Automorphic Cohomology and the Limits of Algebraic Cycles

TL;DR

This work proves an unconditional obstruction showing that a rational Hodge class $\Omega_E$ in $\mathit{IH}^{26}(\overline{X}^{\mathrm{BB}}, \mathbb{Q}) \cap \mathit{IH}^{13,13}$ for a Shimura variety of type $\mathrm{SO}(2,26)$ can originate from a stable residual automorphic representation via a theta lift, yet cannot be realized by any known algebraic-cycle constructions (special cycles, theta lifts from smaller varieties, endoscopic transfers, or boundary push-forwards). The obstruction hinges on a key interior/non-interior dichotomy: algebraic cycles define interior classes, while $\Omega_E$ lies outside interior cohomology due to its residual automorphic origin. The argument leverages Arthur’s classification for orthogonal groups, Vogan–Zuckerman theory for $(\mathfrak{g},K)$-cohomology, the fundamental lemma, and Zucker’s conjecture to produce a fully unconditional result. Consequently, the paper refines the Hodge conjecture by shifting focus from mere existence to constructive tractability, revealing an asymmetry between automorphic cohomology and geometric access to cycles on Shimura varieties. It also indicates that the Kudla program does not capture all Hodge classes in this setting and motivates the search for new geometric constructions aligned with stable residual data.

Abstract

This paper establishes an explicit obstruction to constructing algebraic cycles from automorphic cohomology classes on Shimura varieties. We produce a rational Hodge class $Ω_E$ 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 $Ω_E$ 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.

Theorems & Definitions (99)

• Conjecture 1.1: Hodge, 1950
• Theorem 1.3: Implicit in Arthur's work
• Definition 1.4
• Theorem 1.6: Main Theorem
• Theorem 1.7: Obstruction Theorem
• Theorem 1.8: Unconditional Result
• Remark 1.9: Limitations
• Proposition 1.11: Rationality
• Theorem 1.14: Rallis non-vanishing
• Theorem 1.15: Stable residual nature