Everytopic Seminar

 April 4th, 2008, 1:40 pm – 3:00pm, Goldsmith 226

 

 

Title: The Donaldson-Thomas instantons on compact Kahler threefolds

Speaker: Yuuji Tanaka, MIT

 

Abstract:

In the late 90's, S. K. Donaldson and R. P. Thomas

suggested a program on gauge theory in higher dimensions.

In alignment with this program, Thomas introduced

the holomorphic Casson invariant for a smooth projective Calabi-Yau

threefold.

 

The holomorphic Casson invariant is a deformation invariant of a smooth

projective Calabi-Yau threefold.

Thomas defined it by constructing algebro-geometrically a virtual moduli

cycle

of the moduli space of semi-stable sheaves under some technical assumptions.

This can be viewed as a complex analogy of the Taubes-Casson invariant

defined by C. H. Taubes.

However, it is not achieved as yet to construct the invariant

in a "gauge theoretic", or an analytic way.

 

In this seminar, we exhibit an analytic approach to this invariant

by using perturbed Hermitian-Einstein equations over a compact Kahler

threefold,

which we call Donaldson-Thomas equations.

We begin this seminar with some backgrounds on the holomorphic Casson

invariant

and the Hermitian-Einstein equations.

Subsequently, we introduce the Donaldson-Thomas equations

and describe local structures of the moduli space of solutions

to the Donaldson-Thomas equations.

Furthermore, we discuss analytic results on solutions to the equations,

aiming at constructing the holomorphic Casson invariant

for compact Kahler threefolds.