Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-negative extremal face $F$ of $\overline{NE}(X)$.

Assuming $f$ is a birational morphism, does $Z$ have rational singularities?

If $f$ can be factored as a sequence of divisorial contractions then this is well-known, but I do not know whether this is true if $f$ is a small contraction.

I do not even know if $Z$ is always Cohen-Macaulay (which would be implied by $Z$ having rational singularities).

(As for motivation, the question probably has little or no direct relevance to the MMP; I am interested in rational singularities for their own sake.)


1 Answer 1


In characteristic 0, the answer is well known. By assumption there is an ample divisor $A$ such that $K_X+\Delta+A$ cuts out $F$ and hence by the BPF theorem $K_X+\Delta+A\sim _{\mathbb Q,f}0$ and in fact $K_X+\Delta +A\sim _{\mathbb Q}f^*(K_Z+B_Z)$ where $(Z,B_Z)$ is klt; in the birational case $B_Z=f_*(\Delta+A)$ and otherwise you need to use the canonical bundle formula see Theorem 0.2 of https://arxiv.org/pdf/math/0308143.pdf. But then $Z$ has rational singularities by Thm 5.22 in Koll'ar-Mori.

In characteristic $p>0$ see https://arxiv.org/abs/2006.03571 for some recent results, but note that klt singularities are not always rational or CM (counterexamples in dimension 3 and char 5 are discussed in thm 1.6 of this paper). You may also be interested in https://arxiv.org/pdf/2206.02674.pdf and https://arxiv.org/abs/1703.02269.


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