Do equivariant crepant resolutions always exist?

Let $X_t$ be a family of algebraic varieties (my interest is Calabi-Yau varieties, but I don't think that's important) over $\mathbb{C}$, smooth for $t \neq 0$, on which a group $G$ acts fibre-wise. Suppose further that $X_0$ admits at least one crepant resolution. Does there always exist an equivariant crepant resolution? If not, are there conditions under which such exists?

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1 Answer

Consider $\mathbb Z/2$ acting on $\{xy-zw=t\}$ by $x\leftrightarrow y$. This swaps the two small resolutions of the central fibre (the 3-fold ordinary double point $xy=zw$ in $\mathbb C^4$). So there can't be an equivariant small resolution.

(A formal proof might go along these lines: $\mathbb Z/2$ does act on the blow up of the ODP, swapping the two rulings of the $\mathbb P^1\times\mathbb P^1$ exceptional divisor. The small resolutions are contractions of this blow up. If $\mathbb Z/2$ acted on one of them, it would act on its $H^2$. Pulling back, its action on $H^2(\mathbb P^1\times\mathbb P^1)\cong\mathbb Z\oplus\mathbb Z$ would be the identity on the contracted $\mathbb Z$ summand, contradicting the fact that it swaps the summands.)

However if your (finite?) group action permutes the singular loci with no stabilisers then there would surely be a crepant resolution.

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By the way, this example doesn't preserve the holomorphic 3-form, so might not interest you. Can anyone think of an example where the group preserves the 3-form ? – Richard Thomas Apr 17 '12 at 21:39
Thanks Richard, that's a very clear counter-example. It's not a problem that the holomorphic 3-form is non-invariant; in fact, for the work which prompted this question, I'm interested in exactly such examples. – Rhys Davies Apr 18 '12 at 10:52