Hodge decomposition of L_2-cohomology and intersection cohomology of a Shimura variety

TL;DR

This paper demonstrates that the Hodge decomposition for L_2-cohomology and intersection cohomology of Shimura varieties can be established using existing proofs of the Zucker conjecture, linking classical and intersection cohomology.

Contribution

It shows that the Hodge decompositions for these cohomologies can be obtained from proofs of the Zucker conjecture, clarifying their relationship.

Findings

Hodge decomposition exists for L_2-cohomology of Shimura varieties.

Hodge decomposition exists for intersection cohomology of Baily-Borel compactification.

Existing proofs of the Zucker conjecture imply these Hodge decompositions.

Abstract

Classical Hodge theory endows the square integrable cohomology of a Shimura variety X with values in a locally homogeneous polarized variation of Hodge structure E with a natural Hodge decomposition. The theory of Morihiko Saito does the same for the E-valued intersection cohomology of the Baily-Borel compactification of X. Existing proofs of the Zucker conjecture identify these cohomology groups, but do not claim this for their Hodge decompositions. We show that one of the proofs yields that as well.