Higher Complex Structures

Abstract

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.