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Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the relative dimension is at least 1. Do we know anything about the singularities of Z? Z is normal more or less by construction and we do know that if X was Q-factorial then so is Z. Can we say anything more? Eg: is there a $\Delta'$ on Z with (Z, $\Delta'$) klt? Does Z have rational singularities? What if $\Delta$ = 0, is Z terminal? Canonical?

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up vote 5 down vote accepted

Under the assumption that $X$ is $\mathbb{Q}$-factorial, section 5 of the paper addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the pair $(X,\Delta)$ is klt, so is the base $(Z,\Delta_Z)$ for some $\Delta_Z$. As the example of Prokhorov shows, this is optimal, namely, the base may not be terminal (canonical) even you assume the $(X,\Delta)$ is. For lc case, I think dlt modification+perturbation reduce the question to the klt case.

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To Sandor: surely we can forget $\Delta_Z$. But I write $\Delta_Z$ because I want to emphasize in the proof the discriminant divisor will appear, and for many questions we should take it into account. – CYXU Oct 15 '10 at 17:32
Hi Chenyang! Sure that makes sense, I just meant that for the original question. I actually thought about adding that nevertheless there might be natural divisor that should be added and that I can imagine that it is easier or more natural to prove that a certain pair is klt (etc) than proving it for the ambient variety directly. – Sándor Kovács Oct 16 '10 at 7:15
Nice paper by the way! – Sándor Kovács Oct 16 '10 at 7:16

Dear Eugene,

I think I can answer one of your questions, namely about $Z$ having rational singularities. It does and here is why:

$Z$ is indeed normal by construction.

Since $f$ is a Mori-contraction, $-K_X$ is $f$-nef and $f$-big. (In fact, since the relative Picard number is $1$ it is $f$-ample). I claim that $R^if_*\mathcal O_X=0$ for all $i>0$. If $f$ were birational, this would be a simple consequence of Kawamata-Viehweg vanishing. In this case we need to work a little more.

First, by Kollár's torsion-freeness theorem (see [Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987.] for the format needed here and see the references there for proofs) these sheaves are torsion-free. (Not in general, just in this case!).

Second, let $U\subseteq Z$ be a non-empty subset over which $f$ is flat. The fibers over $U$ are Fano varieties and hence $H^i(F,\mathcal O_F)=0$ for $i>0$ for a fiber $F$. I probably should have taken a resolution to start with and then one would not have to worry about singularities, but at least for a general $F$ one can say that $(F,\Delta_{|F})$ is klt and hence Kawamata-Viehweg vanishing applies, so we get the above vanishing. Anyway, then by Grauert's theorem [Hartshorne, III.12] it follows that $(R^if_*\mathcal O_X)_{|U}=0$ for all $i>0$. However, we have already established that these sheaves are torsion-free, so if they are zero on a non-empty open set then they are zero everywhere.

OK, so we get that $R^if_*\mathcal O_X=0$ for all $i>0$. Now we are only almost there because the original definition of rational singularities would require this for a birational morphism and $f$ is decidedly not birational. But this still implies that $Z$ has rational singularities by SJK: A characterization of rational singularities.

Now, regarding whether $Z$ can be klt, canonical, etc. One simplification I can suggest is that you do not need the $\Delta'$ there. If you find a $\Delta'$ such that $(Z,\Delta')$ is klt, dlt, lc, etc, then so is $Z$ since by the $\mathbb Q$-factoriality $\Delta'$ will be $\mathbb Q$-Cartier. See [Kollár-Mori, 2.35].

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