Enriques–Kodaira classification
In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space.
For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
Max Noether began the systematic study of algebraic surfaces, and Guido Castelnuovo proved important parts of the classification.
Federigo Enriques (1914, 1949) described the classification of complex projective surfaces.
Kunihiko Kodaira (1964, 1966, 1968a, 1968b) later extended the classification to include non-algebraic compact surfaces.
The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the global spherical shell conjecture, still unproved in 2024).
For surfaces of general type not much is known about their explicit classification, though many examples have been found.
, and Igusa showed that even when they are equal they may be greater than the irregularity (the dimension of the Picard variety).
The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups.
The basic ones are the plurigenera and the Hodge numbers defined as follows: There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows: There are further invariants of compact complex surfaces that are not used so much in the classification.
By Castelnuovo's contraction theorem, this is equivalent to saying that X has no (−1)-curves (smooth rational curves with self-intersection number −1).
(In the more modern terminology of the minimal model program, a smooth projective surface X would be called minimal if its canonical line bundle KX is nef.
A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative.)
If q = 0 this argument does not work as the Albanese variety is a point, but in this case Castelnuovo's theorem implies that the surface is rational.
Any ruled surface is birationally equivalent to P1 × C for a unique curve C, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves.
Hodge diamond: These surfaces are classified by starting with Noether's formula
Most solutions to these conditions correspond to classes of surfaces, as in the following table: These are the minimal compact complex surfaces of Kodaira dimension 0 with q = 0 and trivial canonical line bundle.
Invariants: The second cohomology group H2(X, Z) is isomorphic to the unique even unimodular lattice II3,19 of dimension 22 and signature −16.
Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century.
They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles.
Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z.
These are the complex surfaces such that q = 0 and the canonical line bundle is non-trivial, but has trivial square.
Hodge diamond: Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.
Over the complex numbers these are quotients of a product of two elliptic curves by a finite group of automorphisms.
Hodge diamond: Over fields of characteristics 2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article on hyperelliptic surfaces for details.
An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1).
Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c21 and c2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers.
Invariants: There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy: Most pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
However, there is no known construction that can produce "typical" surfaces of general type for large Chern numbers; in fact it is not even known if there is any reasonable concept of a "typical" surface of general type.
