# Analytic surface (in algebraic geometry)

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.

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### 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

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.