I have a holomorphic principal bundle,
$$E\xrightarrow{H} B$$ defined by an action of a contractible (non-compact) Lie group $H$ (in my case $H\cong\mathbb{C}^l$). Here E and B are complex manifolds, meaning that $H$ acts properly. We can assume that $B$ is a complete manifold.


Is there any sufficient condition to say that any automorphism of $B$ lifts to an automorphism of $E$?


I believe the question to be quite natural and classical, but I was unable to find any reference on it :(



Lifting holomorphic automorphisms along the principal bundle
An automorphism $f:B \to B$ lifts to an automorphism of $E$ if and only if the identity of $B$ lifts to an isomorphism from $E$ to $f^\*E$, i.e. if and only if $E$ and $f^\*E$ are isomorphic as principal bundles over $B$.


Defining principal $H$-bundles by transition functions, one sees that isomorphism classes of (holomorphic) principal $H$-bundles are classified by the "first Cech cohomology" of the sheaf of holomorphic functions with values in $H$. I'm putting quotation marks because, when $H$ is not abelian, this is not a cohomology group in the classical sense (it is not a group, and there is no higher cohomology). Still, it is a set $Y$ with an action of $\mathrm{Aut}(B)$, and $f\in \mathrm{Aut}(B)$ lifts to $E$ if and only if $f$ fixes the point corresponding to $E$ in $Y$.


In the case $H= \mathbb C^l$, you really have $Y = H^1(\mathcal O^l)= \left(H^{0,1}(B)\right)^l$ where $\mathcal O$ is the sheaf of holomorphic functions (the second equality is the Dolbeault isomorphism). In conclusion:


* The isomrphism class of $E$ is given by a vector $v$ in $\left(H^{0,1}(B)\right)^l$
* $f$ lifts to $E$ if and only if $v$ is fixed by $f^\*$.


**Example 1:** If $H^{0,1}(B)$ is $0$, then every principal $\mathbb C^l$ bundle is trivial and every automorphism of $B$ lifts trivially.


**Example 2:** Say $B$ is compact Kähler and $f$ belongs to the identity component of $\mathrm{Aut}(B)$. Then $f$ acts trivially on $H^1(B,\mathbb Z)$, hence also on $H^{0,1}(B)$ by Hodge (I wonder if there is a counterexample for $B$ non-kähler by the way). Hence $f$ lifts to an automorphim of any principal $\mathbb C^l$-bundle.


