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Instanton sheaves and representations of quivers

Part of: Surfaces and higher-dimensional varieties Families, fibrations Representation theory of rings and algebras

Published online by Cambridge University Press:  04 September 2020

M. Jardim  and
D. D. Silva
M. Jardim
Affiliation:
Departamento de Matemática, IMECC - UNICAMP, Rua Sérgio Buarque de Holanda, 651, Campinas, São Paulo13083-970, Brazil (jardim@ime.unicamp.br)
D. D. Silva
Affiliation:
DMA – UFS, Avenida Marechal Rondon S/N, São Cristovão, Sergipe, Brazil (ddsilva@ufs.br)
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Abstract

We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.

Keywords

instanton sheavesrepresentations of quiversmoduli spaceswall-crossing

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 16G20: Representations of quivers and partially ordered sets
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 984 - 1004
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Atiyah, M. F., Hitchin, N. J., Drinfeld, V. G. and Manin, Yu. I., Construction of instantons, Phys. Lett. A 65(3) (1978), 185–187.CrossRefGoogle Scholar
Bessenrodt, C. and Le Bruyn, L., Stable rationality of certain PGLn-quotients, Invent. Math. 104(1) (1991), 179–199.CrossRefGoogle Scholar
Costa, L. and Ottaviani, G., Nondegenerate multidimensional matrices and instanton bundles, Trans. Amer. Math. Soc. 355(1) (2003), 49–55.CrossRefGoogle Scholar
Gargate, M. and Jardim, M., Singular loci of instanton sheaves on projective space, Internat. Math. J. 27 (2016), 1640006.CrossRefGoogle Scholar
Hauzer, M. and Langer, A., Moduli space of framed perverse instantons on ℙ3, Glasgow Math. J. 53 (2011), 51–96.CrossRefGoogle Scholar
Henni, A. A., Jardim, M. and Vidal Martins, R., ADHM construction of perverse instanton sheaves, Glasgow Math. J. 57 (2015), 285–321.CrossRefGoogle Scholar
Jardim, M., Instanton sheaves on complex projective spaces, Collect. Math. 57(1) (2006), 69–91.Google Scholar
Jardim, M. and da Silva, V. M. F., Decomposability criterion for linear sheaves, Cent. Eur. J. Math. 10(4) (2012), 1292–1299.CrossRefGoogle Scholar
Jardim, M. and Prata, D. M., Representations of quivers on abelian categories and monads on projective varieties, São Paulo J. Math. Sci. 4(3) (2010), 399–423.CrossRefGoogle Scholar
Jardim, M., Maican, M. and Tikhomirov, A. S., Moduli spaces of rank 2 instanton sheaves on the projective space, Pacific J. Math. 291(2) (2017), 399–424.CrossRefGoogle Scholar
Jardim, M., Markushevich, D. and Tikhomirov, A. S., Two infinite series of moduli spaces of rank 2 sheaves on ℙ3, Ann. Mat. Pura Appl. (4) 196(4) (2017), 1573–1608.CrossRefGoogle Scholar
Jardim, M., Markushevich, D. and Tikhomirov, A. S., New divisors in the boundary of the instanton moduli space, Mosc. Math. J. 18(1) (2018), 117–148.Google Scholar
King, A. D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45(180) (1994), 515–530.CrossRefGoogle Scholar
Kirillov, A. Jr., Quiver representations and quiver varieties, Graduate Studies in Mathematics, Volume 174 (American Mathematical Society, Providence, RI, 2016)Google Scholar
Marchesi, S., Macias Marques, P. and Soares, H., Monads on Projective varieties, Pacific J. Math. 296 (2018), 155–180.CrossRefGoogle Scholar
Maruyama, M. and Trautmann, G., Limits of instantons, Int. J. Math. 3(2) (1992), 213–276.CrossRefGoogle Scholar
Narasimhan, M. S. and Trautmann, G., Compactification of $M_{{P}_{3}}(0,2)$ and Poncelet pairs of conics, Pacific J. Math. 145(2) (1990), 255–365.CrossRefGoogle Scholar
Okonek, C. and Spindler, H., Mathematical instanton bundles on P 2n + 1, J. Reine Angew. Math. 364 (1986), 35–50.Google Scholar
Okonek, C., Schneider, M. and Spindler, H., Vector bundles on complex projective spaces, Progress in Mathematics, Volume 3 (Birkhäuser, Boston, MA, 1980).CrossRefGoogle Scholar
Perrin, N., Deux composantes du bord de I 3, Bull. Soc. Math. France 130(4) (2002), 537–572.CrossRefGoogle Scholar
Tikhomirov, A. S., Moduli of mathematical instanton vector bundles with odd c 2 on projective space, Izv. Math. 76(5) (2012), 991–1073; translated from Izv. Ross. Akad. Nauk Ser. Mat. 76(5) (2012), 143–224.CrossRefGoogle Scholar
Tikhomirov, A. S., Moduli of mathematical instanton vector bundles with even c 2 on projective space, Izv. Math. 77(6) (2013), 1195–1223; translated from Izv. Ross. Akad. Nauk Ser. Mat. 77(6) (2013), 139–168.CrossRefGoogle Scholar