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

TL;DR

The paper addresses whether the natural Hodge decomposition on $L^2$-cohomology $H^{\bullet}_{(2)}(X,{\mathbb E})$ of a Shimura variety matches the Hodge structure on intersection cohomology ${\operatorname{IH}}^{\bullet}(X^{*},{\mathbb E})$ provided by Hodge modules. It extends Zucker’s isomorphism to a filtered, Hodge-theoretic setting by showing that the local Zucker-type complexes carry a compatible Hodge filtration and represent the intersection complex ${\mathscr I}{\mathscr C}^{\bullet}_{X^{*}}({\mathbb E}_{\mathbb C})$ in the category ${\rm MF}(X^{*},{\mathbb C})[-m]$. The main result verifies this in the Shimura case, proving that the $L^{2}$-filtered complex and its Hodge filtration realize the same Hodge structure as Saito’s Hodge module on ${\operatorname{IH}}^{\bullet}({X^{*}},{\mathbb E}_{\mathbb C})$. This yields a filtered refinement of Zucker’s conjecture with a canonical compatibility between harmonic forms and Hodge module decompositions, enhancing the understanding of Hodge theory on Shimura varieties and their compactifications.

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.

Theorems & Definitions (2)

• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:main']}