# The shape of an algebraic variety

An algebraic variety X over the complex numbers has, as one of its main facets, a topological space $X$. The study of $X{\rm top}$ has played an important role in the history of algebraic geometry. We will present a way of measuring the “shape” of $X$ by considering maps from it into different targets. The targets T, which are like spaces, are also profitably viewed as n-stacks, a notion from higher category theory. The complex algebraic structure of X leads to a number of different structures on $Hom(X{\rm top},T)$. For example when $T=BG$, the mapping stack $Hom(X^{\rm top},BG)$ may be viewed as the moduli space of G-bundles with integrable connection, or principal G-Higgs bundles. These fit together into Hitchin’s twistor space. Consideration of these structures is a good way of organizing the investigation of the topology of complex algebraic varieties.

This talk is part of the Kuwait Foundation Lectures series.