I'm reading kollar's paper about higher direct image of dualizing sheaf. Suppose f: X-Y is morphism, X smooth,Y normal. He mentioned usually the higher direct image of structure sheaf is "bad," and higher direct image of dualizing sheaf is "better." I have following vague question. 1) When I think about higher direct image, I see it in the analytic topology. I don't quite understand why we could have torsion for the higher direct image of structure sheaf. Roughly, if product of two holomorphic object is 0,then one of them must be zero. Is there any good example will help me to understand it? 2) Is there any reason intuition for the torsionfreeness of direct image of dualizing sheaf. 3) If f above is resolution of singularity, then higher direct image of canonical sheaf vanish by Grauert's theorem. Is there a good explanation.

Thanks in advance!

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    $\begingroup$ For 1) if you take a normal surface $Y$ with a nonrational singularity, e.g. a cone over a high genus curve, and resolve it $f:X\to Y$, then $R^1 f_*O_X$ will be a sky scraper sheaf supported on the singular set. So it will be a torsion $O_Y$-module. The other questions are harder to answer. $\endgroup$ Mar 10, 2018 at 3:57


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