Cohomology of configuration spaces of punctured varieties

Abstract: Given a smooth complex variety $X$ (not necessarily compact), consider the unordered configuration space $Conf^n(X)$ defined as ${(x_1,...,x_n)\in X^n: x_i \neq x_j\ \text{for}\ i\neq j} / S_n$. The singular cohomology of $Conf^n(X)$ has long been an active area of research. In this talk, we investigate the following phenomenon: "puncturing once more" seems to have a very predictable effect on the cohomology of configuration spaces when the variety we start with is noncompact. In specific, a formula of Napolitano determines the Betti numbers of $Conf^n(X - {P})$ from the Betti numbers of $Conf^m(X)$ $(m \leq n)$ if $X$ is a smooth *noncompact* algebraic curve and $P$ is a point. We present a new proof using an explicit algebraic method, which also upgrades this formula about Betti numbers into a formula about mixed Hodge numbers and generalizes this formula to certain cases where $X$ is of higher dimension.

algebraic geometry

Audience: researchers in the topic

UBC Vancouver Algebraic Geometry Seminar

Series comments: Recordings and slides are available on the seminar webpage:

wiki.math.ubc.ca/mathbook/alggeom-seminar/Main_Page