Title and Abstracts

Aug.9

Kenji Hashimoto (Univ. of Tokyo)

Title: Mirror symmetry for complete intersection K3 surfaces in weighted projective spaces

Abstract: We discuss mirror symmetry for some special complete intersection K3 surfaces in four dimensional weighted projective spaces. In particular, we are interested in Dolgachev mirror symmetry.

Atsushi Kanazawa (Kyoto Univ.)

Title: Tyurin degenerations and Lagrangian fibrations of Calabi-Yau manifolds

Abstract:A Tyurin degeneration is a degeneration of a Calabi-Yau manifold to a union of two quasi-Fano manifolds intersecting along a common smooth anti canonical divisor. This is known to be an analogue of a Heegaard decomposition of a compact oriented 3-manifold into two handlebodies. I will discuss some interplay between Tyurin degenerations and Lagrangian fibrations, with particular emphasis on SYZ and DHT conjectures.

Hisanori Ohashi (Tokyo Univ. of Sci.)

Title: Topological classification of automorphisms on Enriques surfaces of order 4

Abstract: In a joint work with H. Ito, we classified involutions on Enriques surfaces some years ago. Based on this, we extend the result to order 4 including both semi-symplectic and non-semi-symplectic cases. This is a joint work with H. Ito (Nagoya).

Aug.10

Shigeyuki Kondo (Nagoya Univ.)

Title: Borcherds products and K3 surfaces

Abstract: I will give a survey on an application of the theory of automorphic forms on bounded symmetric domains of type IV due to Borcherds to moduli spaces of lattice polarized K3 surfaces. I will discuss, for example, moduli of marked cubic surfaces, plane quartics, Enriques surfaces, ordered 6 points and 8 points on the projective line. All results in this talk are not new.

Kenichiro Kimura (Tsukuba Univ.)

Title: The Abel-Jacobi map for higher Chow cycles

Abstract: The Abel-Jacobi map $\Phi$ is a morphism from the group of homologically trivial algebraic cycles to a complex torus called the intermideate Jacobian. This map was defined by Griffiths, and its image is  described  as certain currents. Later Bloch generalized the map $\Phi$ to higher chow cycles. His definition was given in terms of Deligne-Beilinson cohomology, and currents are implicitly used. We will explain how to describe the image of $\Phi$ for higher Chow cycles as currents. The main ingredient is admissible semi-algebraic chains constructed together with Terasoma and Hanamura. As an application, we show that the Hodge realization of the polylog cycles can be identified with $\Phi(\rho)$ for a certain relative higher Chow cycle $\rho$.

Titile: The elliptic modular surface of level 4 and its reduction modulo 3

Abstract: The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the N\'eron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives rise to the Enriques surface of type IV in Nikulin-Kondo-Martin's  classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3.

Tomohide Terasoma (Univ. of Tokyo)

Title: Comodules over Bloch Hopf algebra associated to Aomoto polylogarithms

Abstract: In the paper by Beilinson-Goncharov-Schechtman-Varchenko, they consider a motives associated to Aomoto polylogarithms associated to hyperplane arrangement in affine spaces. In this talk, we construct a comodule over the Hopf algebra defined by Bloch using his cycle complexes in normal corssing case. In this talk, we use higher homotopy construction.

Noriko Yui (Queen’s Univ.)

Title: Four-dimensional Galois representations of certain Calabi-Yau threefolds over $\QQ$

Abstract: I will consider the four-dimensional Galois representations arising from Calabi--Yau threefolds over $\QQ$ with all the Hodge numbers of the third cohomology groups equal to one. There are many examples of (families) of such Calabi--Yau threefolds. The modularity/automorphy of such Calabi-Yau threefolds will be the main topic of this talk. Conjecturally, such Galois representations should be associated to Siegel modular forms of weight $3$ and genus $2$ on some subgroups of $Sp_4(\ZZ)$. This is a preliminary report on this project.