Bubble tree compactification of moduli spaces of vector bundles on surfaces

Dimitri Markushevich; Alexander Tikhomirov; Günther Trautmann

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1331-1355
  • ISSN: 2391-5455

Abstract

top
We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.

How to cite

top

Dimitri Markushevich, Alexander Tikhomirov, and Günther Trautmann. "Bubble tree compactification of moduli spaces of vector bundles on surfaces." Open Mathematics 10.4 (2012): 1331-1355. <http://eudml.org/doc/269293>.

@article{DimitriMarkushevich2012,
abstract = {We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.},
author = {Dimitri Markushevich, Alexander Tikhomirov, Günther Trautmann},
journal = {Open Mathematics},
keywords = {Instantons; Vector bundles; Coherent sheaves; Moduli spaces of sheaves; Donaldson-Uhlenbeck compactification; Monads; Serre construction; Fulton-McPherson compactification; instantons; vector bundles; coherent sheaves; moduli spaces of sheaves; monads},
language = {eng},
number = {4},
pages = {1331-1355},
title = {Bubble tree compactification of moduli spaces of vector bundles on surfaces},
url = {http://eudml.org/doc/269293},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Dimitri Markushevich
AU - Alexander Tikhomirov
AU - Günther Trautmann
TI - Bubble tree compactification of moduli spaces of vector bundles on surfaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1331
EP - 1355
AB - We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
LA - eng
KW - Instantons; Vector bundles; Coherent sheaves; Moduli spaces of sheaves; Donaldson-Uhlenbeck compactification; Monads; Serre construction; Fulton-McPherson compactification; instantons; vector bundles; coherent sheaves; moduli spaces of sheaves; monads
UR - http://eudml.org/doc/269293
ER -

References

top
  1. [1] Barth W., Moduli of vector bundles on the projective plane, Invent. Math., 1977, 42 63–91 http://dx.doi.org/10.1007/BF01389784 Zbl0386.14005
  2. [2] Buchdahl N.P., Sequences of stable bundles over compact complex surfaces, J. Geom. Anal., 1999, 9(3), 391–428 http://dx.doi.org/10.1007/BF02921982 Zbl0964.32016
  3. [3] Buchdahl N.P., Blowups and gauge fields, Pacific J. Math., 2000, 196(1), 69–111 http://dx.doi.org/10.2140/pjm.2000.196.69 Zbl1073.32506
  4. [4] Donaldson S.K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc., 1985, 50(1), 1–26 http://dx.doi.org/10.1112/plms/s3-50.1.1 Zbl0529.53018
  5. [5] Donaldson S.K., Compactification and completion of Yang-Mills moduli spaces, In: Differential Geometry, Peñíscola, 1988, Lecture Notes in Math., 1410, Springer, Berlin, 1989, 145–160 
  6. [6] Feehan P.M.N., Geometry of the ends of the moduli space of anti-self-dual connections, J. Differential Geom., 1995, 42(3), 465–553 Zbl0856.58007
  7. [7] Fulton W., MacPherson R., A compactification of configuration spaces, Ann. of Math., 1994, 139(1), 183–225 http://dx.doi.org/10.2307/2946631 Zbl0820.14037
  8. [8] Gieseker D., On the moduli of vector bundles on an algebraic surface, Ann. of Math., 1977, 106(1), 45–60 http://dx.doi.org/10.2307/1971157 Zbl0381.14003
  9. [9] Gieseker D., A construction of stable bundles on an algebraic surface, J. Differential Geom., 1988, 27(1), 137–154 Zbl0648.14008
  10. [10] 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
  11. [11] Kirwan F.C., Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math., 1985, 122(1), 41–85 http://dx.doi.org/10.2307/1971369 Zbl0592.14011
  12. [12] Kollár J., Projectivity of complete moduli, J. Differential Geom., 1990, 32(1), 235–268 Zbl0684.14002
  13. [13] Kollár J., Quotient spaces modulo algebraic groups, Ann. of Math., 1997, 145(1), 33–79 http://dx.doi.org/10.2307/2951823 Zbl0881.14017
  14. [14] Li J., Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom., 1993, 37(2), 417–466 Zbl0809.14006
  15. [15] Lübke M., Teleman A., The Kobayashi-Hitchin Correspondence, World Scientific, River Edge, 1995 http://dx.doi.org/10.1142/2660 Zbl0849.32020
  16. [16] Maruyama M., Singularities of the curve of jumping lines of a vector bundle of rank 2 on ℙ2, In: Algebraic Geometry, Tokyo, Kyoto, October 5–14, 1982, Lecture Notes in Math., 1016, Springer, Berlin-New York, 1983, 370–411 
  17. [17] Maruyama M., Trautmann G., On compactifications of the moduli space of instantons, Internat. J. Math., 1990, 1(4), 431–477 http://dx.doi.org/10.1142/S0129167X90000228 Zbl0727.14007
  18. [18] Maruyama M., Trautmann G., Limits of instantons, Internat. J. Math., 1992, 3(2), 213–276 http://dx.doi.org/10.1142/S0129167X92000072 Zbl0770.14012
  19. [19] Nagaraj D.S., Seshadri C.S., Degenerations of the moduli spaces of vector bundles on curves. I, Proc. Indian Acad. Sci. Math. Sci., 1997, 107(2), 101–137 Zbl0922.14023
  20. [20] Nagaraj D.S., Seshadri C.S., Degenerations of the moduli spaces of vector bundles on curves. II, Proc. Indian Acad. Sci. Math. Sci., 1999, 109(2), 165–201 http://dx.doi.org/10.1007/BF02841533 Zbl0957.14021
  21. [21] Okonek C., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces, Progr. Math., 3, Birkhäuser, Boston, 1980 Zbl0438.32016
  22. [22] 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 
  23. [23] Taubes C.H., A framework for Morse theory for the Yang-Mills functional, Invent. Math., 1988, 94(2), 327–402 http://dx.doi.org/10.1007/BF01394329 Zbl0665.58006
  24. [24] Timofeeva N.V., Compactification of the moduli variety of stable 2-vector bundles on a surface in the Hilbert scheme, Math. Notes, 2007, 82(5–6), 667–690 Zbl1167.14028
  25. [25] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface, Sb. Math., 2008, 199(7–8), 1051–1070 http://dx.doi.org/10.1070/SM2008v199n07ABEH003953 
  26. [26] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface. II, Sb. Math., 2009, 200(3–4), 405–427 http://dx.doi.org/10.1070/SM2009v200n03ABEH004002 
  27. [27] Timofeeva N.V., On a new compactification of the moduli of vector bundles on a surface. III: A functorial approach, Sb. Math., 2011, 202(3–4), 413–465 http://dx.doi.org/10.1070/SM2011v202n03ABEH004151 
  28. [28] Trautmann G., Moduli spaces in algebraic geometry (manuscript) 
  29. [29] Uhlenbeck K.K., Removable singularities in Yang-Mills fields, Comm. Math. Phys., 1982, 83(1), 11–29 http://dx.doi.org/10.1007/BF01947068 Zbl0491.58032
  30. [30] Viehweg E., Quasi-Projective Moduli for Polarized Manifolds, Ergeb. Math. Grenzgeb., 30, Springer, Berlin, 1995 http://dx.doi.org/10.1007/978-3-642-79745-3 Zbl0844.14004

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.