Noncommutative Hodge conjecture

TL;DR

This paper formulates a noncommutative version of the Hodge conjecture for certain categories and algebras, providing evidence and proving it for smooth proper connective dg algebras using noncommutative geometry techniques.

Contribution

It introduces a noncommutative Hodge conjecture, establishes its equivalence to existing versions, and proves it for smooth proper connective dg algebras.

Findings

Evidence supporting the noncommutative Hodge conjecture

Equivalence with previously proposed noncommutative Hodge conjecture

Proof of the conjecture for smooth proper connective dg algebras

Abstract

The paper provides a version of the rational Hodge conjecture for $\3\dg$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed in \cite{perry2020integral} for admissible subcategories. We obtain examples of evidence of the Hodge conjecture by techniques of noncommutative geometry. Finally, we show that the noncommutative Hodge conjecture for smooth proper connective $\3\dg$ algebras is true.