Bubble tree compactification of moduli spaces of vector bundles on surfaces

Dimitri Markushevich ; Alexander Tikhomirov ; Günther Trautmann

Open Mathematics, Tome 10 (2012), p. 1331-1355 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2012-01-01

Zbl 1303.14049

EUDML-ID : urn:eudml:doc:269293

Bibliographie

[1] Barth W., Moduli of vector bundles on the projective plane, Invent. Math., 1977, 42 63–91 http://dx.doi.org/10.1007/BF01389784 | Zbl 0386.14005

[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 | Zbl 0964.32016

[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 | Zbl 1073.32506

[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 | Zbl 0529.53018

[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] 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 | Zbl 0856.58007

[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 | Zbl 0820.14037

[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 | Zbl 0381.14003

[9] Gieseker D., A construction of stable bundles on an algebraic surface, J. Differential Geom., 1988, 27(1), 137–154 | Zbl 0648.14008

[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 | Zbl 1206.14027

[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 | Zbl 0592.14011

[12] Kollár J., Projectivity of complete moduli, J. Differential Geom., 1990, 32(1), 235–268 | Zbl 0684.14002

[13] Kollár J., Quotient spaces modulo algebraic groups, Ann. of Math., 1997, 145(1), 33–79 http://dx.doi.org/10.2307/2951823 | Zbl 0881.14017

[14] Li J., Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom., 1993, 37(2), 417–466 | Zbl 0809.14006

[15] Lübke M., Teleman A., The Kobayashi-Hitchin Correspondence, World Scientific, River Edge, 1995 http://dx.doi.org/10.1142/2660 | Zbl 0849.32020

[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] 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 | Zbl 0727.14007

[18] Maruyama M., Trautmann G., Limits of instantons, Internat. J. Math., 1992, 3(2), 213–276 http://dx.doi.org/10.1142/S0129167X92000072 | Zbl 0770.14012

[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 | Zbl 0922.14023

[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 | Zbl 0957.14021

[21] Okonek C., Schneider M., Spindler H., Vector Bundles on Complex Projective Spaces, Progr. Math., 3, Birkhäuser, Boston, 1980 | Zbl 0438.32016

[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] 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 | Zbl 0665.58006

[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 | Zbl 1167.14028

[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] 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] 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] Trautmann G., Moduli spaces in algebraic geometry (manuscript)

[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 | Zbl 0491.58032

[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 | Zbl 0844.14004