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 described the classification of complex projective surfaces. Kunihiko Kodaira later extended the classification to include non-algebraic compact surfaces.

Statement of the classification

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, 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. For surfaces of general type not much is known about their explicit classification, though many examples have been found.

Invariants of surfaces

Hodge numbers and Kodaira dimension

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:

K is the canonical line bundle whose sections are the holomorphic 2-forms.
are called the plurigenera. They are birational invariants, i.e., invariant under blowing up. Using Seiberg–Witten theory, Robert Friedman and John Morgan showed that for complex manifolds they only depend on the underlying oriented smooth 4-manifold. For non-Kähler surfaces the plurigenera are determined by the fundamental group, but for Kähler surfaces there are examples of surfaces that are homeomorphic but have different plurigenera and Kodaira dimensions. The individual plurigenera are not often used; the most important thing about them is their growth rate, measured by the Kodaira dimension.
is the Kodaira dimension: it is if the plurigenera are all 0, and is otherwise the smallest number such that is bounded. Enriques did not use this definition: instead he used the values of and. These determine the Kodaira dimension given the following correspondence:
where is the sheaf of holomorphic i-forms, are the Hodge numbers, often arranged in the Hodge diamond:
Invariants related to Hodge numbers

There are many invariants that can be written as linear combinations of the Hodge numbers, as follows:

Betti numbers: defined by
Euler characteristic or Euler number:
The irregularity is defined as the dimension of the Picard variety and the Albanese variety and denoted by q. For complex surfaces
The geometric genus:
The arithmetic genus:
The holomorphic Euler characteristic of the trivial bundle :
The signature of the second cohomology group for complex surfaces is denoted by :
are the dimensions of the maximal positive and negative definite subspaces of so:
c₂ = e and are the Chern numbers, defined as the integrals of various polynomials in the Chern classes over the manifold.
Other invariants

There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the Picard group Pic of divisors modulo linear equivalence, its quotient the Néron–Severi group NS with rank the Picard number ρ, topological invariants such as the fundamental group π₁ and the integral homology and cohomology groups, and invariants of the underlying smooth 4-manifold such as the Seiberg–Witten invariants and Donaldson invariants.

Minimal models and blowing up

Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces.
Given any point on a surface, we can form a new surface by blowing up this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface X is called minimal if it cannot be obtained from another non-singular surface by blowing up a point. By Castelnuovo's contraction theorem, this is equivalent to saying that X has no -curves.
Every surface X is birational to a minimal non-singular surface, and this minimal non-singular surface is unique if X has Kodaira dimension at least 0 or is not algebraic. Algebraic surfaces of Kodaira dimension may be birational to more than one minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example, P¹ × P¹ blown up at a point is isomorphic to P² blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is enough to classify the minimal non-singular ones.

Surfaces of Kodaira dimension −∞

Algebraic surfaces of Kodaira dimension can be classified as follows. If q > 0 then the map to the Albanese variety has fibers that are projective lines so the surface is a ruled surface. 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.
For non-algebraic surfaces Kodaira found an extra class of surfaces, called type VII, which are still not well understood.

Rational surfaces

means surface birational to the complex projective plane P². These are all algebraic. The minimal rational surfaces are P² itself and the Hirzebruch surfaces Σ_n for n = 0 or n ≥ 2. + O
Invariants: The plurigenera are all 0 and the fundamental group is trivial.
Hodge diamond:
Examples: 'P², P'¹ × P¹ = Σ₀, Hirzebruch surfaces Σ_n, quadrics, cubic surfaces, del Pezzo surfaces, Veronese surface. Many of these examples are non-minimal.

Ruled surfaces of genus > 0

Ruled surfaces of genus g have a smooth morphism to a curve of genus g whose fibers are lines P¹. They are all algebraic. 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. A ruled surface not isomorphic to P¹ × P¹ has a unique ruling.
Invariants: The plurigenera are all 0.
Hodge diamond:
Examples: The product of any curve of genus > 0 with P¹.

Surfaces of class VII

These surfaces are never algebraic or Kähler. The minimal ones with b₂ = 0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue surfaces. Examples with positive second Betti number include Inoue-Hirzebruch surfaces, Enoki surfaces, and more generally Kato surfaces. The global spherical shell conjecture implies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces.
Invariants: q = 1, h^1,0 = 0. All plurigenera are 0.
'''Hodge diamond:'''

Surfaces of Kodaira dimension 0

These surfaces are classified by starting with Noether's formula For Kodaira dimension 0, K has zero intersection number with itself, so Using
we arrive at:
Moreover since κ = 0 we have:
combining this with the previous equation gives:
In general 2h^0,1 ≥ b₁, so three terms on the left are non-negative integers and there are only a few solutions to this equation.

For algebraic surfaces 2h^0,1 − b₁ is an even integer between 0 and 2p_g.
For compact complex surfaces 2h^0,1 − b₁ = 0 or 1.
For Kähler surfaces 2h^0,1 − b₁ = 0 and h^1,0 = h^0,1.

Most solutions to these conditions correspond to classes of surfaces, as in the following table:

b₂	b₁	h^0,1	p_g = h^0,2	h^1,0	h^1,1	Surfaces	Fields
22	0	0	1	0	20	K3	Any. Always Kähler over the complex numbers, but need not be algebraic.
10	0	0	0	0	10	Classical Enriques	Any. Always algebraic.
10	0	1	1			Non-classical Enriques	Only characteristic 2
6	4	2	1	2	4	Abelian surfaces, tori	Any. Always Kähler over the complex numbers, but need not be algebraic.
2	2	1	0	1	2	Hyperelliptic	Any. Always algebraic
2	2	1 or 2	0 or 1			Quasi-hyperelliptic	Only characteristics 2, 3
4	3	2	1	1	2	Primary Kodaira	Only complex, never Kähler
0	1	1	0	0	0	Secondary Kodaira	Only complex, never Kähler