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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?

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  • $\begingroup$ If $U$ is an open substack of a proper algebraic stack with finite inertia over $\mathbb{C}$ and $X$ is a proper scheme over $\mathbb{C}$, then probably any morphism $X^{an}\to U^{an}$ is algebraic. $\endgroup$ – Ariyan Javanpeykar Dec 17 '18 at 23:02
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    $\begingroup$ ...or if $U=[Y/G]$ is the quotient stack for an action of a finite group $G$. $\endgroup$ – Piotr Achinger Dec 18 '18 at 9:20
  • $\begingroup$ @PiotrAchinger Probably one needs some mild "separatedness" condition on the action of $G$ on $Y$... $\endgroup$ – Ariyan Javanpeykar Dec 18 '18 at 13:41
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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 $(g(B)=1)$.

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  • $\begingroup$ Thanks! Is there a good criterion for when such a result does hold? $\endgroup$ – Dmitry Vaintrob Dec 17 '18 at 21:25
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    $\begingroup$ Other examples are given by complex tori: Shafarevich constructed an extension of elliptic curves $0 \to E_1 \to X \to E_2 \to 0$ where $X$ is not algebraic. More generally, given two abelian varieties $A_1,A_2$ of dim $>0$, almost all extensions $0 \to A_1 \to X \to A_2 \to 0$ are not abelian varieties. $\endgroup$ – François Brunault Dec 17 '18 at 21:27

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