Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 044, 22 pages      arXiv:1812.10688      https://doi.org/10.3842/SIGMA.2021.044
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

Wahei Hara
The Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow, G12 8QQ, UK

Received September 30, 2020, in final form April 18, 2021; Published online April 30, 2021

Abstract
The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

Key words: derived category; non-commutative crepant resolution; flop; tilting bundle.

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