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Register a new userNon-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops
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Authors
Wahei Hara
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In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawas derived equivalence between crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ in terms of an NCCR. We also show that the P-twist on the derived category of $Y$ corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama-Wemysss mutations.
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Recently, Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuafs flop. Concretely, we study the equivalence for Abuafs flop by using Toda-Ueharas tilting bundles and Iyama-Wemysss mutation functors. In addition, we observe a flop-flop=twist result and a multi-mutation=twist result for Abuafs flop.
We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group. We apply our results to toric singularities and dimer models.
Associated to a Mukai flop X ---> X is on the one hand a sequence of equivalences D(X) -> D(X), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of D(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlovs derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.
In this paper, we study splitting (or toric) non-commutative crepant resolutions (= NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group $mathbb{Z}^2$ have a splitting NCCR. In the appendix, we also discuss Gorenstein toric rings with class group $mathbb{Z}$, in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange graph.
We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
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