Arithmetic and Geometry Seminar

Date: 11/17

Time: 1-2:30

Place: SH 4607

Speaker: Noriko Yui

Affiliation: Queen's University, visiting MSRI

Title: Arithmetic of Calabi--Yau varieties and Mirror Symmetry

Abstract

A smooth projective variety $X$ of dimension $d$ is called a {\it Calabi--Yau} variety if $H^i(X,{\mathcal O}_X)=0$ for every $i, 0 The dimension three ones are Calabi--Yau threefolds where the full picture of {\it mirror symmetry} comes into the center screen. Roughly speaking, the mirror symmetry conjecture for families of Calabi--Yau threefolds asserts that Calabi--Yau threefolds appear in pairs, that is, given a Calabi--Yau threefolds $X$, there is its mirror partner $X^*$, and both $(X, X^*)$ should give rise to isomorphic physical theories.

In this lecture, I will try to convey the excitement surrounding mirror symmetry from the point of view of arithmetic and geometry.

I will start with rigid Calabi--Yau threefolds. The mirror symmetry conjecture fails for this class of Calabi--Yau threefolds! Rigid Calabi--Yau threefolds are therefore objects governed by algebraic and arithmetic properties. I will discuss the modularity conjecture of all rigid Calabi--Yau threefolds defined over ${\Bbb Q}$.