Enriques–Kodaira Classification of Compact Complex Surfaces

Home , Kodaira dimension

© 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Enriques–Kodaira classification of compact complex surfaces – An Analysis *Ramesha. H.G. Asst Professor of Mathematics. Govt First Grade College, Tiptur.

Abstract

This paper attempts to study the Enriques–Kodaira classification a classification of compact complex surfaces into ten classes. 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. The classification of a collection of objects generally means that a list has been constructed with exactly one member from each isomorphism type among the objects, and that tools and techniques can effectively be used to identify any combinatorially given object with its unique representative in the list. Examples of mathematical objects which have been classified include the finite simple groups and 2- manifolds but not, for example, knots. A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.

It has been established that for given n the n-dimensional compact, connected complex manifolds X can be classified according to their Kodaira dimension kod(X), which can assume the values -00,0,1, ... , n. In the case n = 2 the surfaces in the classes kod(X) = -00 or kod(X) = 0, and to a lesser extent those with kod(X) = 1 can be classified in much more detail. Thus, starting from the rough classification by Kodaira dimension, surfaces are divided into ten classes. This classification is called the Enriques-Kodaira classification and is embodied in the following central result. The word "fibre space" will be used only for smooth surfaces fibred over smooth curves. Thus a fibre space is a triple (X , f , Y) , where X is a smooth surface, Y a smooth curve and f : X - Y a surjective morphism. The fibre space is called (relatively) minimal if no fibre contains an exceptional curve. It is called an elliptic fibre space if almost all fibres are elliptic curves. Sometimes a surface admitting at least one such elliptic fibration is called an elliptic surface. Algebraic surfaces X into four classes, mentioned in section ~, is the division according to the value of Kod(X) , which can be - - , 0 , 1 or 2 (we don’t need the interpretation given for the last two classes in section 1).

Key words: Elliptic Curve, Elliptic Surface, Effective Divisor, Kodaira Dimension.

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An Enriques surface is a smooth compact complex surface having irregularity and nontrivial canonical sheaf such that (Endraß). Such surfaces cannot be embedded in projective three-space, but there nonetheless exist transformations onto singular surfaces in projective three-space. There exists a family of such transformed surfaces of degree six which passes through each edge of a tetrahedron twice. A subfamily with tetrahedral symmetry is given by the two-parameter ( ) family of surfaces. For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.

For any positive integer , a distinct nonorientable surface can be produced by replacing disks with Möbius strips. In particular, replacing one disk with a Möbius strip produces a cross surface and replacing two disks produces the Klein bottle. The sphere, the -holed tori, and this sequence of nonorientable surfaces form a complete list of compact, boundaryless two-dimensional manifolds.

and the polynomial is a sphere with radius ,

(Endraß).

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"

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 j "i","j" *τ is the signature (of the second cohomology group) and is equal to 4χ−"e", which is Σi,j(−1) "h" . *"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. 2 2 *"c"2 = "e" and "c"1 = "K" = 12χ − "e" are

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

Objective:

This paper intends to explore classification that goes under the name of Enriques- Kodaira birational classification of compact complex surfaces. For each of these classes, the surfaces in the class can be parametrized by a moduli space

Blow-ups and birational maps

Let be a surface. The structure of rational maps between and any projective variety is simple by the following

Theorem 1 (Elimination of indeterminacy) Let be a rational map. Then there exists a composite of finite many blow-ups and a morphism such that .

Proof We have a bijection between nondegenerate rational maps and linear systems on of dimension which has no fixed components. We use this correspondence and induct on the dimension of the linear systems. Each blow-up drops the dimension and this process terminates using the intersection pairing. □

The structure of birational morphisms between surfaces is also rather simple. Suppose is a birational morphism of surfaces. Then can be decomposed as a sequence of blow-ups , and an isomorphism . Combining this fact with the elimination of indeterminacy, we obtain that

Theorem 2 Let be a birational map between surfaces. Then there exists morphisms and such that , where each of can be decomposed as a sequence of blow-ups and an isomorphism.

Neron-Severi groups and minimal models

The exponential exact sequence

induces a long exact sequence Hodge theory shows that is a lattice in , so the Picard variety is a complex torus (and indeed an abelian variety). Since , we obtain an exact sequence where is the image of .

Definition 1 The group is called the Neron-Severi group of .

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Since is finitely generated, the Neron-Severi group is also finitely generated. The rank of is called the Picard number of . The Neron-Severi group can also be described as the group of divisors on modulo algebraic equivalence ([3, V 1.7]).

The behavior of the Neron-Severi group under a blow-up can be easily seen.

Theorem 3 Let be a blow-up at a point with exceptional divisor . Then there is a canonical isomorphism .

Proof The canonical isomorphism given by descends to the Neron-Severi groups. □

In other words, a blow-up increases the Picard number by 1. Thus the Neron-Severi group gives us a canonical order on the set of isomorphism classes of surfaces birationally equivalent to a given surface .

Definition 2 Let . We say that dominates if there exists a birationally morphism (in particular, ). We say that is minimal if every birational morphism is an isomorphism.

In particular, every surface dominates a minimal surface since the Picard number is finite. Conversely, every surface is obtained by a sequence of blow-ups of a minimal surface. So the problem of birational classification boils down to classifying minimal surfaces.

We can characterize minimal surfaces as those without exceptional curves. By the very definition of blow-ups, an exceptional curve is isomorphic to and has self-intersection number . This is actually a useful numerical criterion of minimality of surfaces ([1, II. 17]).

Theorem 4 (Castelnuovo's contractibility criterion) Suppose a curve is isomorphic to with . Then is an exceptional curve on .

Ruled surfaces

In most cases, has a unique minimal element. However, the situation is not that simple for a large class of surfaces — ruled surfaces.

Definition 3 A surface is called ruled if it is birational to for a smooth projective curve.

As a little digression, we can calculate several birational invariants for ruled surfaces easily.

Definition 4 Let be any surface. We define

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 the irregularity . It is equal to by Hodge theory. It is the dimension of the Picard variety and the Albanese variety of . It is also the difference of the geometric genus and the arithmetic genus, hence its name.

 the geometric genus . It is equal to by Serre duality and by Hodge theory.

 the plurigenus ( ).

Theorem 5 Let be a ruled surface over . Then , and for all .

Proof Since these are birational invariants, we may assume . Using the isomorphism , we know that . Using the Künneth formula , we know that for all . □

Now let us step back to the problem of finding the minimal models of ruled surfaces. This is closely related to the notion of geometrically ruled surfaces.

Definition 5 A geometrically ruled surface over is a surface together with a smooth morphism whose fibers are isomorphic to .

The first thing to notice is that geometrically ruled surfaces form a subclass of ruled surfaces due to the following theorem ([1, III.4]).

Theorem 6 (Noether-Enriques) Let be a surface together with a smooth morphism . If for some point , is smooth over and is isomorphic to . Then there exists an open neighborhood of such that is a trivial bundle. In particular, every geometrically ruled surface is ruled.

Also from the Noether-Enriques theorem, we know that every geometrically ruled surface over admits local trivializations as a -bundle, thus they are are classified by the cohomology group . The exact sequence gives a long exact sequence

Since , classifies rank 2 vector bundle over and , one knows that every geometrically ruled surface over is actually -isomorphic to for some rank two vector bundle over . Moreover, such and are -isomorphic if and only if for some line bundle over .

Now let us see how geometrically ruled surfaces play a significant role in classifying minimal ruled surfaces.

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Proof Suppose is a minimal surface and is a birational map. Then by elimination of indeterminacy, we can find another surface fitting in the following diagram

where is the projection onto the first factor and is a composition of blow-ups. Suppose is the smallest such integer. If , let be the exceptional curve of the -th blow-up, then must be a point since is irrational by assumption. So we can eliminate the -th blow-up, which contradicts the minimality of . Therefore and is actually a morphism with its generic fiber isomorphic to . Hence ([1, III.8]) is a geometrically ruled surface over . □

In other words, for an irrational ruled surface, its minimal models are not unique and are classified by those projective bundles : the theory of rank two vector bundles over a curve is delicate, but more or less understood.

Rational surfaces and Castelnuovo's theorem

The ruled surfaces with base curve are called Hirzebruch surfaces. In particular, they are rational surfaces. Among these, the only geometrically ruled ones are ( ) since every vector bundle over is a direct sum of line bundles. The above classification of minimal models of ruled surfaces fails for Hirzebruch surfaces. A calculation of intersection numbers on ruled surface implies the following result ([1, IV.1]).

Proposition 1 If , then there is a unique irreducible curve on with negative self-intersection. Moreover, its self-intersection is .

It follows that 's are distinct and minimal for . However, it also follows that there is an irreducible curve on with . Hence by the uniqueness, coincides with the blow-up of at one point, hence is not minimal.

In order to find minimal models for rational surfaces, we need the following nontrivial fact ([1, V.6]). Notice that any rational surface satisfies ( ).

Lemma 1 Let be a minimal surface with . Then there exists a smooth rational curve on such that .

Now we can deduce the following classification of minimal models of rational surfaces.

Theorem 8 The minimal rational surfaces are the Hirzebruch surfaces ( ) and itself.

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© 2018 JETIR December 2018, Volume 5, Issue 12 www.jetir.org (ISSN-2349-5162) Proof Let be a minimal rational surfaces. By the lemma, there exists smooth curves on with the least nonnegative . Choose such a curve with the least , where is a hyperplane section of . Then using the minimality and Riemann-Roch, one can show that every divisor is a smooth rational curve. Since the linear system of curves of passing through with multiplicity has codimension in . We know that . Suppose , then for any , the exact sequence implies that has no base point and as . Therefore . When , the morphism is geometrically ruled over , hence is for some . When , each fiber of the morphism is the intersection of two distinct rational curves, hence a point. Therefore . □

A similar argument using above useful lemma implies the following numerical characterization of rational surfaces ([1, V.1]).

Theorem 9 (Castelnuovo's Rationality Criterion) Let be a surface with . Then is rational.

Castelnuovo's Rationality Criterion together with the usage of Albanese varieties will enable us to finally show the uniqueness of the minimal models of all non-ruled surfaces ([1, V.19]).

Theorem 10 Let be two minimal non-ruled surfaces. Then every birational map between and is an isomorphism. In particular, every non-ruled surface admits a unique minimal model.

Hence we have found a complete list of minimal surfaces by now.

Kodaira dimension

Castelnuovo's Rationality Criterion provides a handy numerical tool to distinguish rational surfaces from others. We would like to see how the birational invariants will help us classifying surfaces. This can be achieved for ruled surfaces as well ([1, VI.18]).

Theorem 11 (Enriques) Let be a surfaces with (or ). Then is ruled. In particular, for all .

In view of the important role played by the plurigenera, we introduce the notion of Kodaira dimension for a smooth projective variety.

Definition 6 Let be a smooth projective variety and be the rational map from to the projective space associated to the complete linear system . We define the Kodaira dimension of to be the maximal dimension of the images of for . We write if for .

In particular, the Kodaira dimension is always no greater than the dimension of . Some examples are in order.

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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 2009). For surfaces of general type not much is known about their explicit classification, though many examples have been found.

The classification of algebraic surfaces in positive characteristic (Mumford 1969, Mumford & Bombieri 1976, 1977) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite group schemes that are not étale.

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