Enriques-Kodaira classification

In mathematics, the Enriques-Kodaira classification is a classification of compact complex surfaces. For complex projective surfaces it was done by Federigo Enriques, and Kunihiko Kodaira later extended it to non-algebraic compact surfaces.

It has also been extended to surfaces in characteristic "p" > 0 by Enrico Bombieri and David Mumford.

tatement of the classification

The Enriques-Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surfaceis 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 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 classification.

Invariants of compact complex surfaces

Basic invariants

The most important invariants of a compact complex surfaces used in the classification can all be given in terms of the dimensions of various cohomology groups of coherent sheaves. The basic ones are the plurigenera and the Hodge numbers defined as follows:

*"K" is the canonical sheaf of holomorphic 2-forms.

*"P"_"n" = dim H⁰("K"^"n") for "n" ≥ 1 are the plurigenera. They are invariant under blowing up, and depend only on the underlying smooth 4-manifold.

*"h"^"i","j" = dim H^j("X", Ω^"i"), where Ω^"i" is the sheaf of holomorphic "i"-forms, are the Hodge numbers, often arranged in the Hodge diamond:

		"h"0,0
	"h"1,0		"h"0,1
"h"2,0		"h"1,1		"h"0,2
	"h"2,1		"h"1,2
		"h"2,2

By Serre duality "h"^"i","j" = "h"^"2-i","2-j", and "h"^0,0 = "h"^2,2 = 1. If the surface is algebraic then "h"^"i","j" = "h"^"j","i", so there are only 3 independent Hodge numbers.In general "h"^"1","0" is either "h"^"0","1" or "h"^"0","1" − 1.The first plurigenus "P"₁ is equal to the Hodge numbers "h"^2,0 and "h"^0,2,and is sometimes called the geometric genus. The Hodge numbers depend only on the orientation and real cohomology ring of the surface, and are invariant under birational transformations except for "h"^1,1 which increases by 1 under blowing up a single point.

The other invariants used in the classification can be expressed in terms of the plurigenera and Hodge numbers as follows.

Kodaira dimension from plurigenera

The individual plurigenera are not often used; the most important thing about them is their growth rate, measured by the Kodaira dimension:

* κ is the Kodaira dimension: it is −∞ (sometimes written −1) if the plurigenera are all 0, and is otherwise the smallest number (0, 1, or 2 for surfaces) such that "P"_"n"/"n"^κ is bounded.

Other invariants from Hodge numbers

There are a large number of invariants that are linear combinations of the Hodge numbers, as follows:

* "b"₀,"b"₁,"b"₂,"b"₃,"b"₄ are the Betti numbers: "b"_i = dim("H"_i("S")). "b"₀ = "b"₄ = 1 and "b"₁ = "b"₃ = "h"^"1","0" + "h"^"0","1" = "h"^"2","1" + "h"^"1","2" and "b"₂ = "h"^"2","0" + "h"^"1","1" + "h"^"0","2"

*"e" = "b"₀ − "b"₁ + "b"₂ − "b"₃ + "b"₄ is the Euler characteristic or Euler number.

*"q" = "h"^"0","1" is the irregularity, dimension of the Picard variety and Albanese variety.

*"p_g" = "h"^"0","2" = "h"^"2","0" = "P"₁is the geometric genus.

*"p_a" = "p_g"-"q" = "h"^"0","2" − "h"^"0","1" is the arithmetic genus.

*χ = "p_g" − "q" + 1 = "h"^"0","2" − "h"^"0","1" + 1 is the holomorphic Euler characteristic of the trivial bundle. (It should not be confused with the Euler number. Add minus signs to the rows for dimensions 1 and 3. This is the sum along any side of the diamond, while the topological Euler-Poincaré characteristic is the sum over the whole diamond.) By Noether's formula it is also equal to the Todd genus ("c"₁² + "c"₂)/12
*τ is the signature (of the second cohomology group) and is equal to 4χ−"e", which is Σ_i,j(−1)^j"h"^"i","j".
*"b"⁺ and "b"⁻ are the dimensions of the maximal positive and negative definite subspaces of "H"², so "b"⁺ + "b" ⁻ = "b"₂ and "b"⁺ − "b"⁻ = τ.
*"c"₂ = "e" and "c"₁² = "K"² = 12χ − "e" are the Chern numbers, defined as the integrals of various polynomials in the Chern classes over the manifold.

Friedman and Morgan proved that the invariants above depend only on the underlying smooth 4-manifold.

non-Hodge invariants

There are further invariants of compact complex surfaces that are not used so much in the classification. These include the Picard group Pic("X") of divisors modulo linear equivalence, the Néron-Severi group NS("X") which is the quotient of the Picard group by an abelian variety and its torsion elements, the Picard number ρ that is the rank of the Néron-Severi group, the fundamental group π₁, the torsion parts of the homology and cohomology groups, Seiberg-Witten invariants and Donaldson invariants of the underlying smooth 4-manifold, and various generalized cohomology groups.

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. A non-singular surface is called minimal if it cannot be obtained from another non-singular surface by blowing up a point. 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 1 minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example, "P"¹×"P"¹ blown up at a point is isomorphic to "P"² blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones.

urfaces 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 (if the surface is minimal) 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 Castelnovo'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

Rational surface means surface birational to the complex projective plane "P"². These are all algebraic. The minimal rational surfaces are "P"² itself and the Hirzebruch surfaces Σ_n for "n" = 0 or "n" ≥ 2;. (The Hirzebruch surface Σ_"n" is the "P"¹ bundle over "P"¹associated to the sheaf O(0)+O(n). The surface Σ₀ is isomorphic to "P"¹×"P"¹, and Σ₁ is isomorphic to "P"² blown up at a point so is not minimal.)

Invariants: The plurigenera are all 0 and the fundamental group is trivial.

Hodge diamond:

		1
	0		0
0		1+"n"		0
	0		0
		1

where "n" is 0 for the projective plane, and 1 for Hirzebruch surfaces(and greater than 1 for other non-minimal rational surfaces).

Examples: "P"², "P"¹×"P"¹ = 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 "P"¹.They are all algebraic. (The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent to "P"¹×"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 "P"¹×"P"¹ has a unique ruling ("P"¹×"P"¹ has two).

Invariants: The plurigenera are all 0.

Hodge diamond:

		1
	g		g
0		2		0
	g		g
		1

Examples: The product of any curve of genus > 0 with "P"¹.

urfaces of class VII

These surfaces are never algebraic. The ones with "b"₂=0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue surfaces. Not much is known about the ones with "b"₂>0, though some examples have been found.

Invariants: "q"=1, "h"^1,0 = 0. All plurigenera are 0.

Hodge diamond:

		1
	0		1
0		"b"2		0
	1		0
		1

Examples: Hopf surfaces are quotients of C²−(0,0) by a discrete group "G" acting freely. The simplest example is to take "G" to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to "S"¹×"S"³.

Inoue surfaces are certain surfaces whose universal cover is C×"H" where "H" is the upper half plane (so they are quotients of this by a group of automorphisms).

urfaces of Kodaira dimension 0

These surfaces are classified by starting with Noether's formula 12χ="c"₂ + "c"₁². For Kodaira dimension 0, "K" has zero intersection number with itself, so "c"₁² = 0. For Kähler surfaces we have "h"^1,0 = "h"^0,1. Using these facts and rewriting Noether's formula in terms of Hodge numbers, we find that 10 + 10"h"^2,0 = 8"h"^1,0 + "h"^1,1. Moreover "h"^2,0 is either 1 (if "K" = 0) or 0 (otherwise) as κ is 0. As all the Hodge numbers are non-negative integers, this implies that there are only five possibilities for "h"^2,0, "h"^1,0, and "h"^1,1.

"h"1,0	"h"2,0	"h"1,1	Type
0	0	10	Enriques
1	0	2	Hyperelliptic
0	1	20	K3
1	1	12	non-classical Enriques surfaces (only possible in characteristic 2)
2	1	4	Tori

For surfaces of positive characteristic, or non-Kähler surfaces, there are a few extra possibilities for the Hodge numbers.

K3 surfaces

These are the minimal compact complex surfaces of Kodaira dimension 0 with "q" = 0 and trivial canonical line bundle. They are all Kähler manifolds. All K3 surfaces are diffeomorphic, and their diffeomorphism class is an important example of a smooth spin simply connected 4-manifold.

Invariants: The second cohomology group "H"²("X", Z) is isomorphic to the unique even unimodular lattice II_3,19 of dimension 22 and signature −16.

Hodge diamond:

		1
	0		0
1		20		1
	0		0
		1

Examples:
*Degree 4 hypersurfaces in P³(C)
*Kummer surfaces. These are obtained by "quotienting out" an abelian surface by the automorphism "a" → −"a", then blowing up the 16 singular points.

A marked K3 surface is a K3 surface together with an isomorphism from II_3,19 to "H"²("X", Z). The moduli space of marked K3 surfaces is connected non-Hausdorff smooth analytic space of dimension 20. The algebraic K3 surfaces form a countable collection of 19-dimensional subvarieties of it.

Abelian surfaces and 2 dimensional complex tori

The two-dimensional complex tori include the abelian surfaces. One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional abelian varieties.Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century.

Invariants: The plurigenera are all 1. The surface is diffeomorphic to "S"¹×"S"¹×"S"¹×"S"¹ so the fundamental group is Z⁴.

Hodge diamond:

		1
	2		2
1		4		1
	2		2
		1

Examples: A product of two elliptic curves. The Jacobian of a genus 2 curve. Any quotient of C⁴ by a lattice.

Kodaira surfaces

These are never algebraic, though they have non-constant meromorphic functions. 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. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.

Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order "k"=1,2,3,4,6, then the plurigenera "P"_n are 1 if "n" is divisible by "k" and 0 otherwise.

Hodge diamond:

		1
	1		2
1		2		1	(Primary)
	2		1
		1

		1
	0		1
0		0		0	(Secondary)
	1		0
		1

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".This gives a primary Kodaira surface.

Enriques surfaces

These are the surfaces such that "q" = 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and therefore Kähler). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces.

Invariants: The plurigenera "P"_n are 1 if "n" is even and 0 if "n" is odd. The fundamental group has order 2. The second cohomology group H²("X", Z) is isomorphic to the sum of the unique even unimodular lattice II_1,9 of dimension 10 and signature -8 and a group of order 2.

Hodge diamond:

		1
	0		0
0		10		0
	0		0
		1

Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.

In characteristic 2 there are some new families of Enriques surfaces.

Hyperelliptic (or bielliptic) surfaces

These are quotients of a product of two elliptic curves by a finite group of automorphisms. The finite group can be Z/2Z, Z/2Z+Z/2Z, Z/3Z, Z/3Z+Z/3Z,Z/4Z, Z/4Z+Z/2Z, or Z/6Z, giving 7 families of such surfaces.

Hodge diamond:

		1
	1		1
0		2		0
	1		1
		1

urfaces of Kodaira dimension 1 (proper elliptic surfaces)

Elliptic surfaces are those that have elliptic fibrations (smooth holomorphic maps to another curve whose fibers are almost all elliptic curves), and are called proper if the base curve has genus at least 2. They can also be thought of as elliptic curves over a field of meromorphic functions on the base curve, and their theory closely resembles the theory of elliptic curves over a ring of algebraic or "p"-adic numbers.

Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular Kodaira gave a complete list of the possible singular fibers.

Some of the surfaces of Kodaira dimension less than 1 are (improper) elliptic surfaces; in particular, all Enriques surfaces, all hyperelliptic surfaces, all Kodaira surfaces, some K3 surfaces, some abelian surfaces, and some rational surfacesare elliptic surfaces.

Invariants: "c"₁² = 0, "c"₂≥ 0.

Examples: Any product of an elliptic curve with a curve of genus at least 2.

In finite characteristic one can also get quasi-elliptic surfaces, whose fibers may almost all be "degenerate elliptic curves".

=Surfaces of Kodaira dimension 2 (surfaces of general type)=

These are all algebraic, and in some sense most surfaces are in this class. 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 "c"₁² and "c"₂, there is a quasi-projective variety classifying the surfaces of general type with those Chern numbers. However it is a very difficult problem to describe these varieties explicitly, and there are very few pairs of Chern numbers for which this has been done (except when the variety is empty!)

Invariants: There are several conditions that the Chern numbers of a minimal surface of general type must satisfy:
*"c"₁² > 0, "c"₂ > 0
*"c"₁² ≤ 3"c"₂ (proved by Miyaoka and Yau)
*5"c"₁² − "c"₂ + 36 ≥ 0 (the Noether inequality)
*"c"₁² + "c"₂ is divisible by 12.

Most pairs of integers satisfying these conditions are the Chern numbers for some surface of general type.

Examples: The simplest examples are the product of two curves of genus at least 2, and a hypersurface of degree at least 5 in "P"³. There are a large number of other constructions known. 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. Hirzebruch has undertaken a large program of research on Hilbert modular surfaces (cusp singularities resolved, parameters a real quadratic field and a level) in pursuit of a 'definitive' result, namely that almost all are of general type.

urfaces in characteristic "p" > 0

The classification of algebraic surfaces in positive characteristics is very similar to that of algebraic surfaces in characteristic 0, except for some extra families of Enriques surfaces in characteristic 2, and some slightly different families of hyperelliptic surfaces in characteristics 2 and 3. These extra families can be understood as follows. In characteristic 0 the surfaces are the quotients of K3 or abelian surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite group schemes that are not discrete finite groups.

Oscar Zariski constructed some surfaces in positive characteristic that are unirational but not rational, derived from inseparable extensions (Zariski surfaces). Serre showed that "h"⁰(Ω) may differ from "h"¹("O"), and Igusa that they may be equal but still exceed the irregularity defined as the dimension of the Picard variety.

ee also

*list of algebraic surfaces

References

*citation|title=Compact Complex Surfaces|first= Wolf P.|last1= Barth|first2=Klaus|last2= Hulek|first3= Chris A.M. |last3=Peters|first4= Antonius |last4=Van de Ven |ISBN=3-540-00832-2|year=2004 This is the standard reference book for compact complex surfaces.

*citation|title=Complex algebraic surfaces|first= Arnaud |last=Beauville| ISBN= 0-521-49842-2|year=1996. This gives a more elementary introduction to the classification.