I have the following question.

Let $f:X\rightarrow Y$ be a surjective projective morphism between smooth projective varieties. I learned that if $\dim Y=1$, then $R^if_*\mathcal O_X$ is torsion free $\forall i\geq0$. I think smoothness of $X,Y$ is important here, and I wonder what about the condition $dim Y=1$?

Q1 Is it still true that $R^if_*\mathcal O$ is torsion free $\forall i\geq0$ when $\dim Y>1$? If false, what's a natural counter example? (To show how dimension of $Y$ matters).

  • 3
    $\begingroup$ As your questions stand, the answer is no for obvious reasons: just take for $f$ a closed immersion. You probably want $f$ surjective, $X$ connected, ...? $\endgroup$
    – abx
    Commented Aug 27, 2020 at 19:18
  • $\begingroup$ @abx Thanks.Yes I need $f$ surjective. $\endgroup$
    – xin fu
    Commented Aug 27, 2020 at 19:42
  • 4
    $\begingroup$ Q1: Take a normal surface $Y'$ with non-rational singularity. Let $g:X \to Y'$ be a resolution, and let $f:X \to Y = \mathbf{P}^2$ be the composition of $g$ with a finite map $Y' \to \mathbf{P}^2$. Then $R^1 f_* \mathcal{O}_X \neq 0$ and is supported at finitely many points. $\endgroup$
    – Anonymous
    Commented Aug 27, 2020 at 19:58
  • $\begingroup$ @Anonymous Nice example, thanks! $\endgroup$
    – xin fu
    Commented Aug 27, 2020 at 20:03
  • 1
    $\begingroup$ If you strengthen your smoothness conditions, then it is true. Assume that $f$ is is smooth over $U=Y-B$ the complement of a snc divisor. Then $Rf_*\omega _X=\sum R^if_*\omega _X[-i]$ and each $R^if_*\omega _X$ and $R^if_*\mathcal O _X$ is locally free (Koll\'ar's Higher Direct Images II Thm 2.6). Note that the $R^if_*\omega _X$ are always torsion free. $\endgroup$
    – Hacon
    Commented Aug 28, 2020 at 15:23


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