__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 $\rho$.

**Ichiro
Shimada (Hiroshima Univ.)**

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 (Queenâ€™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.