Noncommutative Hodge conjecture

Abstract

The paper provides a version of the rational Hodge conjecture for $\mathsf{dg}$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed by Perry (2022) 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 $\mathsf{dg}$ algebras is true.