Date: 2025 July 5th (Sat) - 6th (Sun)
Venue: Tsuda University, Kodaira Campus, Lecture Room 5102
Access Map to the Kodaira Campus
The room is on the first floor of Building 5, which is labeled as number 8 on this map.
::: Program :::
July 5th
10:00-10:45: Yasuhiro Goto (Hokkaido University of Education) slides
11:00-11:45: Ichiro Shimada (Hiroshima University)
13:30-14:00: Rikuto Ito (Nagoya University) slides
14:15-14:45: Hongxia Zhang (Fujian Normal University) slides
15:00-15:30: Yuki Mizuno (Waseda University) slides
free discussion
July 6th
10:00-10:45: Satoshi Minabe (Tokyo Denki University) slides
11:00-11:45: Ken-ichiro Kimura (University of Tsukuba)
13:30-14:15: Hiro-o Tokunaga (Tokyo Metropolitan Univeristy) slides
14:30-15:15: Noriko Yui (Queen's University) slides
free discussion
::: Titles and Abstracts :::
::: Yasuhiro Goto ::: Berglund-H\"ubsch-Krawitz mirror symmetry from a viewpoint of formal groups
The Berglund-H\"ubsch-Krawitz mirror symmetry is constructed from Calabi-Yau threefolds of Delsarte type, $X_A$ and $X_{A^T}$, attached to matrices $A$ and its transpose $A^T$. We consider such threefolds in positive characteristic and show that there is some kind of symmetry between the formal groups of (mirror pairs associated with) $X_A$ and $X_{A^T}$.
::: Ichiro Shimada ::: The automorphism group of an Apéry--Fermi K3 surface
An Apéry--Fermi K3 surface is a complex K3 surface of Picard number 19 that is birational to a general member of a certain one-dimensional family of affine surfaces related to the Fermi surface in solid-state physics. This K3 surface is also linked to a recurrence relation that appears in the famous proof of the irrationality of ζ(3) by Apéry. We compute the automorphism group Aut(X) of the Apéry--Fermi K3 surface X using Borcherds’ method. We describe Aut(X) in terms of generators and relations. Moreover, we determine the action of Aut(X) on the set of ADE-configurations of smooth rational curves on X for some ADE-types.
::: Rikuto Ito ::: On the period map of cubic fourfolds
A cubic fourfold is a smooth cubic hypersurface of P^5. The Hodge structures of Cubic fourfolds have analogies with Hodge structures of K3 surfaces. In this talk, we will consider the period map of cubic fourfolds. For K3 surfaces, Rizov constructed the moduli space M_K with a level structure K, and defined the period map from M_K to the Shimura variety Sh_K(SO(2, 19)), and he showed the period map of K3 surfaces is defined over Q. In this talk, we will consider the moduli space of cubic fourfolds with a level structure K, and define the period map to the Shimura variety Sh_K(SO(2, 20)). Main result is to prove that the period map is defined over Q. If time permits, we want to introduce some arithmetic applications.
::: Hongxia Zhang ::: Geometric model for weighted projective lines of type (p,q)
In this talk, I will construct a geometric model for the category of coherent sheaves on weighted projective lines of type (p,q) using an annulus with marked boundary points. Our main results establish a bijection between indecomposable sheaves and certain homotopy classes of oriented curves, and show that the dimension of extension groups corresponds to the positive intersection number of curves. Using this model, we will describe the tilting graph combinatorially and identify its structure as a line or a union of quadrilaterals. We also prove that the automorphism group of the category is isomorphic to the mapping class group of the annulus and demonstrate the compatibility of their actions.
::: Yuki Mizuno ::: Bondal–Orlov’s reconstruction theorem in noncommutative projective geometry
The (derived) category of coherent sheaves on a scheme encodes rich information about the underlying geometry. P. Gabriel showed that for noetherian schemes X and Y, if Coh(X) and Coh(Y) are equivalent as abelian categories, then X and Y are isomorphic. Furthermore, A. Bondal and D. Orlov proved that for smooth projective schemes X and Y with (anti-)ample canonical bundles, if D^b(Coh(X)) and D^b(Coh(Y)) are equivalent as triangulated categories, then X and Y are isomorphic. On the other hand, J.-P. Serre showed that the category of coherent sheaves on a projective scheme can be described as the quotient category of finitely generated graded modules over the homogeneous coordinate ring by the subcategory of torsion modules. Motivated by the results of Gabriel and Serre, the quotient category of finitely generated graded modules over a (not necessarily commutative) graded ring by the subcategory of torsion modules is called a noncommutative projective scheme. In this talk, I will present an analogue of Bondal–Orlov’s reconstruction theorem in the setting of noncommutative projective geometry.
::: Satoshi Minabe ::: Good basic invariants and flat structure for complex reflection groups
In the end of 1970's, Saito-Yano-Sekiguchi found a distinguished set of basic invariants called flat invariants for real reflection groups. Nowadays, it is understood as the flat coordinates of Dubrovin's Frobenius manifold associated with the real reflection groups.
Around 2020, Satake introduced the notion of good basic invariants for complex reflection groups and showed that in the case of real reflection groups they agree with the flat invarints. Furthermore, he showed that multiplication of the associated Frobenius manifolds can be described by the good basic invariants.
In this talk, I will explain a generalization of these results to a class of complex reflection groups called the duality groups. It is shown that good basic invariants agree with the flat coordinates of the natural Saito structures (without metric) associated with duality groups and that the potentials for the multiplications of the Saito structures are described by good basic invariants. This shows that the natural Saito structures are fully reconstructed via good basic invariants. This talk is based on joint works arXiv:2307.07897, arXiv:2409.00380 with Y. Konishi.
::: Ken-ichiro Kimura ::: On certain complexes associated to categories of mixed Tate motives
Motivated by Zagier's conjecture on special values of Dedekind zeta functions, Goncharov defined a certain complex $B(n,F)$ for any field $F$ and a positive integer $n$. It is conjectured that $B(n,F)$ is quasi-isomorphic to a complex defined in terms of the coLie algebra of the Tannaka group of the conjectural category of mixed Tate motives of $F$. On the other hand Bloch and Kriz defined an explicit candidate of the category of mixed Tate motives of $F$ in terms of algebraic cycles. We discuss the problem of existence of a map from $B(n,F)$ to the complex associated to Bloch-Kriz category.
::: Hiro-o Tokunaga ::: Arithmetic of double covers and the topology of plane curves
Let $B$ and $C$ be reduced (possibly reducible) plane curves with no common components. Assume that $\deg B$ is even and consider the double cover $f'_{B} : S'_{B} \to \PP^2$ branched along ${B}$ and let $\mu_{B} : S_{B} \to S'_{B}$ be the canonical resolution of singularities. Put $\tilde{f}_{B} := f'_{B}\circ \mu_{B}$. By \emph{arithmetic of double covers of $\PP^2$}, we refer to the study of the combinatorics of the preimage $\tilde{f}_{B}^*C$ and their geometric properties, e.g., whether or not the preimage of an irreducible component of $C$ is irreducible, and so on, which can be viewed as an analogue of the arithmetic of the integer ring of a quadratic field. %$\QQ(\sqrt{m})$. We investigate properties of $\tilde{f}_{B}^*C$ and apply our consideration to study the embedded topology of $B + C$.
::: Noriko Yui ::: Modularity of certain families of K3 surfaces
We start with a double sextic family of K3 surfaces with four-parameters with Picard number 16. Then by geometric reduction (top-to-bottom) processes, we obtain three-, two- and one-parameter families of K3 surfaces with Picard number 17, 18 and 19, respectively. All these families turn out to be of hypergeometric type in the sense that their Picard–Fuchs differential equations are given by hypergeometric or Heun functions. It is known that (after suitable specializations of parameters) if these K3 surfaces happen to have CM (complex multiplication), then they will be potentially modular in the sense that the Galois representations of dimension ≤ 6 associated to the transcendental lattices are all induced from one-dimensional representations. However, the potentially modularity does not imply the modularity. In this talk, I try to describe how to determine the associated modular forms, applying arithmetic induction (bottom-to-top) processes, starting with singular K3 surfaces (the Picard number 20) with 2- dimensional Galois representations. Then increading the transcendental ranks (or equivalently, the dimension of the associated Galois representations), we try to determine corresponding modular forms. For instance, when the Picard number is 17 (the associated Galois representation is 5-dimensional), a Siegel modular form of weight 2 arises to desribe the L-function. When the Picard number is 16, it may happen that K3 family acquires RM (real multiplication). In this case, the potential modularity as well as modularity questions are wide open and currently under investigation. This is a joint work with Adrian Clingher (Missouri), Seoyoung Kim (Basel) and Andreas Malmendier (Utah State).
::: Organizers :::
Noriko Yui (Queen's University)
Yukiko Konishi (Tsuda University)
Atsushi Kanazawa (Waseda University)