View source for Analytic surface (in algebraic geometry)

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.

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

===Examples.===


1) Algebraic surfaces. Let

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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.

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]]].

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

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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.

==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.

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" />.

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

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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).

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.

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.

====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 {{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====
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.

====References====
<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>