# Quiver GIT for varieties with tilting bundles

Quiver GIT for varieties with tilting bundles In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra $$A:=\mathrm{End}_{X}(T)^{\mathrm{op}}$$ A : = End X ( T ) op . We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for $$A=\mathrm{End}_{X}(T)^{\mathrm{op}}$$ A = End X ( T ) op then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# Quiver GIT for varieties with tilting bundles

, Volume 154 (2) – Jan 24, 2017
38 pages

/lp/springer_journal/quiver-git-for-varieties-with-tilting-bundles-40utlUrzbg
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-016-0914-3
Publisher site
See Article on Publisher Site

### Abstract

In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra $$A:=\mathrm{End}_{X}(T)^{\mathrm{op}}$$ A : = End X ( T ) op . We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for $$A=\mathrm{End}_{X}(T)^{\mathrm{op}}$$ A = End X ( T ) op then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities.

### Journal

Manuscripta MathematicaSpringer Journals

Published: Jan 24, 2017

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