Rigidity and Schofield's partial tilting conjecture for quiver moduli

[en] We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture, and to show that moduli spaces of quiver representations are (infinitesimally) rigid as varieties.

Disciplines :

Mathematics

Author, co-author :

BELMANS, Pieter ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Brecan, Ana-Maria

Franzen, Hans

PETRELLA, Gianni ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Reineke, Markus

External co-authors :

yes

Language :

English

Title :

Rigidity and Schofield's partial tilting conjecture for quiver moduli

Publication date :

2025

Journal title :

Journal de l'École Polytechnique. Mathématiques

ISSN :

2429-7100

eISSN :

2270-518X

Publisher :

Éditions de l'École Polytechnique, Palaiseau, France

Peer reviewed :

Peer Reviewed verified by ORBi

Commentary :

25 pages, comments welcome

Available on ORBilu :

since 01 December 2023

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Bibliography

J. Adriaenssens & L. Le Bruyn – “Local quivers and stable representations”, Comm. Algebra 31 (2003), no. 4, p. 1777–1797.
K. Altmann & L. Hille – “Strong exceptional sequences provided by quivers”, Algebras Represent. Theory 2 (1999), no. 1, p. 1–17.
T. Beckmann & P. Belmans – “Homological projective duality for the Segre cubic”, 2022, accepted for publication in Proceedings of Nottingham Algebraic Geometry Seminar, arXiv:2202.08601.
P. Belmans, A.-M. Brecan, H. Franzen & M. Reineke – “Vector fields and admissible embeddings for quiver moduli”, Moduli 2 (2025), article no. e3 (23 pages).
P. Belmans, C. Damiolini, H. Franzen, V. Hoskins, S. Makarova & T. Tajakka – “Projectivity and effective global generation of determinantal line bundles on quiver moduli”, 2022, accepted for publication in Algebra & Number Theory, arXiv:2210.00033v1.
P. Belmans & H. Franzen – “On Chow rings of quiver moduli”, Internat. Math. Res. Notices (2024), no. 13, p. 10255–10272.
P. Belmans, H. Franzen & G. Petrella – “Quivertools”, https://quiver.tools.
, “The QuiverTools package for SageMath and Julia”, 2025, arXiv:2506.19432v1.
P. Belmans, L. Fu & T. Raedschelders – “Hilbert squares: derived categories and deformations”, Selecta Math. (N.S.) 25 (2019), no. 3, article no. 37 (32 pages).
P. Belmans & S. Mukhopadhyay – “Admissible subcategories in derived categories of moduli of vector bundles on curves”, Adv. Math. 351 (2019), p. 653–675.
M. Brion – “On linearization of line bundles”, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, p. 113–147.
A. Craw – “Quiver flag varieties and multigraded linear series”, Duke Math. J. 156 (2011), no. 3, p. 469–500.
A. Craw, Y. Ito & J. Karmazyn – “Multigraded linear series and recollement”, Math. Z. 289 (2018), no. 1-2, p. 535–565.
M. Domokos – “Quiver moduli spaces of a given dimension”, J. Comb. Algebra. (2024), online first, arXiv:2303.08522v2.
J.-M. Drezet – “Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur P2(C)”, J. reine angew. Math. 380 (1987), p. 14–58.
, “Groupe de Picard des variétés de modules de faisceaux semi-stables sur P2”, in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lect. Notes in Math., vol. 1273, Springer, Berlin, 1987, p. 337–362.
, “Cohomologie des variétés de modules de hauteur nulle”, Math. Ann. 281 (1988), no. 1, p. 43–85.
A. Fonarev & A. Kuznetsov – “Derived categories of curves as components of Fano manifolds”, J. London Math. Soc. (2) 97 (2018), no. 1, p. 24–46.
H. Franzen & M. Reineke – “Cohomology rings of moduli of point configurations on the projective line”, Proc. Amer. Math. Soc. 146 (2018), no. 6, p. 2327–2341.
H. Franzen, M. Reineke & S. Sabatini – “Fano quiver moduli”, Canad. Math. Bull. 64 (2021), no. 4, p. 984–1000.
D. Halpern-Leistner – “The derived category of a GIT quotient”, J. Amer. Math. Soc. 28 (2015), no. 3, p. 871–912.
D. Happel – “Hochschild cohomology of finite-dimensional algebras”, in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin (Paris, 1987/1988), Lect. Notes in Math., vol. 1404, Springer, Berlin, 1989, p. 108–126.
W. H. Hesselink – “Uniform instability in reductive groups”, J. reine angew. Math. (1978), p. 74–96.
L. Hille – “Tilting line bundles and moduli of thin sincere representations of quivers”, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 4 (1996), no. 2, p. 76–82, Representation theory of groups, algebras, and orders (Constanţa, 1995).
L. Hille & J. A. de la Peña – “Stable representations of quivers”, J. Pure Appl. Algebra 172 (2002), no. 2-3, p. 205–224.
V. Hoskins – “Parallels between moduli of quiver representations and vector bundles over curves”, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), article no. 127 (46 pages).
, “Stratifications for moduli of sheaves and moduli of quiver representations”, Algebraic Geom. 5 (2018), no. 6, p. 650–685.
V. G. Kac – “Infinite root systems, representations of graphs and invariant theory”, Invent. Math. 56 (1980), no. 1, p. 57–92.
G. R. Kempf – “Instability in invariant theory”, Ann. of Math. (2) 108 (1978), no. 2, p. 299–316.
A. D. King – “Moduli of representations of finite-dimensional algebras”, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, p. 515–530.
F. C. Kirwan – Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984.
J. Kollár, Y. Miyaoka & S. Mori – “Rational connectedness and boundedness of Fano manifolds”, J. Differential Geom. 36 (1992), no. 3, p. 765–779.
A. Krug & P. Sosna – “On the derived category of the Hilbert scheme of points on an Enriques surface”, Selecta Math. (N.S.) 21 (2015), no. 4, p. 1339–1360.
K.-S. Lee & H.-B. Moon – “Derived category and ACM bundles of moduli spaces of vector bundles on a curve”, Forum Math. Sigma 11 (2023), article no. e81.
K. L. Martinez Acosta – “Approaches to ample stability for quiver moduli”, PhD Thesis, Ruhr-Universität Bochum, 2023.
M. Musta¸ tă – “Vanishing theorems on toric varieties”, Tohoku Math. J. (2) 54 (2002), no. 3, p. 451–470.
M. S. Narasimhan – “Derived categories of moduli spaces of vector bundles on curves”, J. Geom. Phys. 122 (2017), p. 53–58.
M. S. Narasimhan & S. Ramanan – “Deformations of the moduli space of vector bundles over an algebraic curve”, Ann. of Math. (2) 101 (1975), p. 391–417.
G. Petrella – “Partial semiorthogonal decompositions for quiver moduli”, J. Symbolic Comput. 131 (2025), article no. 102448 (18 pages).
F. Reede – “The Fourier-Mukai transform of a universal family of stable vector bundles”, Internat. J. Math. 32 (2021), no. 2, article no. 2150007 (13 pages).
M. Reineke – “The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli”, Invent. Math. 152 (2003), no. 2, p. 349–368.
, “Moduli of representations of quivers”, in Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2008, p. 589–637.
M. Reineke & S. Schröer – “Brauer groups for quiver moduli”, Algebraic Geom. 4 (2017), no. 4, p. 452–471.
A. Rudakov – “Stability for an abelian category”, J. Algebra 197 (1997), no. 1, p. 231–245.
A. Schofield – “Birational classification of moduli spaces of representations of quivers”, Indag. Math. (N.S.) 12 (2001), no. 3, p. 407–432.
C. Teleman – “The quantization conjecture revisited”, Ann. of Math. (2) 152 (2000), no. 1, p. 1–43.
A. Zamora – “On the Harder-Narasimhan filtration for finite dimensional representations of quivers”, Geom. Dedicata 170 (2014), p. 185–194.