Let $f:X\to Y$ be a holomorphic map of holomorphic manifolds. You can assume that $dimY=1$. Let $\tilde X$ and $\tilde Y$ be universal covers of $X$ and $Y$ with group of holomorphic automorphisms $Aut(\tilde X)$ and $Aut(\tilde Y)$. Do we get a homomorphism $Aut(\tilde X)\to Aut(\tilde Y)$ in general? 



Existence of a map between automorphism group of universal covers
The answer is negative. For instance, let $X$ be a compact quotiont of the unit ball in ${\mathbb C}^2$ by a discrete torsion-free subgroup of $PU(2,1)$, and $Y$ be a hyperbolic Riemann surface. There are many examples where there exists a nonconstant holomorphic map $f: X\to Y$ (say, if $b\_1(X)\ne 0$ then there is always one, for some $Y$), but, clearly, there are no nontrivial homomorphisms $PU(2,1)\to PU(1,1)$. 


