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Surface of class VII

{{Short description|Part of the Kodaira classification}}

In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by {{harvs|last=Kodaira |year1=1964|year2=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

{{harv|Kodaira|1964|loc=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 {{harv|Kodaira|1968|loc=theorem 55}}.

Invariants

The irregularity q is 1, and h1,0 = 0. All plurigenera are 0.

Hodge diamond:

{{Hodge diamond|style=font-weight:bold

| 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 {{harvs|txt|last=Bogomolov|year1=1976|year2=1982}}, and are either Hopf surfaces or Inoue surfaces. Those with b2=1 were classified by {{harvtxt|Nakamura|1984b}} under an additional assumption that the surface has a curve, that was later proved by {{harvtxt|Teleman|2005}}.

A global spherical shell {{Harv|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

{{harvs |txt |last1=Dloussky |first1=Georges |last2=Oeljeklaus |first2=Karl |last3=Toma |first3=Matei |year=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

  • {{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4}}
  • {{Citation | last1=Bogomolov | first1=Fedor A. | title=Classification of surfaces of class VII0 with b2=0 | url=http://www.turpion.org/php/paper.phtml?journal_id=im&paper_id=1688 | mr=0427325 | year=1976 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=10 | issue=2 | pages=273–288}}
  • {{Citation | last1=Bogomolov | first1=Fedor A. | title=Surfaces of class VII0 and affine geometry | doi=10.1070/IM1983v021n01ABEH001640 | mr=670164 | year=1982 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=46 | issue=4 | pages=710–761| bibcode=1983IzMat..21...31B }}
  • {{Citation | last1=Dloussky | first1=Georges | last2=Oeljeklaus | first2=Karl | last3=Toma | first3=Matei | title=Class VII0 surfaces with b2 curves | doi=10.2748/tmj/1113246942 | mr=1979500 | year=2003 | journal=The Tohoku Mathematical Journal |series=Second Series | issn=0040-8735 | volume=55 | issue=2 | pages=283–309| arxiv=math/0201010 | url=http://projecteuclid.org/euclid.tmj/1113246942 }}
  • {{Citation | last1=Kato | first1=Masahide | title=Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) | publisher=Kinokuniya Book Store | location=Tokyo | mr=578853 | year=1978 | chapter=Compact complex manifolds containing "global" spherical shells. I | pages=45–84}}
  • {{Citation | authorlink=Kunihiko Kodaira | last1=Kodaira | first1=Kunihiko | title=On the structure of compact complex analytic surfaces. I | jstor=2373157 | mr=0187255 | year=1964 | journal=American Journal of Mathematics | issn=0002-9327 | volume=86 | pages=751–798 | doi=10.2307/2373157 | issue=4 | publisher=The Johns Hopkins University Press}}
  • {{Citation | last1=Kodaira | first1=Kunihiko | title=On the structure of complex analytic surfaces. IV | jstor=2373289 | mr=0239114 | year=1968 | journal=American Journal of Mathematics | issn=0002-9327 | volume=90 | pages=1048–1066 | doi=10.2307/2373289 | issue=4 | publisher=The Johns Hopkins University Press}}
  • {{Citation | last1=Nakamura | first1=Iku | title=On surfaces of class VII0 with curves | doi=10.1007/BF01388444 | mr=768987 | year=1984a | journal=Inventiones Mathematicae | issn=0020-9910 | volume=78 | issue=3 | pages=393–443| bibcode=1984InMat..78..393N}}
  • {{Citation | last1=Nakamura | first1=Iku | title=Classification of non-Kähler complex surfaces | mr=780359 | year=1984b | journal=Mathematical Society of Japan. Sugaku (Mathematics) | issn=0039-470X | volume=36 | issue=2 | pages=110–124}}
  • {{citation|first=I. |last=Nakamura|url=http://www.math.sci.hokudai.ac.jp/~nakamura/70surfaces080306.pdf |chapter=Survey on VII0 surfaces|title=Recent Developments in NonKaehler Geometry, Sapporo |year=2008}}
  • {{Citation | last1=Teleman | first1=Andrei | title=Donaldson theory on non-Kählerian surfaces and class VII surfaces with b2=1 | doi=10.1007/s00222-005-0451-2 | mr=2198220 | year=2005 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=162 | issue=3 | pages=493–521| arxiv=0704.2638 | bibcode=2005InMat.162..493T }}

Category:Complex surfaces