Self-dual Sheaves on Reductive Borel–Serre Compactifications of Hilbert Modular Surfaces

Abstract

The main theorem of this article asserts that the category of self-dual sheaves compatible with the intersection chain sheaves (for upper/lower middle perversity) on the reductive Borel—Serre compactification \(\bar X\) of a Hilbert modular surface is nonempty. Also we prove that the direct image of such a sheaf under the canonical map to the Baily—Borel compactification is isomorphic (in the derived category) to the intersection chain sheaf for upper and lower middle perversity. As a consequence of the main theorem, there exist characteristic L-classes of these sheaves in the rational homology of \(\bar X\). In fact, these classes do not depend on the choice of a self-dual sheaf and hence are invariants of the compactification \(\bar X\).