Recent results on quiver sheaves

Andreas Laudin; Alexander Schmitt

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1246-1279
  • ISSN: 2391-5455

Abstract

top
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 • ← • → •.

How to cite

top

Andreas Laudin, and Alexander Schmitt. "Recent results on quiver sheaves." Open Mathematics 10.4 (2012): 1246-1279. <http://eudml.org/doc/269116>.

@article{AndreasLaudin2012,
abstract = {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 • ← • → •.},
author = {Andreas Laudin, Alexander Schmitt},
journal = {Open Mathematics},
keywords = {Quiver; Moduli space; Semistability; quiver; moduli space; semistability},
language = {eng},
number = {4},
pages = {1246-1279},
title = {Recent results on quiver sheaves},
url = {http://eudml.org/doc/269116},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Andreas Laudin
AU - Alexander Schmitt
TI - Recent results on quiver sheaves
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1246
EP - 1279
AB - 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 • ← • → •.
LA - eng
KW - Quiver; Moduli space; Semistability; quiver; moduli space; semistability
UR - http://eudml.org/doc/269116
ER -

References

top
  1. [1] Alper J., Good moduli spaces for Artin stacks, preprint available at http://arxiv.org/abs/0804.2242 Zbl1314.14095
  2. [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 Zbl1170.14007
  3. [3] Álvarez-Cónsul L., García-Prada O., Hitchin-Kobayashi correspondence, quivers, and vortices, Comm. Math. Phys., 2003, 238(1–2), 1–33 Zbl1051.53018
  4. [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 Zbl1020.53047
  5. [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 Zbl1111.32012
  6. [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 Zbl1137.14026
  7. [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 Zbl1092.16001
  8. [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 Zbl0509.14014
  9. [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 Zbl0424.14004
  10. [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 Zbl0362.14004
  11. [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 Zbl0275.14007
  12. [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 Zbl1041.32008
  13. [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 Zbl1057.14041
  14. [14] Derksen H., Weyman J., Quiver representations, Notices Amer. Math. Soc., 2005, 52(2), 200–206 Zbl1143.16300
  15. [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 Zbl1271.16016
  16. [16] Donaldson S.K., Instantons and geometric invariant theory, Comm. Math. Phys., 1984, 93(4), 453–460 http://dx.doi.org/10.1007/BF01212289 Zbl0581.14008
  17. [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 Zbl1145.14017
  18. [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. [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 Zbl1316.14060
  20. [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 Zbl1116.14007
  21. [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 Zbl0860.14030
  22. [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 Zbl1095.14012
  23. [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 Zbl0324.14006
  24. [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. [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 Zbl1213.14020
  26. [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 Zbl1043.14011
  27. [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 Zbl1238.14010
  28. [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 Zbl0634.53045
  29. [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 Zbl1206.14027
  30. [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 Zbl1131.14046
  31. [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 Zbl0837.16005
  32. [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. [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. [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 Zbl0688.14023
  35. [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 Zbl0693.16018
  36. [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. [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 Zbl0797.14004
  38. [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 Zbl1116.14003
  39. [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 Zbl0411.14003
  40. [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 Zbl0733.14005
  41. [41] Okonek Ch., Spindler H., Mathematical instanton bundles on ℙ2n+1, J. Reine Angew. Math., 1986, 364, 35–50 
  42. [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. [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. [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 Zbl1076.14019
  45. [45] Schmitt A.H.W., Geometric Invariant Theory and Decorated Principal Bundles, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2008 Zbl1159.14001
  46. [46] Schmitt A.H.W., A remark on semistability of quiver bundles, Eurasian Math. J. (in press) Zbl1272.14011
  47. [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. [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 Zbl1131.16001
  49. [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 Zbl0882.14003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.