Instantons and curves on class VII surfaces

Abstract

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for $b_2=2$, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case $b_2=1$ was solved in a previous article. The fundamental object intervening in our strategy is the moduli space ${\mathcal M}^{\rm pst}(0,{\mathcal K})$ of polystable bundles ${\mathcal E}$ with $c_2({\mathcal E})=0$, $\det({\mathcal E})={\mathcal K}$. For large $b_2$ the geometry of this moduli space becomes very complicated. The case $b_2=2$ treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature. We explain the substantial obstacles which must be overcome in order to extend our methods to the case $b_2\geq 3$.

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      VOLUME = {271},
      YEAR = {1985},
      NUMBER = {4},
      PAGES = {493--526},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {32G13 (14C40 32C35 32J25)},
      MRNUMBER = {87h:32045},
      MRREVIEWER = {Autorreferat},
      DOI = {10.1007/BF01456132},
      ZBLNUMBER = {0539.14005},
      }
  • [Te1] Go to document A. D. Teleman, "Projectively flat surfaces and Bogomolov’s theorem on class ${ VII}_0$ surfaces," Internat. J. Math., vol. 5, iss. 2, pp. 253-264, 1994.
    @article {Te1, MRKEY = {1266285},
      AUTHOR = {Teleman, Andrei Dumitru},
      TITLE = {Projectively flat surfaces and {B}ogomolov's theorem on class {${\rm VII}\sb 0$} surfaces},
      JOURNAL = {Internat. J. Math.},
      FJOURNAL = {International Journal of Mathematics},
      VOLUME = {5},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {253--264},
      ISSN = {0129-167X},
      MRCLASS = {32L07 (32J15 53C07)},
      MRNUMBER = {95e:32031},
      MRREVIEWER = {Zhuang Dan Guan},
      DOI = {10.1142/S0129167X94000152},
      ZBLNUMBER = {0803.53038},
      }
  • [Te2] Go to document A. D. Teleman, "Donaldson theory on non-Kählerian surfaces and class VII surfaces with $b_2=1$," Invent. Math., vol. 162, iss. 3, pp. 493-521, 2005.
    @article {Te2, MRKEY = {2198220},
      AUTHOR = {Teleman, Andrei Dumitru},
      TITLE = {Donaldson theory on non-{K}ählerian surfaces and class {VII} surfaces with {$b\sb 2=1$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {162},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {493--521},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32J15 (57R57)},
      MRNUMBER = {2006i:32020},
      MRREVIEWER = {Vicente Mu{ñ}oz},
      DOI = {10.1007/s00222-005-0451-2},
      }
  • [Te3] Go to document A. D. Teleman, "The pseudo-effective cone of a non-Kählerian surface and applications," Math. Ann., vol. 335, iss. 4, pp. 965-989, 2006.
    @article {Te3, MRKEY = {2232025},
      AUTHOR = {Teleman, Andrei Dumitru},
      TITLE = {The pseudo-effective cone of a non-{K}ählerian surface and applications},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {335},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {965--989},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {32J15 (32G13 32L05 32Q57)},
      MRNUMBER = {2007b:32034},
      MRREVIEWER = {Vicente Mu{ñ}oz},
      DOI = {10.1007/s00208-006-0782-3},
      }
  • [Te4] Go to document A. D. Teleman, "Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds," Geom. Topol., vol. 11, pp. 1681-1730, 2007.
    @article {Te4, MRKEY = {2350464},
      AUTHOR = {Teleman, Andrei Dumitru},
      TITLE = {Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
      VOLUME = {11},
      YEAR = {2007},
      PAGES = {1681--1730},
      ISSN = {1465-3060},
      MRCLASS = {57R57 (53C07 58D27)},
      MRNUMBER = {2008g:57036},
      MRREVIEWER = {Vicente Mu{ñ}oz},
      DOI = {10.2140/gt.2007.11.1681},
      ZBLNUMBER = {1138.57030},
      }
  • [Te5] Go to document A. D. Teleman, "Families of holomorphic bundles," Commun. Contemp. Math., vol. 10, iss. 4, pp. 523-551, 2008.
    @article {Te5, MRKEY = {2444847},
      AUTHOR = {Teleman, Andrei Dumitru},
      TITLE = {Families of holomorphic bundles},
      JOURNAL = {Commun. Contemp. Math.},
      FJOURNAL = {Communications in Contemporary Mathematics},
      VOLUME = {10},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {523--551},
      ISSN = {0219-1997},
      MRCLASS = {32L05 (32G08 32G13 53C07)},
      MRNUMBER = {2009d:32021},
      MRREVIEWER = {Vicente Mu{ñ}oz},
      DOI = {10.1142/S0219199708002892},
      ZBLNUMBER = {1159.32011},
      }

Authors

Andrei Teleman

LATP, UMR 6632; CMI, Université de Provence
Centre de Mathématiques et Informatique, Université de Provence
39 Rue F. Joliot-Curie
13453 Marseille Cedex 13
France