Moduli Spaces of PU (2)-Instantons on Minimal Class VII Surfaces with b 2 =1
Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1691-1722.

We describe explicitly the moduli spaces g pst (S,E) of polystable holomorphic structures with det𝒦 on a rank two vector bundle E with c 1 (E)=c 1 (K) and c 2 (E)=0 for all minimal class VII surfaces S with b 2 (S)=1 and with respect to all possible Gauduchon metrics g. These surfaces S are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When S is a half or parabolic Inoue surface, g pst (S,E) is always a compact one-dimensional complex disc. When S is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when g varies in the space of Gauduchon metrics. g pst (S,E) can be identified with a moduli space of PU (2)-instantons. The moduli spaces of simple bundles of the above type lead to interesting examples of non-Hausdorff singular one-dimensional complex spaces.

Nous décrirons explicitement les espaces de modules g pst (S,E) de structures holomorphes polystables avec det𝒦 sur un fibré vectoriel E de rang deux avec c 1 (E)=c 1 (K) et c 2 (E)=0 pour toutes les surfaces S minimales de la classe VII avec b 2 (S)=1 et par rapport à toutes les métriques de Gauduchon g. Ces surfaces S sont des surfaces complexes non-elliptiques et non-Kählériennes et ont récemment été complètement classifiées. Si S est une demi-surface d’Inoue ou une surface d’Inoue parabolique, g pst (S,E) est toujours un disque complexe compact de dimension un. Si S est une surface d’Enoki, on obtient un disque complexe avec un nombre fini d’auto-intersections transverses, arbitrairement grand quand g varie dans l’espace des métriques de Gauduchon. g pst (S,E) peut être identifié à un espace de modules de PU (2)-instantons. Les espaces de modules de fibrés simples du type considéré mènent à des exemples intéressants d’espaces complexes singuliers non-Hausdorff de dimension un.

DOI: 10.5802/aif.2395
Classification: 14J60, 14J25, 57R57
Keywords: Moduli spaces, holomorphic bundles, complex surfaces, instantons
Mot clés : espaces de modules, fibrés holomorphes, surfaces complexes, instantons
Schöbel, Konrad 1

1 Université de Provence Centre de Mathématiques et Informatique Laboratoire d’Analyse, Topologie et Probabilités 39 rue F. Joliot Curie 13453 Marseille cedex 13 (France)
@article{AIF_2008__58_5_1691_0,
     author = {Sch\"obel, Konrad},
     title = {Moduli {Spaces} of ${\rm PU}(2)${-Instantons}  on {Minimal} {Class~VII} {Surfaces} with $b_2=1$},
     journal = {Annales de l'Institut Fourier},
     pages = {1691--1722},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     doi = {10.5802/aif.2395},
     zbl = {1159.14022},
     mrnumber = {2445830},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2395/}
}
TY  - JOUR
AU  - Schöbel, Konrad
TI  - Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 1691
EP  - 1722
VL  - 58
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2395/
DO  - 10.5802/aif.2395
LA  - en
ID  - AIF_2008__58_5_1691_0
ER  - 
%0 Journal Article
%A Schöbel, Konrad
%T Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$
%J Annales de l'Institut Fourier
%D 2008
%P 1691-1722
%V 58
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2395/
%R 10.5802/aif.2395
%G en
%F AIF_2008__58_5_1691_0
Schöbel, Konrad. Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$. Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1691-1722. doi : 10.5802/aif.2395. http://archive.numdam.org/articles/10.5802/aif.2395/

[1] Barth, W. P.; Hulek, K.; Peters, C. A. M.; de Ven, A. Van Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 4, Springer-Verlag, Berlin, 2004 | MR | Zbl

[2] Bogomolov, F. A. Surfaces of class VII 0 and affine geometry, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, Volume 46 (1982) no. 4, p. 710-761, 896 English translation: Math. USSR Izv. 21 (1983), no. 1, 31–73 | MR | Zbl

[3] Braam, P. J.; Hurtubise, J. Instantons on Hopf surfaces and monopoles on solid tori, Journal für reine und angewandte Mathematik, Volume 400 (1989), pp. 146-172 | MR | Zbl

[4] Brînzănescu, V.; Moraru, R. Stable bundles on non-Kähler elliptic surfaces, Communications in Mathematical Physics, Volume 254 (2005) no. 3, pp. 565-580 | DOI | MR | Zbl

[5] Buchdahl, N. P. Stable 2-bundles on Hirzebruch surfaces, Mathematische Zeitschrift, Volume 194 (1987) no. 1, pp. 143-152 | DOI | MR | Zbl

[6] Buchdahl, N. P. A Nakai-Moishezon criterion for non-Kähler surfaces, Université de Grenoble. Annales de l’Institut Fourier, Volume 50 (2000) no. 5, pp. 1533-1538 | DOI | Numdam | Zbl

[7] Dloussky, G. Structure des surfaces de Kato, Mémoires de la Société Mathématique de France. Nouvelle Série, Volume 112 (1984) no. 14, pp. 1-120 | Numdam | MR | Zbl

[8] Dloussky, G.; Oeljeklaus, K.; Toma, M. Class VII 0 surfaces with b 2 curves, The Tôhoku Mathematical Journal. Second Series, Volume 55 (2003) no. 2, pp. 283-309 | DOI | MR | Zbl

[9] Donagi, R. Y. Principal bundles on elliptic fibrations, The Asian Journal of Mathematics, Volume 1 (1997) no. 2, pp. 214-223 | MR | Zbl

[10] Donaldson, S. K. Irrationality and the h-cobordism conjecture, Journal of Differential Geometry, Volume 26 (1987) no. 1, pp. 141-168 | MR | Zbl

[11] Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1990 (Oxford Science Publications) | MR | Zbl

[12] Elencwajg, G.; Forster, O. Vector bundles on manifolds without divisors and a theorem on deformations, Université de Grenoble. Annales de l’Institut Fourier, Volume 32 (1982) no. 4, p. 25-51 (1983) | DOI | EuDML | Numdam | Zbl

[13] Enoki, I. On surfaces of class VII 0 with curves, Japan Academy. Proceedings. Series A. Mathematical Sciences, Volume 56 (1980) no. 6, pp. 275-279 | DOI | MR | Zbl

[14] Enoki, I. Surfaces of class VII 0 with curves, The Tôhoku Mathematical Journal. Second Series, Volume 33 (1981) no. 4, pp. 453-492 | DOI | MR | Zbl

[15] Friedman, R. Rank two vector bundles over regular elliptic surfaces, Inventiones Mathematicae, Volume 96 (1989), pp. 283-332 | DOI | EuDML | MR | Zbl

[16] Friedman, R.; Morgan, J.; Witten, E. Vector bundles over elliptic fibrations, Journal of Algebraic Geometry, Volume 8 (1999) no. 2, pp. 279-401 | MR | Zbl

[17] Friedman, R.; Morgan, J. W. Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 27, Springer-Verlag, Berlin, 1994 | MR | Zbl

[18] Gauduchon, P. La 1-forme de torsion d’une variété hermitienne compacte, Mathematische Annalen, Volume 267 (1984) no. 4, pp. 495-518 | DOI | EuDML | Zbl

[19] Griffiths, P.; Harris, J. Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994 | MR | Zbl

[20] Inoue, M. On surfaces of class VII 0 , Inventiones Mathematicae, Volume 24 (1974), pp. 269-310 | DOI | EuDML | MR | Zbl

[21] Inoue, M. New surfaces with no meromorphic functions. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 91-106 | MR | Zbl

[22] Kato, M. Compact complex manifolds containing “global” spherical shells. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, Kyoto, 1977), Kinokuniya Book Store, Tokyo (1978), pp. 45-84 | Zbl

[23] Kobayashi, S. Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, Princeton, NJ, 1987 | MR | Zbl

[24] Kodaira, K. On the structure of compact complex analytic surfaces. I, American Journal of Mathematics, Volume 86 (1964), pp. 751-798 | DOI | MR | Zbl

[25] Kodaira, K. On the structure of compact complex analytic surfaces. II, American Journal of Mathematics, Volume 88 (1966), pp. 682-721 | DOI | MR | Zbl

[26] Kotschick, D. On manifolds homeomorphic to CP 2 #8CP ¯ 2 , Inventiones Mathematicae, Volume 95 (1989) no. 3, pp. 591-600 | DOI | EuDML | MR | Zbl

[27] Li, J. Algebraic geometric interpretation of Donaldson’s polynomial invariants, Journal of Differential Geometry, Volume 37 (1993) no. 2, pp. 417-466 | Zbl

[28] Li, J.; Yau, S. T.; Zheng, F. On projectively flat Hermitian manifolds, Communications in Analysis and Geometry, Volume 2 (1994) no. 1, pp. 103-109 | MR | Zbl

[29] Lübke, M.; Teleman, A. The Kobayashi-Hitchin correspondence, World Scientific Publishing Co. Inc., River Edge, NJ, 1995 | MR | Zbl

[30] Moraru, R. Integrable systems associated to a Hopf surface, Canadian Journal of Mathematics, Volume 55 (2003) no. 3, pp. 609-635 | DOI | MR | Zbl

[31] Nakamura, I. On surfaces of class VII 0 with curves, Inventiones Mathematicae, Volume 78 (1984) no. 3, pp. 393-443 | DOI | EuDML | MR | Zbl

[32] Okonek, C.; de Ven, A. Van Stable bundles and differentiable structures on certain elliptic surfaces, Inventiones Mathematicae, Volume 86 (1986) no. 2, pp. 357-370 | DOI | EuDML | MR | Zbl

[33] Okonek, C.; de Ven, A. Van Γ-type-invariants associated to PU (2)-bundles and the differentiable structure of Barlow’s surface, Inventiones Mathematicae, Volume 95 (1989) no. 3, pp. 601-614 | DOI | EuDML | Zbl

[34] Teleman, A. Projectively flat surfaces and Bogomolov’s theorem on class VII 0 surfaces, International Journal of Mathematics, Volume 5 (1994) no. 2, pp. 253-264 | DOI | Zbl

[35] Teleman, A. Moduli spaces of stable bundles on non-Kählerian elliptic fibre bundles over curves, Expositiones Mathematicae. International Journal, Volume 16 (1998) no. 3, pp. 193-248 | MR | Zbl

[36] Teleman, A. Donaldson theory on non-Kählerian surfaces and class VII surfaces with b 2 =1, Inventiones Mathematicae, Volume 162 (2005) no. 3, pp. 493-521 | DOI | MR | Zbl

[37] Teleman, A. The pseudo-effective cone of a non-Kählerian surface and applications, Mathematische Annalen, Volume 335 (2006) no. 4, pp. 965-989 | DOI | MR | Zbl

[38] Teleman, A. Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds, Geometry and Topology, Volume 11 (2007), pp. 1681-1730 | DOI | MR | Zbl

[39] Teleman, A. Instantons and curves on class VII surfaces (2007) (arXiv:0704.2634) | Zbl

[40] Toma, M. Compact moduli spaces of stable sheaves over non-algebraic surfaces, Documenta Mathematica, Volume 6 (2001), pp. 11-29 (Electronic) | EuDML | MR | Zbl

[41] Toma, M. Vector bundles on blown-up Hopf surfaces (2006) (http://www.iecn.u-nancy.fr/~toma/eclahopf.pdf)

Cited by Sources: