scholarly journals Stability conditions and related filtrations for (G,h)-constellations

2017 ◽  
Vol 28 (14) ◽  
pp. 1750098
Author(s):  
Ronan Terpereau ◽  
Alfonso Zamora
Keyword(s):  
Moduli Spaces ◽  
Derived Categories ◽  
Irreducible Representations ◽  
Quiver Representations ◽  
Geometric Invariant ◽  
Reductive Algebraic Group ◽  
Coherent Sheaves ◽  
Finite Dimensional ◽  
Git Quotient ◽  
Stability Notions

Given an infinite reductive algebraic group [Formula: see text], we consider [Formula: see text]-equivariant coherent sheaves with prescribed multiplicities, called [Formula: see text]-constellations, for which two stability notions arise. The first one is analogous to the [Formula: see text]-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.[Formula: see text]2) 45(180) (1994) 515–530] and for [Formula: see text]-constellations by Craw and Ishii [Flops of [Formula: see text]-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124(2) (2004) 259–307], but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for [Formula: see text]-constellations, and depends on some finite subset [Formula: see text] of the isomorphy classes of irreducible representations of [Formula: see text]. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of [Formula: see text]-constellations, Transform. Groups 20(2) (2015) 335–366]. Also, we construct Harder–Narasimhan filtrations for [Formula: see text]-constellations with respect to both stability notions (namely, the [Formula: see text]-HN and [Formula: see text]-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the [Formula: see text]-HN filtration is a subfiltration of the [Formula: see text]-HN filtration, and the polygons of the [Formula: see text]-HN filtrations converge to the polygon of the [Formula: see text]-HN filtration when [Formula: see text] grows.

scholarly journals Variation of geometric invariant theory quotients and derived categories

2019 ◽  
Vol 2019 (746) ◽  
pp. 235-303 ◽  
Author(s):  
Matthew Ballard ◽  
David Favero ◽  
Ludmil Katzarkov
Keyword(s):  
Moduli Spaces ◽  
Invariant Theory ◽  
General Framework ◽  
Derived Categories ◽  
Rational Curves ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Coherent Sheaves ◽  
Orthogonal Decompositions ◽  
The Relationship

Abstract We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov’s σ-model/Landau–Ginzburg model correspondence.

RELATIVE GEOMETRIC INVARIANT THEORY AND UNIVERSAL MODULI SPACES

1996 ◽  
Vol 07 (02) ◽  
pp. 151-181 ◽  
Author(s):  
YI HU
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Invariant Theory ◽  
Hilbert Scheme ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Coherent Sheaves ◽  
Ample Cone

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over [Formula: see text] of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over [Formula: see text] to a compactified universal Picard.

scholarly journals Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

2021 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Spaces ◽  
Blow Up ◽  
Derived Categories ◽  
Model Program ◽  
Minimal Model Program ◽  
Blow Down ◽  
Coherent Sheaves ◽  
Framed Sheaves ◽  
Wall Crossing Formula ◽  
Torsion Free

AbstractIn order to study the wall-crossing formula of Donaldson type invariants on the blown-up plane, Nakajima–Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima–Yoshioka’s diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

Piecewise Hereditary Triangular Matrix Algebras

2021 ◽  
Vol 28 (01) ◽  
pp. 143-154
Author(s):  
Yiyu Li ◽  
Ming Lu
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Derived Categories ◽  
Matrix Algebras ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Abelian Categories ◽  
Upper Triangular Matrix ◽  
Orbit Categories ◽  
Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

scholarly journals Perverse Schobers and Wall Crossing

2017 ◽  
Vol 2019 (18) ◽  
pp. 5777-5810 ◽  
Author(s):  
W Donovan
Keyword(s):  
Invariant Theory ◽  
Local System ◽  
Vector Spaces ◽  
Geometric Invariant ◽  
Perverse Sheaf ◽  
Derived Equivalences ◽  
Grothendieck Groups ◽  
Wall Crossing ◽  
Git Quotient ◽  
Cohomology Complex

Abstract For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

scholarly journals Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle

2019 ◽  
Vol 2019 (749) ◽  
pp. 87-132
Author(s):  
Laurent Meersseman
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Classical Case ◽  
Analytic Space ◽  
Fundamental Result ◽  
Compact Complex Manifold ◽  
Complex Structures ◽  
Analogous Statement ◽  
Finite Dimensional ◽  
Cr Structures

Abstract Kuranishi’s fundamental result (1962) associates to any compact complex manifold {X_{0}} a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to {X_{0}} . In this paper, we give an analogous statement for Levi-flat CR-manifolds fibering properly over the circle by associating to any such {\mathcal{X}_{0}} the loop space of a finite-dimensional analytic space which serves as a local moduli space of CR-structures close to {\mathcal{X}_{0}} . We then develop in this context a Kodaira–Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

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