I have been trying (without success) to determine the following. Let $P$ denote the space of monic polynomials of degree $n$ with complex coefficients, which have distinct roots. It is known that the fundamental group of this space is the braid group $B_n$. Let $X \rightarrow P$ denote the universal cover. Then $B_n\subset Aut (X)$ where $Aut (X)$ is the group of holomorphic automorphisms of $X$. For $n\geq 4$, is it true that $Aut (X)=B_n$? I do not see any other automorphisms (but I am a novice in the area).