Does a resolution of a rational singularity have rationally connected fibers?

A rational singularity is a singularity of a complex variety $$X$$ such that for any resolution $$\pi:\; \tilde X\rightarrow X$$ the higher direct images $$R^i\pi_*(O_{\tilde X})$$ vanish for all $$i>0$$. Suppose that $$X$$ has isolated rational singularity, and $$\tilde X\rightarrow X$$ its resolution. I expect that the fiber of $$\pi$$ over the singular point is rationally connected; I would be very grateful for any reference to this. I need to apply this to the local situation, so it would be especially nice if the argument does not use projectivity.

• Do you want to assume that the singularities are also Gorenstein? Then it should follow from Elkik together with Hacon-McKernan's proof of Shokurov's conjecture. Sep 24 at 1:02
• Many thanks, yes, the examples I meant have crepant resolutions Sep 24 at 12:28

• The rationality of singularity condition is $H^{>0}(S, \oplus_{k \ge 0} L^k) = 0$, where $L$ is the ample line bundle defining the embedding. In particular, $H^{>0}(S,\mathcal{O}_S) = 0$ is necessary, so K3 surface won't work. But some surface of general type, such as fake projective planes, do work. Sep 23 at 18:32