Abstract
In most cases where it has been shown to exist the derived McKay correspondence \({D(Y) \xrightarrow{\sim} D^G(\mathbb{C}^n)}\) can be written as a Fourier–Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in \({D^G(\mathbb{C}^n)}\) . We give a sufficient condition for \({E \in D^G(Y \times \mathbb{C}^n)}\) to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of \({\mathbb{C}^3/G}\) . Along the way we extract more geometrical meaning out of the Intersection Theorem and learn to compute θ-stable families of G-constellations and their direct transforms.
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Logvinenko, T. Derived McKay correspondence via pure-sheaf transforms. Math. Ann. 341, 137–167 (2008). https://doi.org/10.1007/s00208-007-0186-z
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DOI: https://doi.org/10.1007/s00208-007-0186-z