102 citations · 388 across the 40 of their papers we have counts for
26 papers · 1 filter
Groups generated by spherical twists on K3 surfaces and full exceptional collections on Fano threefolds
Anya Nordskova, Michel Van den Bergh
Let Y be a smooth K3 surface of Picard rank 1. We prove that the subgroup G of Aut D^b(Y) generated by spherical twists with respect to all spherical objects is free. Moreover, we…
HMS symmetries of toric boundary divisors
Špela Špenko, Michel Van den Bergh
Let be a projective crepant resolution of a Gorenstein affine toric variety and let be the LG-model which is the Hori-Vafa mirror dual of . Let …
Non-commutative crepant resolutions, an overview
Michel Van den Bergh
Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of the usual crepant resolutions that appear in algebraic geometry. In this paper we survey some results a…
On the GKZ discriminant locus
Špela Špenko, Michel Van den Bergh
Let be an integral matrix and let be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal -determinant locus is equal…
Motivic Atiyah-Segal completion theorem
Gonçalo Tabuada, Michel Van den Bergh
Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg cate…
Perverse schobers and GKZ systems
Špela Špenko, Michel Van den Bergh
Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS)…
Comparing the Kirwan and noncommutative resolutions of quotient varieties
Špela Špenko, Michel Van den Bergh
Let a reductive group act on a smooth variety such that a good quotient exists. We show that the derived category of a noncommutative crepant resolution (NCCR)…
A k-linear triangulated category without a model
Alice Rizzardo, Michel Van den Bergh
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triang…
Non-commutative crepant resolutions for some toric singularities II
Špela Špenko, Michel Van den Bergh
Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give…
A non-commutative Bertini theorem
Jørgen Vold Rennemo, Ed Segal, Michel Van den Bergh
We prove a version of the classical 'generic smoothness' theorem with smooth varieties replaced by non-commutative resolutions of singular varieties. This in particular implies a n…
Non-commutative crepant resolutions for some toric singularities I
Špela Špenko, Michel Van den Bergh
We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensio…
A note on non-unique enhancements
Alice Rizzardo, Michel Van den Bergh
We give an easy example of a triangulated category, linear over a field , with two different enhancements, linear over , answering a question of Canonaco and Stellari.
Additive invariants of orbifolds
Goncalo Tabuada, Michel Van den Bergh
In this article, using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely "fixed-…
On an analogue of the Markov equation for exceptional collections of length 4
Louis de Thanhoffer de Volcsey, Michel Van den Bergh
We classify the solutions to a system of equations, introduced by Bondal, which encode numerical constraints on full exceptional collections of length 4 on surfaces. The correspond…
Semi-orthogonal decompositions of GIT quotient stacks
Špela Špenko, Michel Van den Bergh
If G is a reductive group which acts on a linearized smooth scheme then we show that under suitable standard conditions the derived category of coherent sheaves of the correspo…
Comparing the commutative and non-commutative resolutions for determinantal varieties of skew symmetric and symmetric matrices
Špela Špenko, Michel Van den Bergh
Let Y be the variety of (skew) symmetric nxn-matrices of rank less than or equal to r. In paper we construct a full faithful embedding between the derived category of a non-commuta…
The Gysin triangle via localization and A1-homotopy invariance
Goncalo Tabuada, Michel Van den Bergh
Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-homotopy invariant of dg categories E, we construct an associated Gysi…
Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups
Alexander Polishchuk, Michel Van den Bergh
We consider the derived category of coherent sheaves on a complex vector space equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl g…
Non-commutative resolutions of quotient singularities
Špela Špenko, Michel Van den Bergh
In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that…
Noncommutative motives of separable algebras
Goncalo Tabuada, Michel Van den Bergh
In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the fu…
Noncommutative versions of some classical birational transformations
Michel Van den Bergh, Dennis Presotto
In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commuta…
An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves
Alice Rizzardo, Michel Van den Bergh, Amnon Neeman
Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a…
Scalar extensions of derived categories and non-Fourier-Mukai functors
Alice Rizzardo, Michel Van den Bergh
Orlov's famous representability theorem asserts that any fully faithful functor between the derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai…
Noncommutative motives of Azumaya algebras
Goncalo Tabuada, Michel Van den Bergh
Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the c…