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Instanton sheaves and representations of quivers. (English) Zbl 1468.14021

A rank \(2\) instanton sheaf on the complex projective space \({\mathbb P} ^3\) is a torsion-free rank \(2\) sheaf \(E\) with \(\det E\simeq {\mathcal O}_{{\mathbb P}^3}\) and such that \[H^0(E(-1))=H^1(E(-2))=H^2(E(-2))=H^3(E(-3))=0.\] Such sheaves have only one non-trivial Chern class \(c_2(E)\), which can be identified with an integer via the canonical isomorphism \(H^4({\mathbb P}^3, {\mathbb Z})\simeq {\mathbb Z}\).
One can consider the moduli space \({\mathcal I} (n)\) of rank \(2\) locally free instanton sheaves \(E\) with \(c_2(E)=n\) and the moduli space \({\mathcal L} (n)\) of rank \(2\) instanton sheaves \(E\) with \(c_2(E)=n\). Whereas \({\mathcal I} (n)\) is a smooth affine irreducible variety, \({\mathcal L} (n)\) can be singular with several irreducible components.
Every instanton sheaf corresponds to a representation of some special quiver with relations, consisting of \(3\) vertices and \(4\) arrows between two pairs of vertices. The main aim of the paper is to study compactifications of \({\mathcal I} (n)\) and \({\mathcal L} (n)\) within King’s moduli spaces of \(\theta\)-semistable representations of this quiver. The main theorem achieves this aim for \(n=1\), in which case the authors obtain a complete description of such compactifications.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
16G20 Representations of quivers and partially ordered sets

References:

[1] Atiyah, M. F., Hitchin, N. J., Drinfeld, V. G. and Manin, Yu. I., Construction of instantons, Phys. Lett. A65(3) (1978), 185-187. · Zbl 0424.14004
[2] Bessenrodt, C. and Le Bruyn, L., Stable rationality of certain PGL_n-quotients, Invent. Math. 104(1) (1991), 179-199. · Zbl 0741.14032
[3] Costa, L. and Ottaviani, G., Nondegenerate multidimensional matrices and instanton bundles, Trans. Amer. Math. Soc. 355(1) (2003), 49-55. · Zbl 1031.14004
[4] Gargate, M. and Jardim, M., Singular loci of instanton sheaves on projective space, Internat. Math. J. 27 (2016), 1640006. · Zbl 1360.14061
[5] Hauzer, M. and Langer, A., Moduli space of framed perverse instantons on ℙ^3, Glasgow Math. J. 53 (2011), 51-96. · Zbl 1238.14010
[6] Henni, A. A., Jardim, M. and Vidal Martins, R., ADHM construction of perverse instanton sheaves, Glasgow Math. J. 57 (2015), 285-321. · Zbl 1316.14024
[7] Jardim, M., Instanton sheaves on complex projective spaces, Collect. Math. 57(1) (2006), 69-91. · Zbl 1095.14040
[8] Jardim, M. and Da Silva, V. M. F., Decomposability criterion for linear sheaves, Cent. Eur. J. Math. 10(4) (2012), 1292-1299. · Zbl 1278.14015
[9] 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. · Zbl 1259.14016
[10] 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. · Zbl 1457.14025
[11] Jardim, M., Markushevich, D. and Tikhomirov, A. S., Two infinite series of moduli spaces of rank 2 sheaves on ℙ^3, Ann. Mat. Pura Appl. (4)196(4) (2017), 1573-1608. · Zbl 1481.14020
[12] 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. · Zbl 1439.14047
[13] King, A. D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2)45(180) (1994), 515-530. · Zbl 0837.16005
[14] Kirillov, A. Jr., Quiver representations and quiver varieties, Graduate Studies in Mathematics, Volume 174 (American Mathematical Society, Providence, RI, 2016) · Zbl 1355.16002
[15] Marchesi, S., Macias Marques, P. and Soares, H., Monads on Projective varieties, Pacific J. Math. 296 (2018), 155-180. · Zbl 1412.14012
[16] Maruyama, M. and Trautmann, G., Limits of instantons, Int. J. Math. 3(2) (1992), 213-276. · Zbl 0770.14012
[17] 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. · Zbl 0753.14004
[18] Okonek, C. and Spindler, H., Mathematical instanton bundles on P^2n + 1, J. Reine Angew. Math. 364 (1986), 35-50. · Zbl 0568.14009
[19] Okonek, C., Schneider, M. and Spindler, H., Vector bundles on complex projective spaces, Progress in Mathematics, Volume 3 (Birkhäuser, Boston, MA, 1980). · Zbl 0438.32016
[20] Perrin, N., Deux composantes du bord de I_3, Bull. Soc. Math. France130(4) (2002), 537-572. · Zbl 1065.14055
[21] 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. · Zbl 1262.14053
[22] 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. · Zbl 1308.14045
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