A purity theorem for configuration spaces of smooth compact algebraic varieties

TL;DR

This paper extends Totaro's result by showing that the rational cohomology of configuration spaces of smooth complex projective varieties aligns with the Leray spectral sequence and respects mixed Hodge structures, revealing their purity.

Contribution

It demonstrates that the mixed Hodge structures on these configuration spaces are direct sums of pure Hodge structures, enhancing understanding of their algebraic and geometric properties.

Findings

Rational cohomology is compatible with mixed Hodge structures.

Configuration spaces have pure Hodge structures.

The isomorphism respects the algebraic structure.

Abstract

B. Totaro showed \cite{totaro} that the rational cohomology of configuration spaces of smooth complex projective varieties is isomorphic as an algebra to the $E_2$ term of the Leray spectral sequence corresponding to the open embedding of the configuration space into the Cartesian power. In this note we show that the isomorphism can be chosen to be compatible with the mixed Hodge structures. In particular, we prove that the mixed Hodge structures on the configuration spaces of smooth complex projective varieties are direct sums of pure Hodge structures.