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2012 | 10 | 4 | 1246-1279
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Recent results on quiver sheaves

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In this article, we survey recent work on the construction and geometry of representations of a quiver in the category of coherent sheaves on a projective algebraic manifold. We will also prove new results in the case of the quiver • ← • → •.
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Bibliografia
  • [1] Alper J., Good moduli spaces for Artin stacks, preprint available at http://arxiv.org/abs/0804.2242
  • [2] Álvarez-Cónsul L., Some results on the moduli spaces of quiver bundles, Geom. Dedicata, 2009, 139, 99–120 http://dx.doi.org/10.1007/s10711-008-9327-0
  • [3] Álvarez-Cónsul L., García-Prada O., Hitchin-Kobayashi correspondence, quivers, and vortices, Comm. Math. Phys., 2003, 238(1–2), 1–33
  • [4] Álvarez-Cónsul L., García-Prada O., Dimensional reduction and quiver bundles, J. Reine Angew. Math., 2003, 556, 1–46 http://dx.doi.org/10.1515/crll.2003.021
  • [5] Álvarez-Cónsul L., García-Prada O., Schmitt A.H.W., On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces, IMRP Int. Math. Res. Pap., 2006, #73597
  • [6] Álvarez-Cónsul L., King A.D., A functorial construction of moduli of sheaves, Invent. Math., 2007, 168(3), 613–666 http://dx.doi.org/10.1007/s00222-007-0042-5
  • [7] Assem I., Simson D., Skowronski A., Elements of the Representation Theory of Associative Algebras. I, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006
  • [8] Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, 1983, 308(1505), 523–615 http://dx.doi.org/10.1098/rsta.1983.0017
  • [9] Atiyah M.F., Hitchin N.J., Drinfeld V.G., Manin Yu.I., Construction of instantons, Phys. Lett. A, 1978, 65(3), 185–187 http://dx.doi.org/10.1016/0375-9601(78)90141-X
  • [10] Atiyah M.F., Ward R.S., Instantons and algebraic geometry, Comm. Math. Phys., 1977, 55(2), 117–124 http://dx.doi.org/10.1007/BF01626514
  • [11] Białynicki-Birula A., Theorems on actions of algebraic groups, Ann. of Math., 1973, 98(3), 480–497 http://dx.doi.org/10.2307/1970915
  • [12] Bradlow S.B., García-Prada O., Gothen P.B., Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann., 2004, 328(1–2), 299–351 http://dx.doi.org/10.1007/s00208-003-0484-z
  • [13] Bradlow S.B., García-Prada O., Muñoz V., Newstead P.E., Coherent systems and Brill-Noether theory, Internat. J. Math., 2003, 14(7), 683–733 http://dx.doi.org/10.1142/S0129167X03002009
  • [14] Derksen H., Weyman J., Quiver representations, Notices Amer. Math. Soc., 2005, 52(2), 200–206
  • [15] Derksen H., Weyman J., The combinatorics of quiver representations, Ann. Inst. Fourier (Grenoble), 2011, 61(3), 1061–1131 http://dx.doi.org/10.5802/aif.2636
  • [16] Donaldson S.K., Instantons and geometric invariant theory, Comm. Math. Phys., 1984, 93(4), 453–460 http://dx.doi.org/10.1007/BF01212289
  • [17] Frenkel I.B., Jardim M., Complex ADHM equations and sheaves on ℙ3, J. Algebra, 2008, 319(7), 2913–2937 http://dx.doi.org/10.1016/j.jalgebra.2008.01.016
  • [18] García-Prada O., Moduli spaces and geometric structures, Appendix to: Wells R.O., Differential Analysis on Complex Manifolds, 3rd ed., Grad. Texts in Math., 65, Springer, New York, 2008, 241–283
  • [19] García-Prada O., Heinloth J., Schmitt A., On the motives of moduli of chains and Higgs bundles, preprint available at http://arxiv.org/abs/1104.5558
  • [20] Ginzburg V., The global nilpotent variety is Lagrangian, Duke Math. J., 2001, 109(3), 511–519 http://dx.doi.org/10.1215/S0012-7094-01-10933-2
  • [21] Gothen P.B., The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface, Internat. J. Math., 1994, 5(6), 861–875 http://dx.doi.org/10.1142/S0129167X94000449
  • [22] Gothen P.B., King A.D., Homological algebra of twisted quiver bundles, J. London Math. Soc., 2005, 71(1), 85–99 http://dx.doi.org/10.1112/S0024610704005952
  • [23] Harder G., Narasimhan M.S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann., 1975, 212(3), 215–248 http://dx.doi.org/10.1007/BF01357141
  • [24] Hausel T., Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve, In: Geometric Methods in Algebra and Number Theory, Miami, December 16–20, 2003, Progr. Math., 235, Birkhäuser, Boston, 2005, 193–217 http://dx.doi.org/10.1007/0-8176-4417-2_9
  • [25] Hausel T., Rodriguez-Villegas F., Mixed Hodge polynomials of character varieties, Invent. Math., 2008, 174(3), 555–624 http://dx.doi.org/10.1007/s00222-008-0142-x
  • [26] Hausel T., Thaddeus M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math., 2003, 153(1), 197–229 http://dx.doi.org/10.1007/s00222-003-0286-7
  • [27] Hauzer M., Langer A., Moduli spaces of framed perverse instantons on ℙ3, Glasg. Math. J., 2011, 53(1), 51–96 http://dx.doi.org/10.1017/S0017089510000558
  • [28] Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc., 1987, 55(1), 59–126 http://dx.doi.org/10.1112/plms/s3-55.1.59
  • [29] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010 http://dx.doi.org/10.1017/CBO9780511711985
  • [30] Jardim M., Miró-Roig R.M., On the semistability of instanton sheaves over certain projective varieties, Comm. Algebra, 2008, 36(1), 288–298 http://dx.doi.org/10.1080/00927870701665503
  • [31] King A.D., Moduli of representations of finite-dimensional algebras, Q. J. Math., 1994, 45(180), 515–530 http://dx.doi.org/10.1093/qmath/45.4.515
  • [32] Lang S., Algebra, 3rd rev. ed., Grad. Texts in Math., 211, Springer, New York, 2002 http://dx.doi.org/10.1007/978-1-4613-0041-0
  • [33] Laudin A., Über die Geometrie der Modulräume von 3-er-Casimiten über einer kompakten Riemannschen Fläche vom Geschlecht g ≥ 2, Diploma thesis, Berlin, 2011
  • [34] Laumon G., Un analogue global du cône nilpotent, Duke Math. J., 1988, 57(2), 647–671 http://dx.doi.org/10.1215/S0012-7094-88-05729-8
  • [35] Le Bruyn L., Procesi C., Semisimple representations of quivers, Trans. Amer. Math. Soc., 1990, 317(2), 585–598 http://dx.doi.org/10.2307/2001477
  • [36] Le Potier J., À propos de la construction de l’espace de modules des faisceaux semi-stables sur le plan projectif, Bull. Soc. Math. France, 1994, 122(3), 363–369
  • [37] Mumford D., Fogarty J., Kirwan F., Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb., 34, Springer, Berlin, 1994 http://dx.doi.org/10.1007/978-3-642-57916-5
  • [38] Nevins T.A., Stafford J.T., Sklyanin algebras and Hilbert schemes of points, Adv. Math., 2007, 210(2), 405–478 http://dx.doi.org/10.1016/j.aim.2006.06.009
  • [39] Newstead P.E., Introduction to Moduli Problems and Orbit Spaces, Tata Inst. Fund. Res. Lectures on Math. and Phys., 51, Narosa Publishing House, New Delhi, 1978
  • [40] Nitsure N., Moduli space of semistable pairs on a curve, Proc. London Math. Soc., 1991, 62(2), 275–300 http://dx.doi.org/10.1112/plms/s3-62.2.275
  • [41] Okonek Ch., Spindler H., Mathematical instanton bundles on ℙ2n+1, J. Reine Angew. Math., 1986, 364, 35–50
  • [42] Ressayre N., GIT-cones and quivers, Math. Z., 2012, 270(1–2), 263–275 http://dx.doi.org/10.1007/s00209-010-0796-0
  • [43] Rudakov A., Stability for an abelian category, J. Algebra, 1997, 197(1), 231–245 http://dx.doi.org/10.1006/jabr.1997.7093
  • [44] Schmitt A., Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci., 2005, 115(1), 15–49 http://dx.doi.org/10.1007/BF02829837
  • [45] Schmitt A.H.W., Geometric Invariant Theory and Decorated Principal Bundles, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2008
  • [46] Schmitt A.H.W., A remark on semistability of quiver bundles, Eurasian Math. J. (in press)
  • [47] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math., 1994, 79, 47–129 http://dx.doi.org/10.1007/BF02698887
  • [48] Simson D., Skowronski A., Elements of the Representation Theory of Associative Algebras. II, London Math. Soc. Stud. Texts, 71, Cambridge University Press, Cambridge, 2007 http://dx.doi.org/10.1017/CBO9780511619403
  • [49] Thaddeus M., Stable pairs, linear systems and the Verlinde formula, Invent. Math., 1994, 117(2), 317–353 http://dx.doi.org/10.1007/BF01232244
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0007-9
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