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.

• ${\displaystyle P_{n}=\dim H^{0}(K^{n}),n\geqslant 1}$ 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.

• ${\displaystyle \kappa }$ is the Kodaira dimension: it is ${\displaystyle -\infty }$ (sometimes written −1) if the plurigenera are all 0, and is otherwise the smallest number (0, 1, or 2 for surfaces) such that ${\displaystyle P_{n}/n^{\kappa }}$ is bounded. Enriques did not use this definition: instead he used the values of ${\displaystyle P_{12}}$ and ${\displaystyle K\cdot K=c_{1}^{2}}$. These determine the Kodaira dimension given the following correspondence:

${\displaystyle {\begin{aligned}\kappa =-\infty &\longleftrightarrow P_{12}=0\\\kappa =0&\longleftrightarrow P_{12}=1\\\kappa =1&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K=0\\\kappa =2&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K>0\\\end{aligned}}}$

• ${\displaystyle h^{i,j}=\dim H^{j}(X,\Omega ^{i}),}$ where ${\displaystyle \Omega ^{i}}$ is the sheaf of holomorphic i-forms, are the Hodge numbers, often arranged in the Hodge diamond:

${\displaystyle {\begin{matrix}&&h^{0,0}&&\\&h^{1,0}&&h^{0,1}&\\h^{2,0}&&h^{1,1}&&h^{0,2}\\&h^{2,1}&&h^{1,2}&\\&&h^{2,2}&&\\\end{matrix}}}$

By Serre duality ${\displaystyle h^{i,j}=h^{2-i,2-j}}$ and ${\displaystyle h^{0,0}=h^{2,2}=1.}$ The Hodge numbers of a complex surface depend only on the oriented real cohomology ring of the surface, and are invariant under birational transformations except for ${\displaystyle h^{1,1}}$ which increases by 1 under blowing up a single point.

• If the surface is Kähler then ${\displaystyle h^{i,j}=h^{j,i}}$ and there are only three independent Hodge numbers.
• If the surface is compact then ${\displaystyle h^{1,0}}$ equals ${\displaystyle h^{0,1}}$ or ${\displaystyle h^{0,1}-1.}$

Invariants related to Hodge numbers

There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows:

• Betti numbers: defined by ${\displaystyle b_{i}=\dim H^{i}(S),0\leqslant i\leqslant 4.}$

${\displaystyle {\begin{cases}b_{0}=b_{4}=1\\b_{1}=b_{3}=h^{1,0}+h^{0,1}=h^{2,1}+h^{1,2}\\b_{2}=h^{2,0}+h^{1,1}+h^{0,2}\end{cases}}}$

In characteristic p > 0 the Betti numbers are defined using l-adic cohomology and need not satisfy these relations.

• Euler characteristic or Euler number:

${\displaystyle e=b_{0}-b_{1}+b_{2}-b_{3}+b_{4}.}$

• The irregularity is defined as the dimension of the Picard variety and the Albanese variety and denoted by q. For complex surfaces (but not always for surfaces of prime characteristic)

${\displaystyle q=h^{0,1}.}$

• The geometric genus:

${\displaystyle p_{g}=h^{0,2}=h^{2,0}=P_{1}.}$

• The arithmetic genus:

${\displaystyle p_{a}=p_{g}-q=h^{0,2}-h^{0,1}.}$

• The holomorphic Euler characteristic of the trivial bundle (usually differs from the Euler number e defined above):

${\displaystyle \chi =p_{g}-q+1=h^{0,2}-h^{0,1}+1.}$

By Noether's formula it is also equal to the Todd genus ${\displaystyle {\tfrac {1}{12}}(c_{1}^{2}+c_{2}).}$

• The signature of the second cohomology group for complex surfaces is denoted by ${\displaystyle \tau }$:

${\displaystyle \tau =4\chi -e=\sum \nolimits _{i,j}(-1)^{j}h^{i,j}.}$

• ${\displaystyle b^{\pm }}$are the dimensions of the maximal positive and negative definite subspaces of ${\displaystyle H^{2},}$ so:

${\displaystyle {\begin{cases}b^{+}+b^{-}=b_{2}\\b^{+}-b^{-}=\tau \end{cases}}}$

• c₂ = e and ${\displaystyle c_{1}^{2}=K^{2}=12\chi -e}$ 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(X) of divisors modulo linear equivalence, its quotient the Néron–Severi group NS(X) 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.

Rational surfaces

Rational surface 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. (The Hirzebruch surface Σ_n is the P¹ bundle over P¹ associated to the sheaf O(0) + O(n). The surface Σ₀ is isomorphic to P¹ × P¹, and Σ₁ is isomorphic to P² 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		0	(Projective plane)
	0		0
		1

		1
	0		0
0		2		0	(Hirzebruch surfaces)
	0		0
		1

Examples: P², P¹ × 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. (The ones of genus 0 are the Hirzebruch surfaces and are 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. A ruled surface not isomorphic to P¹ × P¹ has a unique ruling (P¹ × P¹ 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 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:

		1
	0		1
0		b2		0
	1		0
		1

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 H²(X, Z) is isomorphic to the unique even unimodular lattice II_3,19 of dimension 22 and signature −16.

Hodge diamond:

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 H²(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 S¹ × S¹ × S¹ × S¹ 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 C² 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 complex 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 extra families of Enriques surfaces called singular and supersingular Enriques surfaces; see the article on Enriques surfaces for details.

Hyperelliptic (or bielliptic) surfaces

Over the complex numbers 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 seven families of such surfaces.

Hodge diamond:

		1
	1		1
0		2		0
	1		1
		1

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.