On the Hodge conjecture for $q$-complete manifolds
  We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}\mathbb{P}^n\setminus A$ of any complex projective manifold defined by $q<n$ independent equations. 
Franc Forstneric, Jaka Smrekar, and Alexandre Sukhov
math.AG math.CV