Let $f: X \to C$ be a fibration from a smooth variety $X$ to a smooth curve $C$ over an algebraically closed field $k$. If $k=\mathbb C$, we know for all $i$, $R^i f_* \omega_{X/C}$ is semipositive, which gives us $\deg R^i f_* \omega_{X/C} \ge 0$. However, if char $k>0$, the semipositivity is not true. But can we still get the weaker conclusion for the degree to be nonnegative?
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$\begingroup$ for relative dimension 1, see Szpiro's Section 3-2 of "Seminaire sur les pinceaux de courbes de genre au moins deux", asterisque 86 (1981). $\endgroup$– Christian LiedtkeCommented Nov 29, 2012 at 8:44
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$\begingroup$ Thanks, Christian. For the fibered surface case, the degree of $f_* \omega_{X/C}$ is non-negative in semistable cases. But is there any result related to the higher dimensional case which involves the higher direct images? $\endgroup$– TongCommented Nov 29, 2012 at 20:25
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$\begingroup$ I don't know of any reference. If I had to guess: probably the semipositivity fails if f is not generically smooth (the Albanese morphism of a surface of general type with c2<0 might provide a counter-example). If f is generically smooth, it might actually be true, except for some very few cases. $\endgroup$– Christian LiedtkeCommented Nov 30, 2012 at 6:36
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