Michel Van Den Bergh
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Michel Van Den Bergh has a total of 108 co-authors. Here are the top co-authors by number of shared publications:
- Goncalo Tabuada
MIT, Dept Math, Cambridge, MA 02139 USA
11 shared publications - Theo Raedschelders
Departement Wiskunde, Vrije Universiteit Brussel, Ixelles, Belgium
8 shared publications - Damien Calaque
IMAG, Univ Montpellier, CNRS, Montpellier, France
7 shared publications - Graham J. Leuschke
Syracuse University
6 shared publications Špela Špenko
Université Libre de Bruxelles, Belgium
6 shared publications- Špela Špenko
Departement Wiskunde, Vrije Universiteit Brussel, Ixelles, Belgium
6 shared publications
Latest Publications
- Open Access
HMS symmetries of toric boundary divisors
Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system.Open AccessHMS symmetries of toric boundary divisors
Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system. - 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (3) , 1433-1458
Catalan numbers and noncommutative Hilbert schemes
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2024
- Published by International Press of Boston in Pure and Applied Mathematics Quarterly
- Vol. 20 (3) , 1433-1458
Catalan numbers and noncommutative Hilbert schemes
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- Open Access
On the GKZ discriminant locus
Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of the Laurent polynomials associated to the faces of $P$ that are hypersurfaces. In this short note we show that it is also the straightforward union of all the discriminant loci, i.e. we may include those of higher codimension, and there is no need to take closures. This answers a question by Kite and Segal.Open AccessOn the GKZ discriminant locus
Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of the Laurent polynomials associated to the faces of $P$ that are hypersurfaces. In this short note we show that it is also the straightforward union of all the discriminant loci, i.e. we may include those of higher codimension, and there is no need to take closures. This answers a question by Kite and Segal. - 23 October 2023
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2024 (4) , 3497-3550
Motivic Atiyah-Segal Completion Theorem
Let $T$ be a torus, $X$ a smooth separated scheme of finite type equipped with a $T$-action, and $[X/T]$ the associated quotient stack. Given any localizing ${\mathbb {A}}^{1}$-homotopy invariant of dg categories $E$ (homotopy $K$-theory, algebraic $K$-theory with coefficients, étale $K$-theory with coefficients, $l$-adic algebraic $K$-theory, $l$-adic étale $K$-theory, semi-topological $K$-theory, topological $K$-theory, periodic cyclic homology, etc), we prove that the derived completion of $E([X/T])$ at the augmentation ideal $I$ of the representation ring $R(T)$ of $T$ agrees with the classical Borel construction associated to the $T$-action on $X$. Moreover, for certain localizing ${\mathbb {A}}^{1}$-homotopy invariants, we extend this result to the case of a linearly reductive group scheme $G$. As a first application, we obtain an alternative proof of Krishna’s completion theorem in algebraic $K$-theory, of Thomason’s completion theorem in étale $K$-theory with coefficients, and also of Atiyah-Segal’s completion theorem in topological $K$-theory (for those topological $M$-spaces $X^{\textrm {an}}$ arising from analytification; $M$ is a(ny) maximal compact Lie subgroup of $G^{\textrm {an}}$). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason’s completion theorem the base field $k$ no longer needs to be separably closed, and in the case of Atiyah-Segal’s completion theorem the topological $M$-space $X^{\textrm {an}}$ no longer needs to be compact and the $M$-equivariant topological $K$-theory groups of $X^{\textrm {an}}$ no longer need to be finitely generated over the representation ring $R(M)$. As a second application, we obtain new completion theorems in $l$-adic étale $K$-theory, in semi-topological $K$-theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, $l$-adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy $K$-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological $K$-theory.- 23 October 2023
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2024 (4) , 3497-3550
Motivic Atiyah-Segal Completion Theorem
Let $T$ be a torus, $X$ a smooth separated scheme of finite type equipped with a $T$-action, and $[X/T]$ the associated quotient stack. Given any localizing ${\mathbb {A}}^{1}$-homotopy invariant of dg categories $E$ (homotopy $K$-theory, algebraic $K$-theory with coefficients, étale $K$-theory with coefficients, $l$-adic algebraic $K$-theory, $l$-adic étale $K$-theory, semi-topological $K$-theory, topological $K$-theory, periodic cyclic homology, etc), we prove that the derived completion of $E([X/T])$ at the augmentation ideal $I$ of the representation ring $R(T)$ of $T$ agrees with the classical Borel construction associated to the $T$-action on $X$. Moreover, for certain localizing ${\mathbb {A}}^{1}$-homotopy invariants, we extend this result to the case of a linearly reductive group scheme $G$. As a first application, we obtain an alternative proof of Krishna’s completion theorem in algebraic $K$-theory, of Thomason’s completion theorem in étale $K$-theory with coefficients, and also of Atiyah-Segal’s completion theorem in topological $K$-theory (for those topological $M$-spaces $X^{\textrm {an}}$ arising from analytification; $M$ is a(ny) maximal compact Lie subgroup of $G^{\textrm {an}}$). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason’s completion theorem the base field $k$ no longer needs to be separably closed, and in the case of Atiyah-Segal’s completion theorem the topological $M$-space $X^{\textrm {an}}$ no longer needs to be compact and the $M$-equivariant topological $K$-theory groups of $X^{\textrm {an}}$ no longer need to be finitely generated over the representation ring $R(M)$. As a second application, we obtain new completion theorems in $l$-adic étale $K$-theory, in semi-topological $K$-theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, $l$-adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy $K$-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological $K$-theory.- 10 October 2023
- Published by Elsevier in Journal of Algebra
- Vol. 639, 440-463
On cohomological and K-theoretical Hall algebras of symmetric quivers
- 10 October 2023
- Published by Elsevier in Journal of Algebra
- Vol. 639, 440-463
On cohomological and K-theoretical Hall algebras of symmetric quivers
- 22 July 2023
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2023 (801) , 1-43
Comparing the Kirwan and noncommutative resolutions of quotient varieties
Let a reductive group G act on a smooth variety X such that a good quotient exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of , obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks.- 22 July 2023
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2023 (801) , 1-43
Comparing the Kirwan and noncommutative resolutions of quotient varieties
Let a reductive group G act on a smooth variety X such that a good quotient exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of , obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of . In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne–Mumford stacks.- 8 March 2023
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 303 (4) , 1-29
The minimal model program for b-log canonical divisors and applications
- 8 March 2023
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 303 (4) , 1-29
The minimal model program for b-log canonical divisors and applications
- Open Access
T-structures on dg-categories and derived deformations
This paper is a sequel to "t-structures and twisted complexes on derived injectives" by the same authors. We develop the foundations of the infinitesimal derived deformation theory of pretriangulated dg-categories endowed with t-structures. This generalizes the deformation theory of abelian categories developed by the last two authors. We show how deformations of dg-categories of derived injectives yield derived deformations of the associated t-structures.
Open AccessT-structures on dg-categories and derived deformations
This paper is a sequel to "t-structures and twisted complexes on derived injectives" by the same authors. We develop the foundations of the infinitesimal derived deformation theory of pretriangulated dg-categories endowed with t-structures. This generalizes the deformation theory of abelian categories developed by the last two authors. We show how deformations of dg-categories of derived injectives yield derived deformations of the associated t-structures.
- Open Access
Motivic Atiyah-Segal completion theorem
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 categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic K-theory, of Thomason's completion theorem in \'etale K-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological K-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in l-adic \'etale K-theory, in (real) semi-topological K-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, l-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy K-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological K-theory.
Open AccessMotivic Atiyah-Segal completion theorem
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 categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic K-theory, of Thomason's completion theorem in \'etale K-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological K-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in l-adic \'etale K-theory, in (real) semi-topological K-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, l-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy K-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological K-theory.
- Open Access
- 1 June 2022
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 158 (6) , 1254-1267
New examples of non-Fourier–Mukai functors
A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension $\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes.- 1 June 2022
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 158 (6) , 1254-1267
Open AccessNew examples of non-Fourier–Mukai functors
A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension $\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes.