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?

  • $\begingroup$ 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
  • 1
    $\begingroup$ ...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

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 $(g(B)=1)$.

| cite | improve this answer | |
  • $\begingroup$ Thanks! Is there a good criterion for when such a result does hold? $\endgroup$ – Dmitry Vaintrob Dec 17 '18 at 21:25
  • 7
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.