On the supports for cohomology classes of complex manifolds

Abstract

Let $X$ be a compact, connected complex manifold, and let $\xi\in H^{i}(X,{\mathbb Q} )$ be a non-trivial class. The paper deals with the possibility to construct a topological cycle $\Gamma$ on $X,$ whose class is the Poincar\'e dual of $\xi\thinspace ,$ which is closely related in a precise sense to the complex structure of $X.$ The desired properties of $\Gamma$ allow to define a differentiable relation into a suitable space of $1$-jets. This relation shows that there is a preliminary topological obstruction to construct such a $\Gamma$. The main result of the paper is that, in a relevant particular case, this obstruction disappears.