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

TL;DR

This paper introduces higher analytic geometry, extending classical complex analytic geometry to derived K"ahler spaces with homotopical structures, and develops associated cohomologies and theorems.

Contribution

It presents a new framework for derived complex geometry, generalizing classical theories and establishing foundational results for derived K"ahler spaces.

Findings

Development of derived de Rham and Dolbeault cohomologies

Hodge decomposition for compact derived K"ahler spaces

Derived Stokes' theorem unifying classical and homotopical results

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.