Timeline for Easiest proof for showing finite etale (analytic) quotients of algebraic varieties are algebraic
Current License: CC BY-SA 3.0
8 events
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| Feb 17, 2017 at 22:36 | comment | added | Jean-Paul | @nfdc23 Thank you for your comment. I corrected my last parenthetical statement. I actually don't remember where I read it. It's just something that I vaguely recall reading somewhere, but I might actually be misremembering. | |
| Feb 17, 2017 at 22:34 | history | edited | Jean-Paul | CC BY-SA 3.0 |
I took into account the comments.
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| Feb 17, 2017 at 7:25 | answer | added | Francesco Polizzi | timeline score: 6 | |
| Feb 16, 2017 at 21:11 | comment | added | nfdc23 | The parenthetical at the end is an incorrect argument, because the Riemann Existence Theorem doesn't ensure in the non-proper case that the unique compatible algebraization of $X^{\rm{an}}$ to a finite etale cover of $V$ is necessarily $X$; it could be another algebraization of the same analytic space. One gets examples of this via the footnote near the end of Chapter 1 of Mumford's book on abelian varieties. If you are requiring that $V$ be somehow "compatible" with $X$ then that is not always possible. Probably you are forgetting some conditions from what you read; where did you read it? | |
| Feb 16, 2017 at 21:08 | comment | added | nfdc23 | Hironaka's example exists as an algebraic space, though not as a scheme, so it does have meaningful algebro-geometric structure. Indeed, his analytic constructions are compact Hausdorff Moishezon spaces, and Artin proved that analytification is an equivalence from the category of proper algebraic spaces over $\mathbf{C}$ to the category of compact Hausdorff Moishezon spaces. | |
| Feb 16, 2017 at 18:33 | comment | added | David E Speyer | You need $Y$ quasi-projective. Otherwise, consider Hironaka's examples (Hartshorne Appendix B.3) of $X$ an algebraic, nonprojective, $3$-fold and $Y$ a non-algebraic $3$-fold. If you make the right choices, there is a $2$-fold unbranched cover $X \to Y$. | |
| Feb 16, 2017 at 18:16 | review | First posts | |||
| Feb 16, 2017 at 18:20 | |||||
| Feb 16, 2017 at 18:10 | history | asked | Jean-Paul | CC BY-SA 3.0 |