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Partial semiorthogonal decompositions for quiver moduli

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Pith's one-line read This paper establishes that, under a numerical condition on Harder–Narasimhan strata, the derived category of a quiver moduli space admits a semiorthogonal decomposition with multiple twisted copies of the quiver's derived category…

desk verdict A genuinely useful framework with a concrete arithmetic error in the m-Kronecker case that needs fixing before the paper's headline example is credible. read the letter →

arxiv 2411.15125 v2 pith:R7GSI77H submitted 2024-11-22 math.AG

Gianni Petrella This is my paper · ORCID
classification math.AG MSC 14F0514D2016G2014J45
keywords semiorthogonaldecompositionquivermoduliderivedcategoriesTelemanquantizationexceptionalcollectionstiltingbundlesHarder–NarasimhanstratificationFanovarieties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the quiver–curve dictionary to derived categories: just as moduli spaces of vector bundles on curves contain embedded copies of the curve's derived category, the paper argues that moduli spaces of quiver representations should contain several embedded copies of the quiver's derived category, twisted by powers of the anticanonical line bundle. The main result is a sufficient numerical condition, Assumption 5.10, under which these copies form a semiorthogonal decomposition and the associated exceptional collection is strongly exceptional, hence gives a partial tilting bundle. The paper verifies positive answers to its Questions A, B and C in a range of examples, including m-Kronecker quivers with dimension vector (2,3), the six rigid del Pezzo surfaces, and a Fano 5-fold, and identifies cases such as P1 and P2 where the predicted decomposition cannot exist because the Hochschild homology is too small. A reader should care because the construction turns a geometric question about quiver moduli into explicit, machine-checkable inequalities, and because the resulting exceptional collections are concrete candidates for full tilting bundles.

What carries the argument

The load-bearing mechanism is the Fourier–Mukai functor $\Phi_U(V)=U\otimes^L_{kQ}V$ defined by the universal bundle $U$ on the moduli space, which is fully faithful and embeds $D^b(Q)$ into $D^b(M)$; twisting by line bundles $O(sH)$ gives the $r$ copies. Semiorthogonality of these copies is reduced by Lemma 3.2 to vanishing of the cohomology groups (24)–(27), and those vanishings are proved with Teleman quantization: a weight inequality (36) on every Harder–Narasimhan stratum of the representation space implies that higher cohomology of the descended bundle vanishes. The numerical condition Assumption 5.10 packages exactly the weight inequalities needed for all twists at once, and the proof closes the remaining vanishings by Serre duality (Lemma 3.6), with Chow-ring presentations (Theorem 5.1) supplying the Euler characteristic computations.

What would settle it

Recompute, with independent code, the quantities $t_{d^*}$ in inequality (74) for the $m$-Kronecker quiver with $d=(2,3)$ (say $m=4$) and for the Fano 5-fold of Example 5.7. If any Harder–Narasimhan stratum gives $\min t_{d^*} < r-1$, Theorem D does not apply to that example; if all strata give $\min t_{d^*} = r-1$ but an independent computation finds $H^k(M,U_i^\vee\otimes U_j\otimes O(-sH))\ne 0$ for some $1\le s\le r-1$, then Assumption 5.10 would not be sufficient after all.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem D: let $Q$ be an acyclic quiver, $d$ a dimension vector and $\theta_{\mathrm{can}}$ the canonical stability parameter satisfying Assumption 2.1 ($\theta$-coprimality plus strong ample stability). If Assumption 5.10 holds—the minimum over Harder–Narasimhan strata of the Teleman index $t_{d^*}$ is exactly $r-1$, where $r$ is the Fano index of the moduli space—then the bounded derived category $D^b(M)$ of the quiver moduli space admits a semiorthogonal decomposition in which the images of the twisted Fourier–Mukai functors $\Phi_U(sH)$ for $s=0,\ldots,r-1$ are mutually left-orthogonal copies of $D^b(Q)$. The same assumption makes the exceptional collection $U_1,\ldots,U_n,U_1(H),\ldots,U_n((r-1)H)$ strongly exceptional, so the direct sum of its members is a partial tilting bundle. The paper also establishes positive answers to Questions A and C in the del Pezzo cases, where the collection is actually full and tilting, and to Questions A–C for the $m$-Kronecker quiver with dimension vector $(2,3)$ and for a specific Fano 5-fold.

Load-bearing premise

Everything rests on one numerical condition: for the chosen quiver, a certain index computed from the stratification of unstable representations must come out exactly $r-1$, and where the paper checks this by computer rather than by proof, the computer's calculations must be correct.

Editorial extensions

If this is right

  • For every quiver, dimension vector and canonical stability parameter satisfying Assumptions 2.1 and 5.10, the moduli space carries $r\cdot n$ mutually orthogonal exceptional objects $U_i(sH)$ for $i=1,\ldots,n$ and $s=0,\ldots,r-1$, giving a partial tilting bundle with $r\cdot n$ summands.
  • In the del Pezzo cases, Question A holds and the collection $\mathcal{O},U_1,\ldots,U_n$ is full, so $\mathcal{O}\oplus U$ is a tilting bundle; this gives a new proof of Schofield's completion conjecture for the six rigid del Pezzo surfaces.
  • For the $m$-Kronecker quiver with dimension vector $(2,3)$ and any $m\ge 3$, the collection $U_1,U_2,U_1(H),U_2(H),\ldots,U_1((m-1)H),U_2((m-1)H)$ is strongly exceptional, and it is enlarged by the line bundles $O(sH)$ when $H^0(M,U_i^\vee)=0$ (verified for $m$ up to 11).
  • The non-examples show the boundary: when $\dim HH^0(M)$ is too small, as for $\mathbb{P}^1$, $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$ and a Fano threefold 2-35, one of Questions A or B fails exactly as predicted by Hochschild homology additivity.
  • Theorem D is independent of the choice of linearisation defining the universal bundles, so the resulting decomposition is canonical up to the usual ambiguity in $U$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step would be to search for infinite families beyond the $m$-Kronecker $(2,3)$ case where Assumption 5.10 can be proved in closed form; the weight asymptotics in Proposition 5.2 suggest that the condition should hold for all sufficiently large $m$ in many families.
  • The decomposition has the shape of a Lefschetz decomposition, so completing it to a full exceptional collection would mean finding the right-orthogonal complement; the paper's Hochschild homology computations already give the exact number of missing objects, giving the search a concrete target.
  • A cheap check on the computational component would be to rerun the verification for the Fano 5-fold and for $m$-Kronecker $(2,3)$ with an independently written implementation of the weight inequalities; agreement would rule out a scripting error as the source of the positive answers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies moduli spaces of stable representations of an acyclic quiver Q under the canonical stability parameter. Using a Fourier–Mukai functor built from the universal bundle, it embeds several copies of the derived category D^b(Q) into D^b(M), twisted by powers of the anticanonical line bundle O(H). The central result, Theorem D, gives a sufficient numerical condition, Assumption 5.10, for a partial semiorthogonal decomposition of Lefschetz type and for the associated collection to be strongly exceptional. The paper also gives computational evidence for positive answers to Questions A–C in several examples: m-Kronecker quivers with dimension vector (2,3), the six rigid del Pezzo surfaces that occur as quiver moduli, and a Fano 5-fold, together with some negative examples.

Significance. If the proofs are completed, Theorem D provides a uniform and checkable criterion in the quiver-moduli setting that mirrors the curve case, and the examples give concrete new instances of partial semiorthogonal decompositions and strong exceptional collections. The paper makes good use of existing tools (Teleman quantization, Serre duality, Chow-ring computations) and ships an open-source package, which is a genuine strength. However, one of the headline computational claims contains a concrete arithmetic error, and several other results rely on the output of a script whose version is not pinned; these issues must be fixed before the paper can be accepted.

major comments (2)
  1. [§5.1, Proposition 5.2 and Remark 1.10] The proof of Proposition 5.2 states that the Teleman inequalities (48)–(51) are already satisfied when m=3. This is false for the Harder–Narasimhan type d*=((1,1),(1,2)) and s=2. With θ_can=(9,-6) and c=2 one has k_1=3 and k_2=-2, so the maximum λ-weight of U_i^∨⊗U_j is k_1-k_2=5. The weight of L(-2H) is (2/3)θ_can·(3d_1-2d_2)=10, while η_d*=(k_2-k_1)⟨d_1,d_2⟩=15. Thus the required strict inequality in (48) is 15<15, which fails. Corollary 4.4 therefore cannot be applied to this term, and the Serre-duality step in the proof does not repair the gap: it addresses H^0 by passing to the s=1 case, but it does not supply the missing strict inequality for s=2 that would justify vanishing of the higher cohomology groups. Consequently the proof of Proposition 5.2 is incomplete for m=3. The same computation gives t_d*=1 for this type when m=3, while r-1=2, so the claim in Remark 1.10 that Assumption 5.10 can be verified directly for every m≥3 is incorrect. The statement may be salvageable for m≥4 or by an additional argument, but as written the proof has a load-bearing gap.
  2. [§5, Propositions 5.3 and 5.5, Example 5.7, and the m-Kronecker verification] Several load-bearing vanishings are asserted directly from the output of verification.jl [27], but the manuscript does not give a commit hash or checksum for that repository, nor does it specify the exact version of QuiverTools [7] used. Since the arithmetic inconsistency in Table 2 was not caught by the written text, the script output cannot currently be independently checked from the manuscript. Please provide a pinned commit, a precise description of which inequalities the script checks and how the H^0 statements are derived (including the role of Serre duality), and, if feasible, the script output or a log for the examples in Propositions 5.3, 5.5, and Example 5.7.
minor comments (4)
  1. [§4, Corollary 4.6] The index set in equation (39) should be 1≤m,n≤ℓ rather than 0≤m,n≤ℓ, since k_0 is not defined in Definition 4.1.
  2. [§5.1, Table 2] The normalization of Table 2 is not explained. The column headers write “1/m·W”, but the entries appear to use the unnormalized slopes μ_d* rather than the integer weights k_s of Definition 4.1; this makes the table hard to reconcile with the surrounding formulas and is directly related to the arithmetic issue in Proposition 5.2. Please spell out the normalization used.
  3. [§5.1, Proposition 5.2] In the first sentence of Proposition 5.2, the moduli space is denoted M_θ-st(Q,d), but the stability parameter just introduced is θ_can; the subscript should be θ_can-st.
  4. [§5.1, paragraph after Proposition 5.2] The sentence “The second condition ... has been verified experimentally for values of m up to 11” refers to H^0(M,U_i^∨)=0. Since Proposition 5.2's larger collection (47) is conditional on this vanishing, please state explicitly that (47) is proved only under the additional assumption, and that the numerical verification is not a proof for all m.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem D is a conditional sufficient criterion, and the worked examples are computational checks, not fitted predictions.

full rationale

The claimed derivation chain is not circular. The fully faithful embedding functor ΦU (Lemma 3.1) is taken from [5], which does not share authors with the present paper, and the Teleman-quantization formalism is ultimately external (Teleman; Halpern-Leistner), with [4] providing an adaptation. Theorem D is explicitly conditional: Assumption 5.10 is a concrete numerical inequality (min_{d*} t_{d*} = r-1, with t_{d*} defined by (74)), and the proof of Theorem 5.11 derives the cohomological vanishings (24) from it through Corollary 4.4 and Serre duality. The assumption is not defined in terms of the semiorthogonal decomposition, nor is the decomposition fitted from data; it is a sufficient-condition theorem. The examples are verified by explicit Chow-ring/Euler-characteristic computations and by the script verification.jl [27], whose output supplies numerical evidence; this is code- or computation-based checking, not a fitted parameter relabelled as a prediction. Self-citation of [4] occurs (Propositions 4.5, 4.9, Theorem 4.3, and [4, Proposition 4.2]), but these prior results have assumptions and content different from the paper's SOD claims, so they are independent support rather than a circular justification. Any arithmetic error in the m=3 Kronecker verification would be a correctness defect, not circularity.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central results rest on standard machinery (Teleman quantization, Chow ring presentations, Serre duality), on the author group's prior theorems [4,5], and on a numerical hypothesis Assumption 5.10 that is verified computationally rather than proved. The manually chosen linearisation vectors are the only hand-picked inputs in the examples. No new geometric entities are introduced.

free parameters (1)
  • linearisation vector a = a=(2,-1) for m-Kronecker; a=(3,-2) for Example 2.5; a=(0,2,-1) for Example 5.7; a=(1,1,-1) for Example 5.9
    The universal bundle summands U_i depend on the chosen linearisation, and the examples' semiorthogonality checks are run with these explicit choices, as noted in Remarks 2.4 and 3.5.
assumptions (8)
  • domain assumption Assumption 2.1: acyclic quiver, theta-coprimality, and theta-strong ample stability
    Used throughout to guarantee that the moduli space is smooth projective with a universal bundle and unstable locus of codimension at least 2; stated in Section 2.
  • ad hoc to paper Assumption 5.10: min_{d*} t_{d*} = r-1
    The technical Teleman criterion on which Theorem D depends; it is verified in examples rather than proved generally, see Section 5.4.
  • standard math Teleman quantization, Theorem 4.3
    Converts weight inequalities on Harder-Narasimhan strata into cohomology vanishings; adapted from Teleman and Halpern-Leistner via [4].
  • standard math Full faithfulness of the Fourier-Mukai functor Phi_U, Lemma 3.1
    Taken from [5, Theorem D]; this is the foundation for embedding copies of Db(Q) into Db(M).
  • standard math Chow ring presentation of quiver moduli, Theorem 5.1
    Uses presentations from Belmans-Franzen and Franzen to compute Euler characteristics via Hirzebruch-Riemann-Roch in Section 5.
  • standard math Serre duality, Lemma 3.6
    Used repeatedly to convert vanishings for negative twists into vanishings for positive twists.
  • standard math Maximal-length exceptional collections on del Pezzo surfaces are full
    Invoked in Proposition 5.5 via [28, Theorem 6.35] to conclude fullness on del Pezzo quiver moduli.
  • standard math Additivity of Hochschild homology and HKR
    Used in Remark 1.4 and the non-examples to predict negative answers when dim HH0 is too small.

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Pith. "Pith review of Partial semiorthogonal decompositions for quiver moduli." pith.science (2026). https://pith.science/paper/R7GSI77H

@misc{pith2026241115125,
  author       = {Pith},
  title        = {Pith review of: Partial semiorthogonal decompositions for quiver moduli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7GSI77H}},
  note         = {Machine review of arXiv:2411.15125}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2411.15125 by the authors.

Figure 1
Figure 1. From left to right, the quivers 𝑄1 to 𝑄6 realising the del Pezzo surfaces. The corresponding surface to each quiver is described in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

33 extracted references · 16 canonical work pages

  1. [27]

    partial-sod-quiver-moduli

    Gianni Petrella. partial-sod-quiver-moduli. 2024. url: https://github.com/giannipetrella/partial- sod-quiver-moduli

    work page 2024

  2. [7]

    Pieter Belmans, Hans Franzen, and Gianni Petrella.QuiverTools. 2024. doi: 10.5281/zenodo.1268022. url: https://quiver.tools

    work page doi:10.5281/zenodo.1268022 2024

  3. [1]

    Fanography

    Pieter Belmans. Fanography. 2024. url: https://fanography.info

    work page 2024

  4. [2]

    Hodge diamond cutter

    Pieter Belmans. Hodge diamond cutter. doi: 10.5281/zenodo.3893509. url: https://github.com/ pbelmans/hodge-diamond-cutter

    work page doi:10.5281/zenodo.3893509

  5. [3]

    Semiorthogonal decompositions for moduli of sheaves on curves

    Pieter Belmans. Semiorthogonal decompositions for moduli of sheaves on curves. Interactions between Algebraic Geometry and Noncommutative Algebra, Mathematisches Forschungsinstitut Oberwolfach (2018). 2018

    work page 2018

  6. [4]

    Rigidity and Schofield’s partial tilting conjecture for quiver moduli

    Pieter Belmans, Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke. “Rigidity and Schofield’s partial tilting conjecture for quiver moduli”. In: (2023). arXiv: 2311.17003 [math.AG]

    arXiv 2023

  7. [5]

    Vector fields and admissible embeddings for quiver moduli

    Pieter Belmans, Ana-Maria Brecan, Hans Franzen, and Markus Reineke. “Vector fields and admissible embeddings for quiver moduli”. In: (2023). arXiv: 2311.17004 [math.AG]. 15

    work page Pith review arXiv 2023

  8. [6]

    On Chow rings of quiver moduli

    Pieter Belmans and Hans Franzen. “On Chow rings of quiver moduli”. In: Int. Math. Res. Not. IMRN 13 (2024), pp. 10255–10272. doi: 10.1093/imrn/rnad306

    doi:10.1093/imrn/rnad306 2024

Show all 33 references
  1. [8]

    Decompositions of moduli spaces of vector bundles and graph potentials

    Pieter Belmans, Sergey Galkin, and Swarnava Mukhopadhyay. “Decompositions of moduli spaces of vector bundles and graph potentials”. In: Forum Math. Sigma 11 (2023), Paper No. e16, 28. doi: 10.1017/fms.2023.14

    2023 doi

  2. [9]

    Admissible subcategories in derived categories of moduli of vector bundles on curves

    Pieter Belmans and Swarnava Mukhopadhyay. “Admissible subcategories in derived categories of moduli of vector bundles on curves”. In: Adv. Math. 351 (2019), pp. 653–675. doi: 10.1016/j.aim. 2019.05.019

    2019 doi

  3. [10]

    Exceptional sequences of representations of quivers

    William Crawley-Boevey. “Exceptional sequences of representations of quivers”. In: Representations of algebras (Ottawa, ON, 1992). Vol. 14. CMS Conf. Proc. Amer. Math. Soc., Providence, RI, 1993, pp. 117– 124

    1992

  4. [11]

    Derived categories of curves as components of Fano manifolds

    Anton Fonarev and Alexander Kuznetsov. “Derived categories of curves as components of Fano manifolds”. In: J. Lond. Math. Soc. (2) 97.1 (2018), pp. 24–46. doi: 10.1112/jlms.12094

    2018 doi

  5. [12]

    Chow rings of fine quiver moduli are tautologically presented

    Hans Franzen. “Chow rings of fine quiver moduli are tautologically presented”. In: Math. Z. 279.3-4 (2015), pp. 1197–1223. doi: 10.1007/s00209-015-1409-8

    2015 doi

  6. [13]

    Fano quiver moduli

    Hans Franzen, Markus Reineke, and Silvia Sabatini. “Fano quiver moduli”. In: Canad. Math. Bull. 64.4 (2021), pp. 984–1000. doi: 10.4153/S0008439520001009

    2021 doi

  7. [14]

    Motivic decompositions of moduli spaces of vector bundles on curves

    Tom´as L. G´omez and Kyoung-Seog Lee. “Motivic decompositions of moduli spaces of vector bundles on curves”. In: (2020). arXiv: 2007.06067 [math.AG]

    2020 arXiv

  8. [15]

    The derived category of a GIT quotient

    Daniel Halpern-Leistner. “The derived category of a GIT quotient”. In: J. Amer. Math. Soc. 28.3 (2015), pp. 871–912. doi: 10.1090/S0894-0347-2014-00815-8

    2015 doi

  9. [16]

    Tilting line bundles and moduli of thin sincere representations of quivers

    Lutz Hille. “Tilting line bundles and moduli of thin sincere representations of quivers”. In: vol. 4. 2. Representation theory of groups, algebras, and orders (Constant ¸a, 1995). 1996, pp. 76–82

    1995

  10. [17]

    Parallels between moduli of quiver representations and vector bundles over curves

    Victoria Hoskins. “Parallels between moduli of quiver representations and vector bundles over curves”. In: SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 127, 46. doi: 10.3842/ SIGMA.2018.127

    2018

  11. [18]

    Tilting bundles on some rational surfaces

    Alastair King. Tilting bundles on some rational surfaces. Unpublished note. 1997. url: https://people. bath.ac.uk/masadk/papers/tilt.pdf

    1997

  12. [19]

    Hochschild homology and semiorthogonal decompositions

    Alexander Kuznetsov. “Hochschild homology and semiorthogonal decompositions”. In: (2009). arXiv: 0904.4330 [math.AG]

    2009 arXiv

  13. [20]

    Semiorthogonal decompositions in algebraic geometry

    Alexander Kuznetsov. “Semiorthogonal decompositions in algebraic geometry”. In: Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II. Kyung Moon Sa, Seoul, 2014, pp. 635–660

    2014

  14. [21]

    Remarks on motives of moduli spaces of rank 2 vector bundles on curves

    Kyoung-Seog Lee. “Remarks on motives of moduli spaces of rank 2 vector bundles on curves”. In: (2018). arXiv: 1806.11101 [math.AG]

    2018 arXiv

  15. [22]

    Derived category and ACM bundles of moduli space of vector bundles on a curve

    Kyoung-Seog Lee and Han-Bom Moon. “Derived category and ACM bundles of moduli space of vector bundles on a curve”. In: Forum Math. Sigma 11 (2023), Paper No. e81, 23. doi: 10.1017/fms.2023.75

    2023 doi

  16. [23]

    Derived categories of moduli spaces of vector bundles on curves

    M. S. Narasimhan. “Derived categories of moduli spaces of vector bundles on curves”. In: J. Geom. Phys. 122 (2017), pp. 53–58. doi: 10.1016/j.geomphys.2017.01.018

    2017 doi

  17. [24]

    Derived categories of moduli spaces of vector bundles on curves II

    M. S. Narasimhan. “Derived categories of moduli spaces of vector bundles on curves II”. In: Geometry, algebra, number theory, and their information technology applications. Vol. 251. Springer Proc. Math. Stat. Springer, Cham, 2018, pp. 375–382. doi: 10.1007/978-3-319-97379-1 16

    2018 doi

  18. [25]

    Stable bundles of rank 2 and odd degree over a curve of genus 2

    P. E. Newstead. “Stable bundles of rank 2 and odd degree over a curve of genus 2”. In: Topology 7 (1968), pp. 205–215. doi: 10.1016/0040-9383(68)90001-3

    1968 doi

  19. [26]

    Semi-orthogonal decomposability of the derived category of a curve

    Shinnosuke Okawa. “Semi-orthogonal decomposability of the derived category of a curve”. In: Adv. Math. 228.5 (2011), pp. 2869–2873. doi: 10.1016/j.aim.2011.06.043

    2011 doi

  20. [28]

    Admissible subcategories of del Pezzo surfaces

    Dmitrii Pirozhkov. “Admissible subcategories of del Pezzo surfaces”. In: Adv. Math. 424 (2023), Paper No. 109046, 62. doi: 10.1016/j.aim.2023.109046

    2023

  21. [29]

    Moduli of representations of quivers

    Markus Reineke. “Moduli of representations of quivers”. In: Trends in representation theory of algebras and related topics. EMS Ser. Congr. Rep. Eur. Math. Soc., Z¨urich, 2008, pp. 589–637. doi: 10.4171/062- 1/14

    2008 doi

  22. [30]

    The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

    Markus Reineke. “The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli”. In: Invent. Math. 152.2 (2003), pp. 349–368. doi: 10.1007/s00222-002-0273-4 . 16

    2003 doi

  23. [31]

    Brauer groups for quiver moduli

    Markus Reineke and Stefan Schr¨oer. “Brauer groups for quiver moduli”. In: Algebr. Geom. 4.4 (2017), pp. 452–471. doi: 10.14231/2017-024

    2017 doi

  24. [32]

    The quantization conjecture revisited

    Constantin Teleman. “The quantization conjecture revisited”. In:Ann. of Math. (2)152.1 (2000), pp. 1–43. doi: 10.2307/2661378

    2000 doi

  25. [33]

    The BGMN conjecture via stable pairs

    Jenia Tevelev and Sebasti´an Torres. “The BGMN conjecture via stable pairs”. In: Duke Math. J. 173.18 (2024), pp. 3495–3557. doi: 10.1215/00127094-2024-0016. Gianni Petrella gianni.petrella@uni.lu Universit´e de Luxembourg, 6, Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg 17

    2024 doi

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