# Does existance of crepant resolution of tangent space imply existance of crepant resolution globally in the algebraic setting?

Suppose $$X$$ is smooth proper algebraic $$\mathbb C$$-variety with algebraic action of a finite abelian group $$G$$. Suppose I know that

1. $$X/G$$ (good geometric quotient) exists and it is normal Gorenstein Calabi-Yau variety (Calabi-Yau means here that dualizing sheaf is trivial).
2. There is exactly one $$\mathbb C$$-point $$p \in X$$ such that stabilizer $$G_p \subset G$$ is non-trivial ($$\neq \{e\}$$). Moreover, $$G_p = G$$, so $$p$$ is a fixed point.
3. Crepant resolution of $$(T_p X)/G_p$$ exists.

Can I deduce that crepant resolution of $$X/G$$ also exists?

The reason why I am asking is that in an analytical setting the question of crepant resolutions of general Calabi-Yau manifolds could be reduced to the question of crepant resolutions of $$\mathbb C^n/G$$. For example, in Crepant Resolutions of Calabi-Yau Orbifolds author says

A Calabi-Yau manifold is a complex Kahler manifold with trivial canonical bundle. ... An orbifold is the quotient of a smooth Calabi-Yau manifold by a discrete group action that generically has fixed points. Locally such an orbifold is modeled on $$\mathbb C^n/G$$... A resolution $$(X, \pi)$$ of $$\mathbb C^n/G$$ is a nonsingular complex manifold X of dimension n with a proper biholomorphic map $$\pi : X → \mathbb C^n/G$$ that induces a biholomorphism between dense open sets.

Сan something similar take place in the algebraic setting as well?

• It exists in the category of algebraic spaces, but not necessarily as an algebraic scheme. Nov 1, 2021 at 9:37
• @JasonStarr Thank you. Could you please provide some references to the fact you mentioned? I am not so familiar with algebraic spaces. Nov 1, 2021 at 10:02
• One reference is Michael Artin, "Algebraization of Formal Moduli, II". Here Artin proves that the category of compact, complex algebraic spaces is equivalent to the category of what are now called compact Moishezon spaces: this was independently proved by Boris Moishezon in the smooth case. So constructions, such as crepant resolutions, that work in the category of Moishezon spaces also work in the category of complex algebraic spaces, by the work of Michael Artin and Boris Moishezon. Nov 1, 2021 at 15:14
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