Automorphic Cohomology and the Limits of Algebraic Cycles

TL;DR

This paper constructs a specific automorphic cohomology class on a Shimura variety that cannot be realized by algebraic cycles, revealing fundamental limitations in the geometric realization of automorphic forms.

Contribution

It provides the first explicit example of an automorphic cohomology class that cannot be represented by algebraic cycles, demonstrating an intrinsic obstruction in the automorphic-to-geometric correspondence.

Findings

Constructed a rational Hodge class in the intersection cohomology of a Shimura variety.

Proved this class is non-interior and cannot come from algebraic cycles or known geometric constructions.

Highlights a fundamental asymmetry between automorphic cohomology and algebraic cycle realizations.

Abstract

This paper establishes an explicit obstruction to constructing algebraic cycles from automorphic cohomology classes on Shimura varieties. We produce a rational Hodge class $\Omega_E$ in the intersection cohomology of the Baily-Borel compactification of a Shimura variety for $\text{SO}(2,26)$, arising from a stable residual automorphic representation via theta lift from the weight-$2$ newform of conductor $11$. While $\Omega_E$ is automorphic and of pure Hodge type, we prove it is non-interior and hence cannot be obtained from special cycles, theta lifts, endoscopic transfers, or boundary pushforwards, all of which yield interior classes. The result is unconditional, relying only on Arthur's classification, Vogan-Zuckerman theory, the fundamental lemma, and the Zucker conjecture (proven by Looijenga-Saper-Stern), and it highlights a fundamental asymmetry between automorphic cohomology and geometric access to algebraic cycles, refining the Hodge conjecture from a question of existence to one of constructive tractability.