# Singularities of contractions of extremal faces

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.)

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.