Silver Workshop : Complex geometry and related topics VII (Series final)

Silver Workshop : Complex geometry and related topics VII (Series final)

(List updated on January 14, 2025)

• Organizer: N. Yui (Queen’s Univ.), K. Saito (RIMS, Kyoto), S. Hikami (OIST)
• Local organizer: N. Kawazumi (Univ. Tokyo)
• Support: OIST funding
• Date: March 11-13, 2025
• Venue: Room123, The University of Tokyo, Math. Department.

Silver Workshop VII (2025, March) Tentative Program

*Confirmed talker

• Conference dinner: March12, 2025 18:00-20:00
  Organized by S.Ishii, M.Oka , K.Watanabe  
• Venue: Lever Son Verre (Komaba Campus)
  Registration : Attendants for this dinner is required for registration. Dead line on March 1, 2025.
  Talkers of Silver workshop VII are invited to this dinner with free charge. Talkers are also required for registration of dinner.

Title and abstract :

Kyoji Saito (RIMS Kyoto University)
Title: Semi-infinite Hodge structure for hyperbolic root systems
Abstract: We construct semi-infinite Hodge structure associated with  hyperbolic root systems of rank 2. As its consequences, we determine
(1) its associated flat structure (the Frobenius manifold structure) and 
(2) the period map associated with the primitive form, both defined on  the extended orbit space of the hyperbolic Weyl group action on a tube domain.

Mutsuo Oka (Tokyo University of Science)
Title:Almost non-degenerate functions and some applications
Abstract: We first introduce a class of functions "Almost Newton non-degenerate functions" which include Newton non-degenerate functions but it contains much wider class of functions.
We give a generalization of Varchenko formula for the zeta functions and then we present two applications.
 

Takuya Sakasai (The University of Tokyo)
Title: On structures of groups of Kim-Manturov

Makiko Mase (Tokyo Metropolitan University）
Title: On K3 surfaces admitting finite symplectic automorphisms
Abstract: We consider an algebraic K3 surface X admitting a finite automorphism group G that symplectically acts on X. It is known that the quotient space X/G contains at most simple singularities, and is birational to a K3 surface Y. Denote by L the lattice generated by all the classes of the irreducible components of the exceptional divisor in Y obtained by resolving the singularities in X/G. Despite the fact that the lattice L, which is a disjoint union of lattices of type ADE, is a sublattice of the K3  lattice, the embedding is not necessarily primitive. We post a problem to determine whether or not there exists a unique primitive sublattice of the K3 lattice that contains the lattice L. If the group G is abelian, then, V.V.Nikulin gives an affirmative answer to the problem as well as presenting the explicit primitive sublattice for each group. In case the group G is simple, U.Whitcher investigates the problem and gives the answers.
In our study, we analyse the problem in all the remaining cases. In the talk, we discuss how to attack the problem, and show how much we have sorted it out.
This is based on an on-going joint work with Kenji Hashimoto.

Todor Milanov (IPMU, University of Tokyo)
Title: Dubrovin conjecture and the second structure connection
Abstract: My talk will be based on my recent paper with John Alexander Cruz Morales. The theory of  Frobenius manifolds was preceded by  Saito's theory of flat structures in singularity theory. In particular, 
many constructions in singularity theory are straightforward to import in the theory of Frobenius manifolds too. In my talk, I would like first to explain how one can introduce the ingredients of twisted 
Picard--Lefschetz theory for an arbitrary semi-simple Frobenius manifold.  Our main result with Alex is a reformulation of the so-called refined Dubrovin conjecture in terms of the monodromy data for the 
second structure connection of quantum cohomology.
 

Tatsuo Suwa (Hokkaido University)
Title:Localized intersection product for maps and applications
Abstract: In his celebrated book, W. Fulton defines intersection products, in the algebraic category, using normal cones. We define the notions in combinatorial topology. In particular, the localized intersection product corresponds to the cup product in relative cohomology via the Alexander duality. This may be extended to the localized intersection product for maps.  Combined with the relative Čech-de Rham cohomology, it is effectively used in the residue theory of vector bundles and coherent sheaves. As an application, we have the functoriality of Baum-Bott residues of singular holomorphic foliations under certain conditions, which yields an answer to an existing problem. 
This includes a joint work with M. Corrêa.
References:
[1] W. Fulton, Intersection Theory, Springer, 1984.
[2] T. Suwa, Complex Analytic Geometry - From the Localization Viewpoint,
     World Scientific, 2024.
[3] M. Corrêa and T. Suwa, On functoriality of Baum-Bott residues, in preparation.
 

Andreani Petrou (OIST)
Title: Knots, links and Harer-Zagier factorisability
Abstract: The focus of this talk will be the HOMFLY-PT polynomial and its Harer-Zagier (HZ) transform, a discrete Laplace transform, which maps it into a rational function. For some special families of knots and links, generated by full twists and Jucys–Murphy braids, the latter has a simple factorised form and hence their HOMFLY-PT polynomial is fully encoded in two sets of integers, corresponding to the numerator and denominator exponents. These exponents turn out to be related to the Khovanov homology and its Euler characteristics. We conjecture that the HZ factorisability is in 1-1 correspondence with a relation between the HOMFLY-PT and Kauffman polynomials, which is proven in some specific cases. The latter is equivalent to the vanishing of the two-crosscap BPS invariants of topological strings.

Shinichi Tajima (Niigata University)
Title: Holonomic D-modules associated to a hypersurface with non-isolated singularities
Abstract: We consider a method for analyzing holonomic D-modules associated to a hypersurface with non-isolated singularities. We start by recalling some basics on solution sheaves to holonomic D-modules and b-functions. Then, we give a method for computing local cohomology solutions to holonomic D-modules associated to roots of b-functions. The key idea is the use of the concept of Noetherian differential operators for local cohomology classes. We study, as an application of our approach, Whitney equisingular deformations of isolated hypersurface singularity. We examine two typical cases, (1) semi-quasi homogeneous singularity and (2) Newton non-degenerate singularity. We show, by using examples, a method to determine the structure of holonomic D-modules associated to the deformations. We consider, as another application, Kashiwara operator by looking at s-parametric annihilators.
Reference:
J. Alvarez Montaner and F. Planas Vilanova, Divisors of expected Jacobian type, Math. Scand. 127 (2021), 161-184.

Shinobu Hikami (OIST)
Title: Residues and Duality  for  HOMFLY-PT polynomial
Abstract: Harer-Zagier transform (HZ) of HOMFLY-PT polynomial is a rational polynomial, which  is  decomposed into the sum of factorised forms with some algebraic structures.   We apply the residues and duality theorem for HZ, and discuss the relation between  sl(N) Khovanov-Rozansky link homology and  N=2 supersymmetric Landau-Ginzburg potential.  By the insertion of the parameter λ as λ= \(q^m\) in this decomposed formula, a new kind of knot invariant polynomial is obtained, which leads  to a factorised form in a single variable q, similar to  Molien series of the singularity theory. This is a joint work with Andreani Petrou.

Takahiro Saito
Title: Monodromic mixed Hodge modules and Hodge microsheaves on plumbings
Abstract: This talk is about an application of my previous talk at Silver workshop V in 2022. Mixed Hodge modules on complex manifolds are objects defined as a tuple of D-modules, perverse sheaves, Hodge and weight filtrations on them, for Hodge theory on constructible sheaves. In general, it is difficult to answer the questions like "What kind of objects exists?" and "What forms do they take?". In this talk, I will present a result that describes mixed Hodge modules under the assumption of being "monodromic" in terms of linear algebraic data. And I will explain that it leads a natural definition of the "Fourier-Laplace transform of monodromic mixed Hodge modules". On the other hand, the category of microsheaves on a plumbing plays an important role for a description of the Fukaya category in the study of mirror symmetry. As an application of the above result, I will introduce a "mixed Hodge structure" on microsheaves. Furthermore, I will discuss an observation that the weight on the Hodge-microlocal skyscrapers on the $A_n$-plumbing induces a certain grading called the Koszul grading of an algebra, which appears in the formulation of "Koszul duality" for the catgeory of microsheaves (or the Fukaya category).
This is a joint work with Tatsuki Kuwagaki (Kyoto University).