Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$

Schöbel, Konrad

Moduli Spaces of

PU (2)

-Instantons on Minimal Class VII Surfaces with

b_{2} = 1

[Espaces de modules de

PU (2)

-instantons sur les surfaces minimales de classe

VII

b_{2} = 1

]

Schöbel, Konrad

Annales de l'Institut Fourier, Tome 58 (2008), p. 1691-1722 / Harvested from Numdam

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Résumé
Abstract

Nous décrirons explicitement les espaces de modules $ℳ_{g}^{pst} (S, E)$ de structures holomorphes polystables $ℰ$ avec $det ℰ ≅ 𝒦$ 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, $ℳ_{g}^{pst} (S, E)$ 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. $ℳ_{g}^{pst} (S, E)$ 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.

We describe explicitly the moduli spaces $ℳ_{g}^{pst} (S, E)$ of polystable holomorphic structures $ℰ$ with $det ℰ ≅ 𝒦$ 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, $ℳ_{g}^{pst} (S, E)$ 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. $ℳ_{g}^{pst} (S, E)$ 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.

Publié le : 2008-01-01

MR 2445830 | Zbl 1159.14022

DOI : https://doi.org/10.5802/aif.2395
Classification: 14J60, 14J25, 57R57
Mots clés: espaces de modules, fibrés holomorphes, surfaces complexes, instantons

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     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},
     volume = {58},
     year = {2008},
     pages = {1691-1722},
     doi = {10.5802/aif.2395},
     zbl = {1159.14022},
     mrnumber = {2445830},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_5_1691_0}
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Schöbel, Konrad. Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$. Annales de l'Institut Fourier, Tome 58 (2008) pp. 1691-1722. doi : 10.5802/aif.2395. http://gdmltest.u-ga.fr/item/AIF_2008__58_5_1691_0/

Bibliographie

[1] Barth, W. P.; Hulek, K.; Peters, C. A. M.; De Ven, A. Van Compact Complex Surfaces, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Tome 4 (2004) | MR 2030225 | Zbl 1036.14016

[2] Bogomolov, F. A. Surfaces of class ${VII}_{0}$ and affine geometry, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, Tome 46 (1982) no. 4, p. 710-761, 896 (English translation: Math. USSR Izv. 21 (1983), no. 1, 31–73) | MR 670164 | Zbl 0527.14029

[3] Braam, P. J.; Hurtubise, J. Instantons on Hopf surfaces and monopoles on solid tori, Journal für reine und angewandte Mathematik, Tome 400 (1989), pp. 146-172 | MR 1013728 | Zbl 0669.32012

[4] Brînzănescu, V.; Moraru, R. Stable bundles on non-Kähler elliptic surfaces, Communications in Mathematical Physics, Tome 254 (2005) no. 3, pp. 565-580 | Article | MR 2126483 | Zbl 1071.32009

[5] Buchdahl, N. P. Stable $2$ -bundles on Hirzebruch surfaces, Mathematische Zeitschrift, Tome 194 (1987) no. 1, pp. 143-152 | Article | MR 871226 | Zbl 0627.14028

[6] Buchdahl, N. P. A Nakai-Moishezon criterion for non-Kähler surfaces, Université de Grenoble. Annales de l’Institut Fourier, Tome 50 (2000) no. 5, pp. 1533-1538 | Article | Numdam | Zbl 0964.32014

[7] Dloussky, G. Structure des surfaces de Kato, Mémoires de la Société Mathématique de France. Nouvelle Série, Tome 112 (1984) no. 14, pp. 1-120 | Numdam | MR 763959 | Zbl 0543.32012

[8] Dloussky, G.; Oeljeklaus, K.; Toma, M. Class ${VII}_{0}$ surfaces with $b_{2}$ curves, The Tôhoku Mathematical Journal. Second Series, Tome 55 (2003) no. 2, pp. 283-309 | Article | MR 1979500 | Zbl 1034.32012

[9] Donagi, R. Y. Principal bundles on elliptic fibrations, The Asian Journal of Mathematics, Tome 1 (1997) no. 2, pp. 214-223 | MR 1491982 | Zbl 0927.14006

[10] Donaldson, S. K. Irrationality and the $h$ -cobordism conjecture, Journal of Differential Geometry, Tome 26 (1987) no. 1, pp. 141-168 | MR 892034 | Zbl 0631.57010

[11] Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1990) (Oxford Science Publications) | MR 1079726 | Zbl 0820.57002

[12] Elencwajg, G.; Forster, O. Vector bundles on manifolds without divisors and a theorem on deformations, Université de Grenoble. Annales de l’Institut Fourier, Tome 32 (1982) no. 4, p. 25-51 (1983) | Article | Numdam | Zbl 0488.32012

[13] Enoki, I. On surfaces of class ${VII}_{0}$ with curves, Japan Academy. Proceedings. Series A. Mathematical Sciences, Tome 56 (1980) no. 6, pp. 275-279 | Article | MR 581470 | Zbl 0462.32012

[14] Enoki, I. Surfaces of class ${VII}_{0}$ with curves, The Tôhoku Mathematical Journal. Second Series, Tome 33 (1981) no. 4, pp. 453-492 | Article | MR 643229 | Zbl 0476.14013

[15] Friedman, R. Rank two vector bundles over regular elliptic surfaces, Inventiones Mathematicae, Tome 96 (1989), pp. 283-332 | Article | MR 989699 | Zbl 0671.14006

[16] Friedman, R.; Morgan, J.; Witten, E. Vector bundles over elliptic fibrations, Journal of Algebraic Geometry, Tome 8 (1999) no. 2, pp. 279-401 | MR 1675162 | Zbl 0937.14004

[17] Friedman, R.; Morgan, J. W. Smooth four-manifolds and complex surfaces, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 27 (1994) | MR 1288304 | Zbl 0817.14017

[18] Gauduchon, P. La $1$ -forme de torsion d’une variété hermitienne compacte, Mathematische Annalen, Tome 267 (1984) no. 4, pp. 495-518 | Article | Zbl 0523.53059

[19] Griffiths, P.; Harris, J. Principles of Algebraic Geometry, John Wiley & Sons Inc., New York, Wiley Classics Library (1994) | MR 1288523 | Zbl 0836.14001

[20] Inoue, M. On surfaces of class ${VII}_{0}$ , Inventiones Mathematicae, Tome 24 (1974), pp. 269-310 | Article | MR 342734 | Zbl 0283.32019

[21] Inoue, M. New surfaces with no meromorphic functions. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), pp. 91-106 | MR 442297 | Zbl 0365.14011

[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 0421.32010

[23] Kobayashi, S. Differential geometry of complex vector bundles, Princeton University Press, Princeton, NJ, Publications of the Mathematical Society of Japan, Tome 15 (1987) | MR 909698 | Zbl 0708.53002

[24] Kodaira, K. On the structure of compact complex analytic surfaces. I, American Journal of Mathematics, Tome 86 (1964), pp. 751-798 | Article | MR 187255 | Zbl 0137.17501

[25] Kodaira, K. On the structure of compact complex analytic surfaces. II, American Journal of Mathematics, Tome 88 (1966), pp. 682-721 | Article | MR 205280 | Zbl 0193.37701

[26] Kotschick, D. On manifolds homeomorphic to $C P^{2} # 8 {\bar{C P}}^{2}$ , Inventiones Mathematicae, Tome 95 (1989) no. 3, pp. 591-600 | Article | MR 979367 | Zbl 0691.57008

[27] Li, J. Algebraic geometric interpretation of Donaldson’s polynomial invariants, Journal of Differential Geometry, Tome 37 (1993) no. 2, pp. 417-466 | Zbl 0809.14006

[28] Li, J.; Yau, S. T.; Zheng, F. On projectively flat Hermitian manifolds, Communications in Analysis and Geometry, Tome 2 (1994) no. 1, pp. 103-109 | MR 1312680 | Zbl 0837.53053

[29] Lübke, M.; Teleman, A. The Kobayashi-Hitchin correspondence, World Scientific Publishing Co. Inc., River Edge, NJ (1995) | MR 1370660 | Zbl 0849.32020

[30] Moraru, R. Integrable systems associated to a Hopf surface, Canadian Journal of Mathematics, Tome 55 (2003) no. 3, pp. 609-635 | Article | MR 1980616 | Zbl 1058.37043

[31] Nakamura, I. On surfaces of class ${VII}_{0}$ with curves, Inventiones Mathematicae, Tome 78 (1984) no. 3, pp. 393-443 | Article | MR 768987 | Zbl 0575.14033

[32] Okonek, C.; De Ven, A. Van Stable bundles and differentiable structures on certain elliptic surfaces, Inventiones Mathematicae, Tome 86 (1986) no. 2, pp. 357-370 | Article | MR 856849 | Zbl 0613.14018

[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, Tome 95 (1989) no. 3, pp. 601-614 | Article | Zbl 0691.57007

[34] Teleman, A. Projectively flat surfaces and Bogomolov’s theorem on class ${VII}_{0}$ surfaces, International Journal of Mathematics, Tome 5 (1994) no. 2, pp. 253-264 | Article | Zbl 0803.53038

[35] Teleman, A. Moduli spaces of stable bundles on non-Kählerian elliptic fibre bundles over curves, Expositiones Mathematicae. International Journal, Tome 16 (1998) no. 3, pp. 193-248 | MR 1630918 | Zbl 0933.14022

[36] Teleman, A. Donaldson theory on non-Kählerian surfaces and class VII surfaces with $b_{2} = 1$ , Inventiones Mathematicae, Tome 162 (2005) no. 3, pp. 493-521 | Article | MR 2198220 | Zbl 1093.32006

[37] Teleman, A. The pseudo-effective cone of a non-Kählerian surface and applications, Mathematische Annalen, Tome 335 (2006) no. 4, pp. 965-989 | Article | MR 2232025 | Zbl 1096.32011

[38] Teleman, A. Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds, Geometry and Topology, Tome 11 (2007), pp. 1681-1730 | Article | MR 2350464 | Zbl 1138.57030 | Zbl pre05220950

[39] Teleman, A. Instantons and curves on class VII surfaces (2007) (arXiv:0704.2634) | Zbl 1231.14028

[40] Toma, M. Compact moduli spaces of stable sheaves over non-algebraic surfaces, Documenta Mathematica, Tome 6 (2001), pp. 11-29 ((Electronic)) | MR 1816525 | Zbl 0973.32007

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