Compact Kahler manifolds have the property that surjective maps induce injections on cohomology with coefficents in $\mathbb{Q}$ (That is, if $X,Y$ compact Kahler, then a surjective map $\phi: X \rightarrow Y$ induces injections $\phi^\*: H^i(Y, \mathbb{Q}) \rightarrow H^i(X, \mathbb{Q})$ for all $i$, [Voisen, Hodge Theory I, p 177]).


Question: I'm wondering if I should think of this as a property of compact Kahler manifolds, or as an instance of something more general. For example, can the Kahler condition be replaced with a more general class of manifolds (not dropping the compactness hypothesis). Perhaps one that includes not just complex manifolds but maybe a few manifolds of odd (topological) dimension? I know that if we require that $\dim X = \dim Y$ then the fact above is true more generally just for compact oriented manifolds, for formal reasons.



Injective maps on cohomology and Kahler manifolds
In order to have this property it is sufficient to require that $\phi^{-1}(y)$ is a non-zero cycle in $H\_\*(X,\mathbb Q)$, where $y$ is a generic point in $Y$. This holds indeed when $X$ and $Y$ are Kahler.


Let me give an example showing that this does not work when $X$ and $Y$ are just complex. 


**Example.** Let $X$ be a Hopf surface $X=(\mathbb C^2\setminus 0)/\mathbb Z$ where $\mathbb Z$ is acting on $\mathbb C^2$ by multiplication by (say) $2$. Then there is a fibration $\phi\colon X\to \mathbb CP^1=Y$. This fibration comes from the standard $\mathbb C^\*$-action on $\mathbb C^2$. Now $\phi^{-1}(y)$ is null-homlogous.


