# Torsion freeness of direct image of structure sheaf?

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).

• 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, ...?
– abx
Commented Aug 27, 2020 at 19:18
• @abx Thanks.Yes I need $f$ surjective. Commented Aug 27, 2020 at 19:42
• 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. Commented Aug 27, 2020 at 19:58
• @Anonymous Nice example, thanks! Commented Aug 27, 2020 at 20:03
• 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. Commented Aug 28, 2020 at 15:23