Sheaves of E-infinity algebras and applications to algebraic varieties and singular spaces

TL;DR

This paper develops E-infinity algebra structures on sheaves related to algebraic varieties and singular spaces, extending mixed Hodge theory and defining new algebraic invariants with Steenrod operations.

Contribution

It introduces E-infinity algebra sheaf structures on singular cochains and intersection complexes, extending mixed Hodge theory and enabling Steenrod operations on intersection cohomology.

Findings

Defined E-infinity algebra structures on singular cochains and intersection complexes.

Extended mixed Hodge structures to p-adic homotopy theory.

Produced new algebraic invariants with Steenrod operations for algebraic varieties.

Abstract

We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold.