# GAGA for stacks

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?

• 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. – Ariyan Javanpeykar Dec 17 '18 at 23:02
• ...or if $U=[Y/G]$ is the quotient stack for an action of a finite group $G$. – Piotr Achinger Dec 18 '18 at 9:20
• @PiotrAchinger Probably one needs some mild "separatedness" condition on the action of $G$ on $Y$... – Ariyan Javanpeykar Dec 18 '18 at 13:41

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 $$(g(B)=1)$$.
• 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. – François Brunault Dec 17 '18 at 21:27