If $f : X \to X$ is an automorphism of a singular variety over $\mathbb C$, does there exist a resolution of singularities $g : Y \to X$ such that the induced birational map on $Y$ is also an automorphism?


1 Answer 1


Edited after Mark's comment. Also after ACL's comment. Cheers.

Take a resolution $g:Y\to X$ and let $Y'=X\times_{f,g} Y$ with projection maps $f':Y'\to Y$ and $g':Y'\to X$. Next let $Y''=X\times_{f^{-1},g'} Y'$ with projection maps $f'':Y''\to Y'$ and $g'':Y''\to X$ and $Y'''=X\times_{f,g''} Y''$ with projection maps $f''':Y'''\to Y''$ and $g''':Y'''\to X$.

Observe that $Y''=X\times_{f^{-1},g'} Y'= X\times_{f^{-1},g'}(X\times_{f,g} Y)\simeq X\times_{\mathrm{id}_X ,g}Y\simeq Y$ where the last isomorphism is given by $f'\circ f''$ and also, clearly, $g''=g$. Similarly $Y'''=X\times_{f,g''} Y''\simeq X\times_{f,g'}(X\times_{f^{-1},g} Y')\simeq X\times_{\mathrm{id}_X ,g'}Y'\simeq Y'$ where the last isomorphism is given by $f''\circ f'''$ and also, clearly, $g'''=g'$ and then $f'''=f'$.

So we get that for an arbitrary resolution $g:Y\to X$ there is another resolution $g':Y'\to X$ such that $f':Y'\to Y$ is an isomorphism with inverse $f'':Y\to Y''$.

Now if we choose $g:Y\to X$ to be a functorial resolution of $X$ (as in 3.4 of Kollár's book), then it follows that $g'=g$ and so $f'$ is the required lift. Functorial resolutions exist by 3.36 of Kollár's book.

  • 2
    $\begingroup$ What properties of $g$ are you using here? It's certainly not true that it lifts to an automorphism on every resolution: maybe $X$ is already smooth, and $g$ blows up $X$ at some point that isn't fixed by $f$. Then the induced birational map contracts the exceptional divisor. $\endgroup$
    – user47305
    Commented Mar 4, 2016 at 2:38
  • $\begingroup$ @Mark: You are right. But (I think) so am I! I wrote this hastily, but it is not wrong, just as it was it did not quite answer the question or more kindly put it did not have quite enough details. I think it is better now. Cheers! $\endgroup$ Commented Mar 4, 2016 at 6:38
  • $\begingroup$ When you write “of Kollár”, do you mean that the existence of functorial resolutions should be attributed to him? I thought it was due to Bierstone-Milman and Encinas-Villamayor. In any case, I agre that Kollár's book is more than a practical and useful reference. $\endgroup$
    – ACL
    Commented Mar 4, 2016 at 12:02
  • $\begingroup$ Come on, ACL, this is not an article, you can't possibly expect me to properly attribute every single notion in an MO answer. The question was not "Who came up with this idea first?". When I write [reference] I mean that you can click on it, it tells you a place where it is stated. I thought it was useful to tell people what I actually mean by that word. I didn't say the result was Kollár's just that it can be found in that book. $\endgroup$ Commented Mar 4, 2016 at 16:38

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