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

Chataur, David; Cirici, Joana

doi:10.1090/tran/8569

Sheaves of $E$-infinity algebras and applications to algebraic varieties and singular spaces
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by David Chataur and Joana Cirici PDF

Trans. Amer. Math. Soc. 375 (2022), 925-960 Request permission

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 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.

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