Derived analytic geometry: Derived K\"ahler space and Hodge theory

Abstract

We introduce higher analytic geometry, a novel framework extending Lurie's derived complex analytic spaces. This theory generalizes classical complex analytic geometry, enabling the study of derived K\"ahler spaces with non-trivial higher homotopy groups. We develop derived de Rham and Dolbeault cohomologies, yielding a Hodge decomposition for compact derived K\"ahler spaces, and establish a derived Stokes' theorem, unifying classical results with homotopical structures.