Complex NonKähler Geometry pp 121161  Cite as
NonKählerian Compact Complex Surfaces
Abstract

The Enriques Kodaira classification for surfaces and the classes of nonKählerian surfaces,

Class VII surfaces and their general properties,

Kato surfaces: construction, classification and moduli.
In Sect. 3.4 we explain the main ideas and techniques used in the proofs of our results on the existence of cycles of curves on class VII surfaces with small b_{2}. Section 3.5 deals with criteria for the existence of smooth algebraic deformations of the singular surface obtained by contracting a cycle of rational curves in a minimal class VII surface. We included an Appendix in which we introduce several fundamental objects in nonKählerian complex geometry (the Picard group of a compact complex manifold, the Gauduchon degree, the KobayashiHitchin correspondence for line bundles, unitary flat line bundles), and we prove basic properties of these objects.
Notes
Acknowledgements
The author thanks Daniele Angella, Leandro Arosio, and Eleonora Di Nezza, the organizers of the “CIME School “Complex nonKähler geometry”, for the invitation to give a lecture series, and to submit a written version of my lectures for publication in the proceedings of the meeting. The author is grateful to Georges Dloussky for his constant help, encouragement and collaboration, and for his useful suggestions and comments on the text.
References
 [Ba]D. Barlet, Convexité de l’espace des cycles. Bull. Soc. Math. France 106, 373–397 (1978)MathSciNetCrossRefGoogle Scholar
 [BHPV]W. Barth, K. Hulek, Ch. Peters, A. Van de Ven, Compact Complex Surfaces (Springer, Berlin, 2004)CrossRefGoogle Scholar
 [Bu1]N. Buchdahl, HermitianEinstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280, 625–648 (1988)MathSciNetCrossRefGoogle Scholar
 [Bu2]N. Buchdahl, Algebraic deformations of compact Kähler surfaces. Math. Z. 253, 453–459 (2006)MathSciNetCrossRefGoogle Scholar
 [Bu3]N. Buchdahl, Algebraic deformations of compact Kähler surfaces II. Math. Z. 258, 493–498 (2008)MathSciNetCrossRefGoogle Scholar
 [Dl1]G. Dloussky, Structure des surfaces de Kato. Mémoires de la Société Mathématique de France, tome 14, 1–120 (1984)MathSciNetzbMATHGoogle Scholar
 [Dl2]G. Dloussky, Sur la classification des germes d’applications holomorphes contractantes. Math. Ann. 289(4), 649–661MathSciNetCrossRefGoogle Scholar
 [DlKo]G. Dloussky, F. Kohler, Classification of singular germs of mappings and deformations of compact surfaces of class VII_{0}. Ann. Polon. Math. LXX, 49–83 (1998)MathSciNetCrossRefGoogle Scholar
 [DlTe1]G. Dloussky, A. Teleman, Infinite bubbling in nonKählerian geometry. Math. Ann. 353(4), 1283–1314 (2012)MathSciNetCrossRefGoogle Scholar
 [DlTe2]G. Dloussky, A. Teleman, Smooth deformations of singular contractions of class VII surfaces. https://arxiv.org/pdf/1803.07631.pdf
 [DOT]G. Dloussky, K. Oeljeklaus, M. Toma, Class VII_{0} surfaces with b _{2} curves. Tohoku Math. J. 55, 283–309 (2003)MathSciNetCrossRefGoogle Scholar
 [En]I. Enoki, Surfaces of class VII_{0} with curves. Tohoku Math. J. 33, 453–492 (1981)MathSciNetCrossRefGoogle Scholar
 [Eng]Ph. Engel, A proof of Looijenga’s conjecture via integralaffine geometry. Ph.D. thesis, Columbia University, 2015. ISBN: 9781321695960. arXiv:1409.7676Google Scholar
 [Fa]Ch. Favre, Classification of 2dimensional contracting rigid germs and Kato surfaces. I. J. Math. Pures Appl. 79(5), 475–514 (2000)MathSciNetCrossRefGoogle Scholar
 [Gau]P. Gauduchon, Sur la 1forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)MathSciNetCrossRefGoogle Scholar
 [GHK]M. Gross, P. Hacking S. Keel, Mirror symmetry for log CalabiYau surfaces I. Publ. Math. IHES 122(1), 65–168 (2015)MathSciNetCrossRefGoogle Scholar
 [In]M. Inoue, New surfaces with no meromorphic functions. Proc. Int. Congr. Math. Vancouver 1974, 423–426 (1976)Google Scholar
 [Ka1]M. Kato, Compact complex manifolds containing “global” spherical shells. Proc. Jpn. Acad. 53(1), 15–16 (1977)MathSciNetCrossRefGoogle Scholar
 [Ka2]M. Kato, Compact complex manifolds containing “global” spherical shells. I, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo (1978), pp. 45–84Google Scholar
 [Ka3]M. Kato, On a certain class of nonalgebraic nonKähler compact complex manifolds, in Recent Progress of Algebraic Geometry in Japan. NorthHolland Mathematics Studies, vol. 73 (NorthHolland, Amsterdam, 1983), pp. 28–50CrossRefGoogle Scholar
 [Loo]E. Looijenga, Rational surfaces with an anticanonical cycle. Ann. Math. 114(2), 267–322 (1981)MathSciNetCrossRefGoogle Scholar
 [LT]M. Lübke, A. Teleman, The KobayashiHitchin Correspondence (World Scientific Publishing Co Pte Ltd, Singapore, 1995), 250 pp.CrossRefGoogle Scholar
 [Ma]M. Manetti, Normal degenerations of the complex projective plane. J. Reine Angew. Math. 419, 89–118 (1991)MathSciNetzbMATHGoogle Scholar
 [Na1]I. Nakamura, On surfaces of class VII_{0} surfaces with curves. Invent. Math. 78, 393–443 (1984)MathSciNetCrossRefGoogle Scholar
 [Na2]I. Nakamura, Towards classification of nonKählerian surfaces. Sugaku Expositions 2(2), 209–229 (1989)zbMATHGoogle Scholar
 [Na3]I. Nakamura, On surfaces of class VII_{0} surfaces with curves II. Tohoku Math. J. 42(4), 475–516 (1990)MathSciNetCrossRefGoogle Scholar
 [OT]K. Oeljeklaus, M. Toma, Logarithmic moduli spaces for surfaces of class VII. Math. Ann. 341(2), 323–345 (2008)MathSciNetCrossRefGoogle Scholar
 [Pl]R. Plantiko, Stable bundles with torsion Chern classes on nonKählerian elliptic surfaces. Manuscripta Math. 87, 527–543 (1995)MathSciNetCrossRefGoogle Scholar
 [Siu]Y.T. Siu, Every K3surface is Kähler. Invent. Math. 73, 139–150 (1983)MathSciNetCrossRefGoogle Scholar
 [Te1]A. Teleman, Projectively flat surfaces and Bogomolov’s theorem on class VII_{0}surfaces. Int. J. Math. 05(02), 253–264 (1994)CrossRefGoogle Scholar
 [Te2]A. Teleman, Donaldson Theory on nonKählerian surfaces and class VII surfaces with b _{2} = 1. Invent. Math. 162, 493–521 (2005)MathSciNetCrossRefGoogle Scholar
 [Te3]A. Teleman, Instantons and holomorphic curves on class VII surfaces. Ann. Math. 172, 1749–1804 (2010)MathSciNetCrossRefGoogle Scholar
 [Te4]A. Teleman, Introduction á la théorie de jauge, Cours Spécialisés, SMF (2012)Google Scholar
 [Te5]A. Teleman, Instanton moduli spaces on nonKählerian surfaces. Holomorphic models around the reduction loci. J. Geom. Phys. 91, 66–87 (2015)zbMATHGoogle Scholar
 [Te6]A. Teleman, Analytic cycles in flip passages and in instanton moduli spaces over nonKählerian surfaces. Int. J. Math. 27(07), 1640009 (2016)MathSciNetCrossRefGoogle Scholar
 [Te7]A. Teleman, A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces, in Geometry, Analysis and Probability: In Honor of JeanMichel Bismut. Progress in Mathematics (Birkhäuser, Basel, 2017)Google Scholar
 [Te8]A. Teleman, Towards the Classification of Class VII Surfaces, in Complex and Symplectic Geometry. Springer INdAM Series, vol. 21 (Springer, Cham, 2017)CrossRefGoogle Scholar
 [Te9]A. Teleman, Donaldson theory in nonKählerian geometry, in Modern Geometry: A Celebration of the Work of Simon Donaldson. Proceedings of Symposia in Pure Mathematics, vol. 99 (2018), pp. 363–392Google Scholar