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A two-dimensional (complex) [[Analytic manifold|analytic manifold]], i.e. a smooth four-dimensional manifold with a complex structure. While the theory of analytic surfaces forms part of the general theory of complex manifolds, the two-dimensional case is treated separately, since much more is known about analytic surfaces than about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124501.png" />-dimensional manifolds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124502.png" />. Moreover, certain facts are specific to the two-dimensional case alone. These results concern the classification of analytic surfaces, which is analogous to that of algebraic surfaces (cf. [[Algebraic surface|Algebraic surface]]) — a fact which largely reduces the theory of analytic surfaces to that of algebraic surfaces. The principal results on the classification of analytic surfaces were obtained by K. Kodaira [[#References|[1]]], , , but his work is based on the results of the classical Italian school of algebraic geometry on the classification of algebraic surfaces.
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A two-dimensional (complex) [[Analytic manifold|analytic manifold]], i.e. a smooth four-dimensional manifold with a complex structure. While the theory of analytic surfaces forms part of the general theory of complex manifolds, the two-dimensional case is treated separately, since much more is known about analytic surfaces than about $  n $-
 +
dimensional manifolds if $  n \geq  3 $.  
 +
Moreover, certain facts are specific to the two-dimensional case alone. These results concern the classification of analytic surfaces, which is analogous to that of algebraic surfaces (cf. [[Algebraic surface|Algebraic surface]]) — a fact which largely reduces the theory of analytic surfaces to that of algebraic surfaces. The principal results on the classification of analytic surfaces were obtained by K. Kodaira [[#References|[1]]], , , but his work is based on the results of the classical Italian school of algebraic geometry on the classification of algebraic surfaces.
  
 
All analytic surfaces discussed below are assumed to be compact and connected.
 
All analytic surfaces discussed below are assumed to be compact and connected.
  
 
===Examples.===
 
===Examples.===
 
  
 
1) Algebraic surfaces. Let
 
1) Algebraic surfaces. Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124503.png" /></td> </tr></table>
+
$$
 +
f _ {i} ( x _ {0} \dots x _ {N} ) ,
 +
\  i = 1 \dots m ,
 +
$$
  
be a set of homogeneous polynomials with complex coefficients. The closed subset of the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124504.png" /> specified by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124505.png" /> is an analytic surface if it is non-singular, connected and has complex dimension two. This is the basic example of an analytic surface.
+
be a set of homogeneous polynomials with complex coefficients. The closed subset of the complex projective space $  P  ^ {N} ( \mathbf C ) $
 +
specified by the equations $  f _ {i} (x) = 0 $
 +
is an analytic surface if it is non-singular, connected and has complex dimension two. This is the basic example of an analytic surface.
  
2) Complex tori. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124506.png" /> be the two-dimensional vector space over the field of complex numbers (as a vector space over the field of real numbers it is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124507.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124508.png" /> be a lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a0124509.png" />. The quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245010.png" /> is an analytic surface. Being a smooth manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245011.png" /> is diffeomorphic to a four-dimensional torus, but the complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245012.png" /> depends on the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245013.png" />. Complex tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245014.png" /> play an important role in analysis, since meromorphic functions on such tori are meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245015.png" /> and are periodic with period lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245016.png" />. Analytic surfaces of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245017.png" /> are not always algebraic. There also exist lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245018.png" /> such that there are no meromorphic functions at all (except for constants) on the corresponding torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245019.png" />. For specific examples of such tori see [[#References|[5]]].
+
2) Complex tori. Let $  \mathbf C  ^ {2} $
 +
be the two-dimensional vector space over the field of complex numbers (as a vector space over the field of real numbers it is isomorphic to $  \mathbf R  ^ {4} $)  
 +
and let $  \Gamma \simeq \mathbf Z  ^ {4} $
 +
be a lattice in $  \mathbf C  ^ {2} $.  
 +
The quotient space $  X = \mathbf C  ^ {2} / \Gamma $
 +
is an analytic surface. Being a smooth manifold, $  X $
 +
is diffeomorphic to a four-dimensional torus, but the complex structure on $  X $
 +
depends on the lattice $  \Gamma $.  
 +
Complex tori $  X = \mathbf C  ^ {2} / \Gamma $
 +
play an important role in analysis, since meromorphic functions on such tori are meromorphic functions on $  \mathbf C  ^ {2} $
 +
and are periodic with period lattice $  \Gamma $.  
 +
Analytic surfaces of the type $  \mathbf C  ^ {2} / \Gamma $
 +
are not always algebraic. There also exist lattices $  \Gamma $
 +
such that there are no meromorphic functions at all (except for constants) on the corresponding torus $  \mathbf C  ^ {2} / \Gamma $.  
 +
For specific examples of such tori see [[#References|[5]]].
  
3) Hopf surfaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245020.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245021.png" /> be a positive number. Consider the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245023.png" /> given by
+
3) Hopf surfaces. Let $  Y = \mathbf C  ^ {2} - \{ 0 \} $
 +
and let $  c $
 +
be a positive number. Consider the action of the group $  \mathbf Z $
 +
on $  Y $
 +
given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245024.png" /></td> </tr></table>
+
$$
 +
( z _ {1} , z _ {2} )  \rightarrow \
 +
( c  ^ {k} z _ {1} , c  ^ {k} z _ {2} ) ,
 +
\  k \in \mathbf Z .
 +
$$
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245025.png" /> acts discretely and without fixed points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245026.png" />, while the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245027.png" /> is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245028.png" />. The quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245029.png" /> has a natural structure of an analytic surface, and is called a Hopf surface.
+
The group $  \mathbf Z $
 +
acts discretely and without fixed points on $  Y $,  
 +
while the quotient space $  X = Y / \mathbf Z $
 +
is diffeomorphic to $  S  ^ {1} \times S  ^ {3} $.  
 +
The quotient space $  X $
 +
has a natural structure of an analytic surface, and is called a Hopf surface.
  
 
==Classification of analytic surfaces.==
 
==Classification of analytic surfaces.==
The principal invariant in the classification of analytic surfaces is the transcendence degree of the field of meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245030.png" /> on the analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245031.png" />. According to Siegel's theorem, for any compact connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245032.png" /> the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245033.png" /> is finitely generated, and its transcendence degree is not larger than the complex dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245034.png" />. Thus, for an analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245035.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245036.png" /> contains two independent meromorphic algebraic functions, or one such function or constants only. These possibilities lead to the following theorems.
+
The principal invariant in the classification of analytic surfaces is the transcendence degree of the field of meromorphic functions $  \mathbf C (X) $
 +
on the analytic surface $  X $.  
 +
According to Siegel's theorem, for any compact connected manifold $  X $
 +
the field $  \mathbf C (X) $
 +
is finitely generated, and its transcendence degree is not larger than the complex dimension of $  X $.  
 +
Thus, for an analytic surface $  X $,  
 +
the field $  \mathbf C (X) $
 +
contains two independent meromorphic algebraic functions, or one such function or constants only. These possibilities lead to the following theorems.
  
For any analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245037.png" /> to be an algebraic surface it is necessary and sufficient that there exist two algebraically independent meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245038.png" />.
+
For any analytic surface $  X $
 +
to be an algebraic surface it is necessary and sufficient that there exist two algebraically independent meromorphic functions on $  X $.
  
If an analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245039.png" /> has a field of meromorphic functions of transcendence degree 1, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245040.png" /> is an elliptic surface, i.e. there is a holomorphic mapping onto an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245042.png" />, such that
+
If an analytic surface $  X $
 +
has a field of meromorphic functions of transcendence degree 1, then $  X $
 +
is an elliptic surface, i.e. there is a holomorphic mapping onto an algebraic curve $  Y $,  
 +
$  P: X \rightarrow Y $,  
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245043.png" /></td> </tr></table>
+
$$
 +
P  ^ {*} \mathbf C ( Y )  = \mathbf C ( X )
 +
$$
  
and all fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245044.png" />, except for a finite number, are elliptic curves (the singular fibres may only have a very special form, which has been thoroughly studied).
+
and all fibres of $  P $,  
 +
except for a finite number, are elliptic curves (the singular fibres may only have a very special form, which has been thoroughly studied).
  
If no meromorphic functions other than constants exist on an analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245045.png" />, and no exceptional curves (cf. [[Exceptional subvariety|Exceptional subvariety]]) exist on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245046.png" />, then the first Betti number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245048.png" /> assumes only three values: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245049.png" /> or 0. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245051.png" /> is a complex torus, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245053.png" /> has a trivial canonical fibration. These analytic surfaces are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245054.png" />-surfaces. They are all mutually homeomorphic. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245055.png" /> has not been studied in detail, but certain examples of analytic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245056.png" /> are obtained by a generalization of the construction of Hopf surfaces.
+
If no meromorphic functions other than constants exist on an analytic surface $  X $,  
 +
and no exceptional curves (cf. [[Exceptional subvariety|Exceptional subvariety]]) exist on $  X $,  
 +
then the first Betti number $  b _ {1} $
 +
of $  X $
 +
assumes only three values: $  4, 1 $
 +
or 0. If $  b _ {1} = 4 $,  
 +
$  X $
 +
is a complex torus, and if $  b _ {1} = 0 $,  
 +
$  X $
 +
has a trivial canonical fibration. These analytic surfaces are called $  K3 $-
 +
surfaces. They are all mutually homeomorphic. The case $  b _ {1} = 1 $
 +
has not been studied in detail, but certain examples of analytic surfaces with $  b _ {1} = 1 $
 +
are obtained by a generalization of the construction of Hopf surfaces.
  
Analytic Kähler surfaces are not always algebraic. They are, however, algebraic if the square of their first [[Chern class|Chern class]] is positive. All analytic Kähler surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245057.png" /> are deformations of algebraic surfaces.
+
Analytic Kähler surfaces are not always algebraic. They are, however, algebraic if the square of their first [[Chern class|Chern class]] is positive. All analytic Kähler surfaces with $  b _ {1} > 0 $
 +
are deformations of algebraic surfaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kodaira,   ''Matematika'' , '''6''' : 6 (1962) pp. 3–17</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> K. Kodaira,   "On compact (complex) analytic surfaces, I" ''Ann. of Math.'' , '''71''' : 1 (1960) pp. 111–152</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> K. Kodaira,   "On compact (complex) analytic surfaces, II" ''Ann. of Math.'' , '''77''' : 3 (1963) pp. 563–626</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> K. Kodaira,   "On compact (complex) analytic surfaces, III" ''Ann. of Math.'' , '''78''' : 1 (1963) pp. 1–40</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact (complex) analytic surfaces II" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact (complex) analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact (complex) analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top"> K. Kodaira,   "On the structure of compact (complex) analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kodaira, ''Matematika'' , '''6''' : 6 (1962) pp. 3–17</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> K. Kodaira, "On compact (complex) analytic surfaces, I" ''Ann. of Math.'' , '''71''' : 1 (1960) pp. 111–152</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> K. Kodaira, "On compact (complex) analytic surfaces, II" ''Ann. of Math.'' , '''77''' : 3 (1963) pp. 563–626 {{MR|165541}} {{ZBL|0193.37701}} {{ZBL|0133.16505}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> K. Kodaira, "On compact (complex) analytic surfaces, III" ''Ann. of Math.'' , '''78''' : 1 (1963) pp. 1–40 {{MR|165541}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact (complex) analytic surfaces II" ''Amer. J. Math.'' , '''86''' (1964) pp. 751–798 {{MR|187255}} {{ZBL|0133.16505}} </TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact (complex) analytic surfaces II" ''Amer. J. Math.'' , '''88''' (1966) pp. 682–721 {{MR|205280}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact (complex) analytic surfaces III" ''Amer. J. Math.'' , '''90''' (1968) pp. 55–83 {{MR|228019}} {{ZBL|0193.37701}} </TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top"> K. Kodaira, "On the structure of compact (complex) analytic surfaces IV" ''Amer. J. Math.'' , '''90''' (1968) pp. 1048–1066 {{MR|239114}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The notion defined above is also called a complex-analytic surface, since one considers complex structures and the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012450/a01245058.png" />. If instead one considers real structures and the field of real numbers, one speaks of real-analytic surfaces. However, an analytic surface is always understood in the sense explained above.
+
The notion defined above is also called a complex-analytic surface, since one considers complex structures and the field of complex numbers $  \mathbf C $.  
 +
If instead one considers real structures and the field of real numbers, one speaks of real-analytic surfaces. However, an analytic surface is always understood in the sense explained above.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth,   C. Peters,   A. van der Ven,   "Compact complex surfaces" , Springer (1984)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) {{MR|0749574}} {{ZBL|0718.14023}} </TD></TR></table>

Latest revision as of 18:47, 5 April 2020


A two-dimensional (complex) analytic manifold, i.e. a smooth four-dimensional manifold with a complex structure. While the theory of analytic surfaces forms part of the general theory of complex manifolds, the two-dimensional case is treated separately, since much more is known about analytic surfaces than about $ n $- dimensional manifolds if $ n \geq 3 $. Moreover, certain facts are specific to the two-dimensional case alone. These results concern the classification of analytic surfaces, which is analogous to that of algebraic surfaces (cf. Algebraic surface) — a fact which largely reduces the theory of analytic surfaces to that of algebraic surfaces. The principal results on the classification of analytic surfaces were obtained by K. Kodaira [1], , , but his work is based on the results of the classical Italian school of algebraic geometry on the classification of algebraic surfaces.

All analytic surfaces discussed below are assumed to be compact and connected.

Examples.

1) Algebraic surfaces. Let

$$ f _ {i} ( x _ {0} \dots x _ {N} ) , \ i = 1 \dots m , $$

be a set of homogeneous polynomials with complex coefficients. The closed subset of the complex projective space $ P ^ {N} ( \mathbf C ) $ specified by the equations $ f _ {i} (x) = 0 $ is an analytic surface if it is non-singular, connected and has complex dimension two. This is the basic example of an analytic surface.

2) Complex tori. Let $ \mathbf C ^ {2} $ be the two-dimensional vector space over the field of complex numbers (as a vector space over the field of real numbers it is isomorphic to $ \mathbf R ^ {4} $) and let $ \Gamma \simeq \mathbf Z ^ {4} $ be a lattice in $ \mathbf C ^ {2} $. The quotient space $ X = \mathbf C ^ {2} / \Gamma $ is an analytic surface. Being a smooth manifold, $ X $ is diffeomorphic to a four-dimensional torus, but the complex structure on $ X $ depends on the lattice $ \Gamma $. Complex tori $ X = \mathbf C ^ {2} / \Gamma $ play an important role in analysis, since meromorphic functions on such tori are meromorphic functions on $ \mathbf C ^ {2} $ and are periodic with period lattice $ \Gamma $. Analytic surfaces of the type $ \mathbf C ^ {2} / \Gamma $ are not always algebraic. There also exist lattices $ \Gamma $ such that there are no meromorphic functions at all (except for constants) on the corresponding torus $ \mathbf C ^ {2} / \Gamma $. For specific examples of such tori see [5].

3) Hopf surfaces. Let $ Y = \mathbf C ^ {2} - \{ 0 \} $ and let $ c $ be a positive number. Consider the action of the group $ \mathbf Z $ on $ Y $ given by

$$ ( z _ {1} , z _ {2} ) \rightarrow \ ( c ^ {k} z _ {1} , c ^ {k} z _ {2} ) , \ k \in \mathbf Z . $$

The group $ \mathbf Z $ acts discretely and without fixed points on $ Y $, while the quotient space $ X = Y / \mathbf Z $ is diffeomorphic to $ S ^ {1} \times S ^ {3} $. The quotient space $ X $ has a natural structure of an analytic surface, and is called a Hopf surface.

Classification of analytic surfaces.

The principal invariant in the classification of analytic surfaces is the transcendence degree of the field of meromorphic functions $ \mathbf C (X) $ on the analytic surface $ X $. According to Siegel's theorem, for any compact connected manifold $ X $ the field $ \mathbf C (X) $ is finitely generated, and its transcendence degree is not larger than the complex dimension of $ X $. Thus, for an analytic surface $ X $, the field $ \mathbf C (X) $ contains two independent meromorphic algebraic functions, or one such function or constants only. These possibilities lead to the following theorems.

For any analytic surface $ X $ to be an algebraic surface it is necessary and sufficient that there exist two algebraically independent meromorphic functions on $ X $.

If an analytic surface $ X $ has a field of meromorphic functions of transcendence degree 1, then $ X $ is an elliptic surface, i.e. there is a holomorphic mapping onto an algebraic curve $ Y $, $ P: X \rightarrow Y $, such that

$$ P ^ {*} \mathbf C ( Y ) = \mathbf C ( X ) $$

and all fibres of $ P $, except for a finite number, are elliptic curves (the singular fibres may only have a very special form, which has been thoroughly studied).

If no meromorphic functions other than constants exist on an analytic surface $ X $, and no exceptional curves (cf. Exceptional subvariety) exist on $ X $, then the first Betti number $ b _ {1} $ of $ X $ assumes only three values: $ 4, 1 $ or 0. If $ b _ {1} = 4 $, $ X $ is a complex torus, and if $ b _ {1} = 0 $, $ X $ has a trivial canonical fibration. These analytic surfaces are called $ K3 $- surfaces. They are all mutually homeomorphic. The case $ b _ {1} = 1 $ has not been studied in detail, but certain examples of analytic surfaces with $ b _ {1} = 1 $ are obtained by a generalization of the construction of Hopf surfaces.

Analytic Kähler surfaces are not always algebraic. They are, however, algebraic if the square of their first Chern class is positive. All analytic Kähler surfaces with $ b _ {1} > 0 $ are deformations of algebraic surfaces.

References

[1] K. Kodaira, Matematika , 6 : 6 (1962) pp. 3–17
[2a] K. Kodaira, "On compact (complex) analytic surfaces, I" Ann. of Math. , 71 : 1 (1960) pp. 111–152
[2b] K. Kodaira, "On compact (complex) analytic surfaces, II" Ann. of Math. , 77 : 3 (1963) pp. 563–626 MR165541 Zbl 0193.37701 Zbl 0133.16505
[2c] K. Kodaira, "On compact (complex) analytic surfaces, III" Ann. of Math. , 78 : 1 (1963) pp. 1–40 MR165541
[3a] K. Kodaira, "On the structure of compact (complex) analytic surfaces II" Amer. J. Math. , 86 (1964) pp. 751–798 MR187255 Zbl 0133.16505
[3b] K. Kodaira, "On the structure of compact (complex) analytic surfaces II" Amer. J. Math. , 88 (1966) pp. 682–721 MR205280 Zbl 0193.37701
[3c] K. Kodaira, "On the structure of compact (complex) analytic surfaces III" Amer. J. Math. , 90 (1968) pp. 55–83 MR228019 Zbl 0193.37701
[3d] K. Kodaira, "On the structure of compact (complex) analytic surfaces IV" Amer. J. Math. , 90 (1968) pp. 1048–1066 MR239114
[4] "Algebraic surfaces" Trudy Mat. Inst. Steklov. , 75 (1965) (In Russian) Zbl 0154.33002 Zbl 0154.21001
[5] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

The notion defined above is also called a complex-analytic surface, since one considers complex structures and the field of complex numbers $ \mathbf C $. If instead one considers real structures and the field of real numbers, one speaks of real-analytic surfaces. However, an analytic surface is always understood in the sense explained above.

References

[a1] W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) MR0749574 Zbl 0718.14023
How to Cite This Entry:
Analytic surface (in algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_surface_(in_algebraic_geometry)&oldid=11330
This article was adapted from an original article by B.B. Venkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article