# Log Hodge Theory and Elliptic Flat Invariants

Date 24 Februaly 2011

Place Balcony A, IPMU, Kashiwa campus, The University of Tokyo

Program

10:30 ー 12:00 Sampei Usui (Osaka university)

13:00 ー 15:00 Ikuo Satake (Osaka university)

15:30 ー 17:30 Atsushi Takahashi (Osaka univrsity)

Titles and Abstracts

Sampei Usui

Title:

Log mixed Hodge theory and applications

(joint with K. Kato and C. Nakayama)

Summary:

We constructed toroidal partial completions of moduli spaces of mixed Hodge structures with polarized graded quotients.

They are moduli spaces of log mixed Hodge structures with polarized graded quotients.

They have applications to Torelli theorem for some CY manifolds, Neron models, ...

References:

K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures,

Ann. of Math. Stud. 169, Princeton Univ. Press, 2009.

（This is a book on log Hodge theory.）

K. Kato, C. Nakayama and S. Usui, SL(2)-orbit theorem for degeneration of mixed Hodge structure, J.

Algebraic Geometry 17 (2008), 401--479.

K. Kato, C. Nakayama and S. Usui,

Classifying spaces of degenerating mixed Hodge structures,

I: Borel--Serre spaces, Advanced Studies in Pure Math. 54: Algebraic

Analysis and Around, 2009, 187--222.

II: Spaces of SL(2)-orbits, Kyoto J. Math. 51-1: Nagata Memorial Issue

(2011), 149--261. (available in arXiv).

III: Spaces of nilpotent orbits, submitted. (available in arXiv).

(Log mixed Hodge theory is developed in these four papers.)

＊ ＊ ＊ ＊ ＊

Ikuo Satake

Title:

Toward the algebraic descriptions of Frobenius manifolds for the elliptic root systemsSummary:

For the elliptic root systems, we could construct certain Frobenius manifolds.

Explicit descriptions of the flat coordinates and the potentials are given by Jacobi forms, theta functions and solutions of the differential

equations.

However it seems that some algebraic data which determine the modular properties and the behavior on the cusps of the flat coordinates are hidden behind the above

explicit descriptions.

We give a trial to describe these algebraic data by use of the notion of the conformal Frobenius structures.

＊ ＊ ＊ ＊ ＊

Atsushi Takahashi

Title:

Genus one potentials for simple elliptic singularitiesAbstract:

A calculation of the G-function for simple elliptic singularities by Ian Strachan (arXiv:1004.2140) will be explained.

The relation of their caluculations and ours (Satake-Takahashi: Gromov-Witten invariants for mirror orbifolds of simple elliptic singularities) may also be discussed.

###############################################################

This workshop is supported by

Kaken-hi Kiban S, Grant #, Daihyousha: Katsura Toshiyuki