manuscripta math. 154, 91–128 (2017) © The Author(s) 2017
Quiver GIT for varieties with tilting bundles
Received: 19 February 2016 / Accepted: 23 December 2016
Published online: 24 January 2017
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 := End
. 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 = End
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.
Any variety X equipped with a tilting bundle T induces a derived equivalence
between the bounded derived category of coherent sheaves on X and the bounded
derived category of ﬁnitely generated left modules for the algebra A := End
This situation is similar to the case of an afﬁne variety Spec(R) where we can con-
struct the commutative algebra R = End
and there is an abelian equiva-
lence between coherent sheaves on Spec(R) and ﬁnitely generated left R-modules.
However, whereas in the afﬁne case we can recover the variety Spec(R) from the
algebra R, it is not so clear how to recover the variety X from the algebra A.One
possibility is to present A as the path algebra of a quiver with relations, construct
a moduli space of quiver representations for some dimension vector and stability
condition, and attempt to relate this moduli space back to X.
J. Karmazyn (
): Mathematical Sciences, University of Shefﬁeld, Shefﬁeld S3 7RH, UK.
Mathematics Subject Classiﬁcation: 14D22 · 16S38 · 14F05 · 16G20