Botong Wang : Deformation theory with cohomology constrains

Posted by Mihnea Popa , part of the Algebraic Geometry Seminar.

At

Sept. 18, 2013, 4 p.m.

In

SEO 427

Abstract

Given a smooth manifold $X$, the set $\mathbf{R}(X, n)\stackrel{\textrm{def}}{=}Hom(\pi_1(X), Gl(n, \mathbb{C}))$ has naturally an algebraic variety structure. Each element in $\mathbf{R}(X, n)$ corresponds to a local system on $X$. The local structure of $\mathbf{R}(X, n)$ at a point $\rho$ can be understood by studying the deformation theory of the associated local system $L_\rho$. As a general principle, such deformation problem is governed by a differential graded Lie algebra (DGLA). In this talk, we will discuss the local structure of some canonically defined subvarieties of $\mathbf{R}(X, n)$: $V^i_k(X, n)=\{\rho\in\mathbf{R}(X, n)|\dim H^i(X, L_\rho)\geq k\}$. This is equivalent to studying deformation problem with cohomology constrains. We will introduce a new principle, that such deformation problem with cohomology constrains is governed by a DGLA together with a module of this DGLA. Particularly nice results can be obtained, when $X$ is a compact K\"ahler manifold. This is joint work with Nero Budur.