Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vector fields and admissible embeddings for quiver moduli

Pieter Belmans, Ana-Maria Brecan, Hans Franzen, Markus Reineke

TL;DR

The paper develops a double framing construction to study moduli spaces X of θ-(semi)stable quiver representations, reducing universal-bundle cohomology to line-bundle cohomology and enabling methods from geometric invariant theory. It proves that, under standing assumptions including θ-strong ample stability, the global endomorphisms of the universal representation U recover the whole path algebra via End_X(U) ≅ kQ, and that the space of vector fields H^0(X, T_X) is isomorphic to the Hochschild cohomology HH^1(kQ). It further shows that the universal representation yields a Fourier–Mukai–like functor that is fully faithful, providing an admissible embedding Db(kQ) → Db(X), with four related functors linked by adjunction and dualities. These results connect symmetries and deformation theory of quiver moduli to the algebraic structure of the path algebra, and extend admissible-embedding techniques beyond toric and Hilbert-scheme settings, highlighting a deep relation between moduli geometry and noncommutative invariants.

Abstract

We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.

Vector fields and admissible embeddings for quiver moduli

TL;DR

The paper develops a double framing construction to study moduli spaces X of θ-(semi)stable quiver representations, reducing universal-bundle cohomology to line-bundle cohomology and enabling methods from geometric invariant theory. It proves that, under standing assumptions including θ-strong ample stability, the global endomorphisms of the universal representation U recover the whole path algebra via End_X(U) ≅ kQ, and that the space of vector fields H^0(X, T_X) is isomorphic to the Hochschild cohomology HH^1(kQ). It further shows that the universal representation yields a Fourier–Mukai–like functor that is fully faithful, providing an admissible embedding Db(kQ) → Db(X), with four related functors linked by adjunction and dualities. These results connect symmetries and deformation theory of quiver moduli to the algebraic structure of the path algebra, and extend admissible-embedding techniques beyond toric and Hilbert-scheme settings, highlighting a deep relation between moduli geometry and noncommutative invariants.

Abstract

We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.
Paper Structure (18 sections, 22 theorems, 81 equations)

This paper contains 18 sections, 22 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Vector fields
  3. Admissible embeddings
  4. Acknowledgements
  5. Quiver moduli
  6. Construction of the moduli space
  7. Ample stability
  8. Standing assumptions
  9. Double framing construction
  10. On fiber products of projective bundles
  11. Double framing construction
  12. Global sections
  13. Global sections of the endomorphism bundle
  14. Global sections of the tangent bundle
  15. Functors associated to the universal representation
  16. ...and 3 more sections

Key Result

proposition 2.1

Let $Q$ be an acyclic quiver, $\mathbf{d}$ a dimension vector, and $\theta$ a stability parameter such that Then $\IfNoValueTF{\theta-\mathrm{st}}{\mathrm{M}{(Q,\mathbf{d})}}{\mathrm{M}^{\theta-\mathrm{st}}(Q,\mathbf{d})}$ is a smooth projective variety, which comes equipped with a universal bundle $\mathcal{U}(\mathbf{a})$ for every $\mathbf{a}\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{Z}^{Q_0},\

Theorems & Definitions (41)

  • proposition 2.1
  • definition 2.2
  • definition 2.3
  • proposition 3.1
  • lemma 3.2
  • proof
  • proof : Proof of \ref{['proposition:semistability-double-framed']}
  • remark 3.3
  • remark 3.4
  • corollary 3.5
  • ...and 31 more