Surface of class VII

In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII₀. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times.

The name "class VII" comes from (Kodaira 1964, theorem 21), which divided minimal surfaces into 7 classes numbered I₀ to VII₀. However Kodaira's class VII₀ did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII₀ also included some other surfaces, such as secondary Kodaira surfaces, that are no longer considered to be class VII as they do not have Kodaira dimension −∞. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces in (Kodaira 1968, theorem 55).

Invariants

The irregularity q is 1, and h^1,0 = 0. All plurigenera are 0.

Hodge diamond:

		1
	0		1
0		b₂		0
	1		0
		1

Examples

Hopf surfaces are quotients of C²−(0,0) by a discrete group G acting freely, and have vanishing second Betti numbers. The simplest example is to take G to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to S¹×S³.

Inoue surfaces are certain class VII surfaces whose universal cover is C×H where H is the upper half plane (so they are quotients of this by a group of automorphisms). They have vanishing second Betti numbers.

Inoue–Hirzebruch surfaces, Enoki surfaces, and Kato surfaces give examples of type VII surfaces with b₂ > 0.

Classification and global spherical shells

The minimal class VII surfaces with second Betti number b₂=0 have been classified by Bogomolov (1976, 1982), and are either Hopf surfaces or Inoue surfaces. Those with b₂=1 were classified by Nakamura (1984b) under an additional assumption that the surface has a curve, that was later proved by Teleman (2005).

A global spherical shell (Kato 1978) is a smooth 3-sphere in the surface with connected complement, with a neighbourhood biholomorphic to a neighbourhood of a sphere in C². The global spherical shell conjecture claims that all class VII₀ surfaces with positive second Betti number have a global spherical shell. The manifolds with a global spherical shell are all Kato surfaces which are reasonably well understood, so a proof of this conjecture would lead to a classification of the type VII surfaces.

A class VII surface with positive second Betti number b₂ has at most b₂ rational curves, and has exactly this number if it has a global spherical shell. Conversely Georges Dloussky, Karl Oeljeklaus, and Matei Toma (2003) showed that if a minimal class VII surface with positive second Betti number b₂ has exactly b₂ rational curves then it has a global spherical shell.

For type VII surfaces with vanishing second Betti number, the primary Hopf surfaces have a global spherical shell, but secondary Hopf surfaces and Inoue surfaces do not because their fundamental groups are not infinite cyclic. Blowing up points on the latter surfaces gives non-minimal class VII surfaces with positive second Betti number that do not have spherical shells.

References

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
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Bogomolov, Fedor A. (1982), "Surfaces of class VII₀ and affine geometry", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 46 (4): 710–761, Bibcode:1983IzMat..21...31B, doi:10.1070/IM1983v021n01ABEH001640, ISSN 0373-2436, MR 0670164
Dloussky, Georges; Oeljeklaus, Karl; Toma, Matei (2003), "Class VII₀ surfaces with b₂ curves", The Tohoku Mathematical Journal, Second Series, 55 (2): 283–309, arXiv:math/0201010, doi:10.2748/tmj/1113246942, ISSN 0040-8735, MR 1979500
Kato, Masahide (1978), "Compact complex manifolds containing "global" spherical shells. I", Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Tokyo: Kinokuniya Book Store, pp. 45–84, MR 0578853
Kodaira, Kunihiko (1964), "On the structure of compact complex analytic surfaces. I", American Journal of Mathematics, 86 (4), The Johns Hopkins University Press: 751–798, doi:10.2307/2373157, ISSN 0002-9327, JSTOR 2373157, MR 0187255
Kodaira, Kunihiko (1968), "On the structure of complex analytic surfaces. IV", American Journal of Mathematics, 90 (4), The Johns Hopkins University Press: 1048–1066, doi:10.2307/2373289, ISSN 0002-9327, JSTOR 2373289, MR 0239114
Nakamura, Iku (1984a), "On surfaces of class VII₀ with curves", Inventiones Mathematicae, 78 (3): 393–443, Bibcode:1984InMat..78..393N, doi:10.1007/BF01388444, ISSN 0020-9910, MR 0768987
Nakamura, Iku (1984b), "Classification of non-Kähler complex surfaces", Mathematical Society of Japan. Sugaku (Mathematics), 36 (2): 110–124, ISSN 0039-470X, MR 0780359
Nakamura, I. (2008), "Survey on VII₀ surfaces", Recent Developments in NonKaehler Geometry, Sapporo (PDF)
Teleman, Andrei (2005), "Donaldson theory on non-Kählerian surfaces and class VII surfaces with b₂=1", Inventiones Mathematicae, 162 (3): 493–521, arXiv:0704.2638, Bibcode:2005InMat.162..493T, doi:10.1007/s00222-005-0451-2, ISSN 0020-9910, MR 2198220