibre dimension is not semicontinuous and the quasi-finite locus is not open. However, if we pass to the completed fibers in a family of modules we can prove that  their fiber dimension  is semicontinuous under some mild conditions. We prove this in a rather general framework by introducing and using the completed and the Henselian tensor product. Finally we apply this to the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections and to the determinacy in a family of ideals.

====================================================================================

========================  Thursday ==========================================

Shihoko Ishii

A new bridge between positive characteristic and characteristic 0

Abstract:

One bridge between singularities in positive characteristic  and the singularities in

characteristic 0 is already established in the following form:

F-singularities in characteristic p for p>>0” <==> birational singularities in characteristic 0”

In my talk I will show a challenge to establish a bridge in a different form:

birational singularities in characteristic p (fixed) "==>

construct a birational singularities in characteristic 0 with the same properties”.

I will show the present status of this direction.

I will also show applications when the challenge is successful.

====================================================================================

Klaus Altmann

Universal extensions of semigroups

Homogeneous deformations of toric singularities can be

understood as extensions of the associated semigroups. They

will form their own category, and the surprising result is

that this category contains an initial object.

That is, unlike to the geometric singularities obtained from

taking the Spec of their semigroup algebras, the semigroups

themseves are not obstructed. We will demonstrate this for

Pinkham's example of the cone over the rational normal curve

of degree 4.

====================================================================================

Stanislaw Janeczko

Geometric and algebraic restrictions of differential forms

Abstract: By algebraic and geometric conormals and tangents we study germs of differen- tial forms over singular varieties. The geometric restriction of differential forms to singular varieties is introduced and algebraic restriction of differential forms with vanishing geometric restrictions, called residual algebraic restrictions, are investi- gated. Residues of plane curve-germs, hypersurfaces, Lagrangian varieties as well as the geometric and algebraic restriction via a mapping were calculated. The natural exact sequence 0 → R•(Z) → A•(Z) → G•(Z) → 0 defines the residues R•(Z) = G•(Z, M)/A•(Z, M) with A•(Z) = Λ•(M)/A•(Z, M) and G•(Z) = Λ•(M)/G•(Z, M), where G•(Z,M) are geometric restrictions and A•(Z,M) are algebraic restrictions to Z.

This is a joint work with Goo Ishikawa.

====================================================================================

Guangfeng Jiang

Free subarrangements of SHI and ISH arrangements

Abstract: The cones over SHI arrangements are proved to be free by Athanasiadis.

Recently Abe, Suyama and Tsujie proved the freeness of the ISH arrangement.

They also showed conditions on the freeness of the deleted ISH arrangements

which contain the braid arrangment as sub-arrangement.

In this talk, we are interested in the sub-arrangements SHI(G) and ISH(G)

of the SHI and ISH arrangements associated with a graph G,which may not contain all hyperplanes of the braid arrangement.We prove necessary and sufficient conditions on the graph G tomake the cones of SHI(G) and ISh(G) free.

====================================================================================

Ngo Viet Trung

Depth functions of homogeneous ideals

Abstract:  A classical result of Brodmann says that the numerical non-negative function depth R/I^t, where I is a homogeneous ideal in a polynomial ring R, is always convergent, i.e. it is constant for t large enough. In 2005, Herzog and Hibi posed the conjecture that the function depth R/I^t can be any numerical non-negative convergent function. This talk will will give an affirmative answer to this conjecture and shows a similar result for the symbolic powers I^(t) of I, namely that for any numerical positive function f(t) which is periodic for t large enough, there is a homogeneous ideal I in a polynomial ring R such that depth R/I^(t) = f(t) for t > 0. It is still an open question whether the function depth R/I^(t) is always periodic for t large enough.

====================================================================================

Antonio Campillo

Poincare series of matroids

Abstract: Multiindex filtrations on the local ring of singularities have an associated Poincaré series which in many cases have direct information on their topology of geometry. These is the case of the natural filtration for plane curve singularities for which the Poincaré series coincides with the Alexander polynomial of the link. We show how for a natural filtration on polynomial rings provides a Poincaré series of matroids which has complete direct information of their discrete structure. This is a joint work with Ricardo Podestá (UNC, Argentina).

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========================  Friday  ===========================================

Xiaotao Sun

A Miyaoka-Yau type inequality of surfaces in characteristic $p>0$

Abstract: For minimal smooth projective surfaces $S$ of general type, we prove $K^2_S\le 32\chi(\sO_S)$ and give examples of $S$ with $$K^2_S=32\chi(\sO_S).$$

This proves that $\chi(\sO_S)>0$ holds for all smooth projective minimal surfaces $S$ of general type, which answers completely a question of Shepherd-Barron.

Our key observation is that such Miyaoka-Yau type inequality follows slope inequalities of a fiberation $f:S\to C$.However, we will gives examples of

$f:S\to C$ with non-smooth generic fibers of arithmetic genus $g\ge 2$ such that

$$K^2_{S/C}<\frac{4g-4}{g}{\rm deg}f_*\omega_{S/C},$$

which are counterexamples of Xiao's slope inequality in case of positive characteristic. This is a joint work with Gu Yi and Zhou Mingshuo.

====================================================================================

Poincaré polynomials of filtrations and algebraic links in the Poincaré sphere and in the links of simple surface singularities.

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the $E_8$-surface singularity $(S,0)$ in $(C^3,0)$ with the 5-dimensional sphere $\S_{\epsilon}^5$ of radius $\epsilon$ in $C^3·. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in$(S,0)$with the sphere$\S_{\epsilon}^5$of radius$\epsilon% small enough. We discuss to which extend the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It turns out that, if the strict transform of a curve in $(S,0)$ does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $E_8$-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. The Poincaré series of a filtration on the ring of germs of functions on a surface can be computed as a certain integral with respect to the Euler characteristic over the space of effective Cartier divisors. One can consider the corresponding integral over the space of effective Weil divisors. It is a power series with rational exponents: "the Weil Poincaré series". We describe to which extend the Weil Poincaré series of a collection of curve filtrations on a simple surface singularity determines the resolution of the curve and thus the topology of the corresponding algebraic link. The corresponding questions are considered for divisorial valuations on the simple surface singularities.

The talk is based on a joint work with A.Campillo and F.Delgado.

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