$(0,2)$-deformations and the $G$-Hilbert scheme

We study first-order deformations of the tangent sheaf of resolutions of Calabi–Yau threefolds that are of the form $\mathbb{C}^3 / \mathbb{Z}_r$, focusing on the cases where the orbifold has an isolated singularity.We prove a lower bound on the number of deformations for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the $G$-Hilbert scheme, and note that this lower bound can be found using methods from string theory. These methods lead us to a new way to construct the $G$-Hilbert scheme using the singlet count.

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Published 17 January 2017