Algebraic Geometry Seminar
Botong Wang
Notre Dame
Deformation theory with cohomology constrains
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.
Wednesday September 18, 2013 at 4:00 PM in SEO 427