I am curious about stacky generalizations of the following GAGA theorem:
If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is algebraic.
There is an established theory of analytic stacks (as well as higher analytic stacks). I am curious about the following question: for what stacks $U$ does the above theorem still hold (with $X$ still assumed to be a proper scheme). One case that is known to hold since the original GAGA is $U = B\mathbb{G}_m$ and I believe it is also true more general affine reductive groups. I'm interested in whether this holds for more exotic stacks, for example $BA$ for $A$ an abelian variety.
In this special case (which I am most curious about) the question can be formulated more classically: suppose $X$ is a proper scheme (say, a curve), $A$ is a polarized abelian variety, and $\mathcal{A}\to X$ is a complex-analytic principal $A$-bundle over $X$. Is the total space $\mathcal{A}$ also algebraic?
Best Answer
For your "special" question, the answer is negative, already when $A$ is an elliptic curve. In fact, a principal $A$-bundle over a smooth projective curve $B$ which is not topologically trivial is never algebraic — see the book by Barth, Hulek, Peters, Van de Ven, ch. V, Proposition 5.3. There are many examples of this situation, for instance Hopf surfaces $(B=\mathbb{P}^1)$ or Kodaira primary surfaces $(g(B)=1)$.