  The paper generalizes the classical Hodge conjecture to smooth and proper dg
categories. The noncommutative Hodge conjecture is proved to be additive for
semi-orthogonal decompositions. 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 dg algebras is
true.
