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Partial semiorthogonal decompositions for quiver moduli

Gianni Petrella

TL;DR

The paper develops a framework to realize partial semiorthogonal decompositions for derived categories of quiver moduli by embedding multiple copies of $\mathbf{D}^{\mathrm{b}}(Q)$ into $\mathbf{D}^{\mathrm{b}}(M^{\\theta-\mathrm{st}}(Q,\\mathbf{d}))$ through the universal bundle $\\mathcal{U}$ and twists by multiples of $H=-\frac{1}{r}K_M$. It introduces a Teleman quantization-based criterion to control higher Ext-vanishings and obtain semiorthogonal blocks, and it verifies positive decompositions in key examples such as $m$-Kronecker quivers with $(2,3)$ and rigid del Pezzo surfaces realized as quiver moduli, while also presenting non-examples explained by Hochschild homology. A weaker, linearisation-independent version of the main result is provided under a $t$-large condition, yielding a strongly exceptional decomposition built from the twisted universal bundles, with proofs aided by Chow-ring computations and Serre duality. Together, these results extend the curve-case SOD picture to quiver moduli, enabling explicit tilting objects and exceptional collections for new Fano varieties and offering computational tools via QuiverTools. The work highlights where full decompositions fail and how a Lefschetz-type structure emerges in the quiver setting, contributing a concrete program for systematic SODs in quiver moduli.

Abstract

We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with QuiverTools, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.

Partial semiorthogonal decompositions for quiver moduli

TL;DR

The paper develops a framework to realize partial semiorthogonal decompositions for derived categories of quiver moduli by embedding multiple copies of into through the universal bundle and twists by multiples of . It introduces a Teleman quantization-based criterion to control higher Ext-vanishings and obtain semiorthogonal blocks, and it verifies positive decompositions in key examples such as -Kronecker quivers with and rigid del Pezzo surfaces realized as quiver moduli, while also presenting non-examples explained by Hochschild homology. A weaker, linearisation-independent version of the main result is provided under a -large condition, yielding a strongly exceptional decomposition built from the twisted universal bundles, with proofs aided by Chow-ring computations and Serre duality. Together, these results extend the curve-case SOD picture to quiver moduli, enabling explicit tilting objects and exceptional collections for new Fano varieties and offering computational tools via QuiverTools. The work highlights where full decompositions fail and how a Lefschetz-type structure emerges in the quiver setting, contributing a concrete program for systematic SODs in quiver moduli.

Abstract

We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with QuiverTools, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.

Paper Structure

This paper contains 14 sections, 19 theorems, 53 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The case of curves
  3. The case of quivers
  4. A conjecture of Schofield
  5. Funding
  6. Moduli spaces of quiver representations
  7. Semiorthogonal embeddings
  8. Stratification of the unstable locus and Teleman quantization
  9. Computations and proofs of our claims
  10. Note
  11. m-Kronecker quivers
  12. del Pezzo surfaces
  13. Non-examples
  14. The proof of Theorem D

Key Result

theorem 1.7

Let $Q, \mathbf{d}$ and $\theta$ satisfy assumptions. The universal representation $\mathcal{U}$ on $\mathop{\mathrm{M}}\nolimits^{\theta-\mathrm{st}}(Q,\mathbf{d})$ is a partial tilting bundle, i.e.,

Figures (1)

  • Figure 1: From left to right, the quivers $Q_{1}$ to $Q_{6}$ realising the del Pezzo surfaces. The corresponding surface to each quiver is described in \ref{['tab:del-pezzos']}.

Theorems & Definitions (50)

  • remark 1.1
  • remark 1.2
  • remark 1.3
  • remark 1.4
  • remark 1.5
  • definition 1.6
  • theorem 1.7: Corollary C, rigidity-paper
  • conjecture 1.8
  • remark 1.9
  • remark 1.10
  • ...and 40 more