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Introduction to Algebraic Surfaces Lecture Notes for the Course at the University of Mainz

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Introduction to algebraic surfaces Lecture Notes for the course at the University of Mainz Wintersemester 2009/2010 Arvid Perego (preliminary draft) October 30, 2009 2 Contents Introduction 5 1 Background material 9 1.1 Complex and projective manifolds . 9 1.1.1 Complex manifolds . 10 1.1.2 The algebraic world . 13 1.2 Vector bundles . 15 1.3 Sheaves and cohomology . 18 1.3.1 Sheaves . 18 1.3.2 Cohomology . 25 1.3.3 Cohomology of coherent sheaves . 29 1.3.4 The GAGA Theorem . 33 1.4 Hodge theory . 35 1.4.1 Harmonic forms . 35 1.4.2 K¨ahlermanifolds and Hodge decomposition . 38 2 Line bundles 41 2.1 The Picard group . 41 2.2 Cartier and Weil divisors . 45 2.2.1 Cartier divisors . 45 2.2.2 Weil divisors . 47 2.3 Ampleness and very ampleness . 52 2.4 Intersection theory on surfaces . 56 2.4.1 The Riemann-Roch Theorem for surfaces . 61 2.5 Nef line bundles . 64 2.5.1 Cup product and intersection product . 64 2.5.2 The N´eron-Severi group . 65 2.5.3 The Nakai-Moishezon Criterion for ampleness . 69 2.5.4 Cones . 72 3 Birational geometry 79 3.1 Contraction of curves . 79 3.1.1 Rational and birational maps . 79 3.1.2 The Zariski Main Theorem . 81 3.1.3 Blow-up of a point . 84 3.1.4 Indeterminacies . 87 3.1.5 Castelnuovo's contraction theorem . 89 3 4 Contents 3.2 Kodaira dimension . 93 3.2.1 Definition and geometrical interpretation . 94 3.2.2 Kodaira dimension and nefness . 100 3.3 Albanese tori . 102 4 The Enriques-Castelnuovo Classification 107 4.1 The Kawamata-Mori Rationality Theorem . 109 4.2 Kodaira dimension −∞ . 111 4.2.1 Ruled surfaces . 111 4.2.2 Invariants of geometrically ruled surfaces . 114 4.2.3 Rational surfaces . 117 4.2.4 Minimal ruled surfaces . 118 4.2.5 Classification for Kodaira dimension −∞ . 122 4.2.6 The Castelnuovo Rationality Criterion . 125 4.3 Non-negative Kodaira dimension . 127 4.3.1 Definitions and examples . 127 4.3.2 Fibrations . 132 4.3.3 Elliptic fibrations . 136 4.3.4 Classification for Kodaira dimension 0 and 1 . 139 5 Further properties and classifications 147 5.1 Ruled surfaces . 148 5.1.1 Classification of non-rational ruled surfaces . 148 5.1.2 Ruled and bielliptic surfaces . 151 5.2 Rational surfaces . 152 5.2.1 Linear systems of conics . 154 5.2.2 Linear systems of cubics . 156 5.3 K3 surfaces . 161 5.3.1 The Kummer surface . 161 5.3.2 Projectivity . 163 5.3.3 The Torelli Theorem for K3 surfaces . 167 5.4 Enriques surfaces . 169 5.4.1 Construction and projectivity . 170 5.4.2 The Torelli Theorem for Enriques surfaces . 171 5.5 Bielliptic surfaces . 173 5.6 Abelian surfaces . 177 5.6.1 Properties of abelian surfaces . 177 5.6.2 The Torelli Theorem for abelian surfaces . 181 5.7 Elliptic surfaces . 183 5.7.1 Classification of singular fibers . 183 5.7.2 Classification of elliptic surfaces . 186 5.8 Surfaces of general type . 194 6 Generalizations of the Enriques-Castelnuovo Classification 199 6.1 Singular projective surfaces . 199 6.2 The Enriques-Kodaira Classification . 200 6.3 Characteristic p ............................205 6.4 Higher dimensional classification . 207 Introduction In this series of lectures I will present an introduction to the classification of compact complex surfaces. This is a quite old subject in geometry, and involves several elements coming from different areas of mathematics, like differential geometry, topology, algebraic geometry (classical and modern). It is, to my opinion, a very beautiful piece of mathematics, which is nowadays considered classical, and which is very useful to modern research in geometry. As usual when dealing with a geometric problem, one needs to define the family of objects one wants to investigate, and to define a notion of equivalence between them. Moreover, in order to understand if two objects of the chosen family are equivalent, one studies their invariants. If one considers as objects topological spaces, i. e. couples (X; τ) where X is a set and τ is a topology on it, then one can talk about continuous maps between two topological spaces, and can introduce the notion of homeomorphism: namely, two topological spaces (X1; τ1) and (X2; τ2) are homeomorphic if there are two continuous maps f1 : X1 −! X2 and f2 : X2 −! X1 such that f2 ◦f1 = idX1 and f1 ◦f2 = idX2 . One can then try to classify topological spaces up to homeomorphism, even if this problem seems somehow impossible to solve, as every set can be considered as a topological space (e.g. with the discrete topology). One can even change the notion of equivalence, introducing, as instance, the homotopic equivalence: first of all, one defines the notion of homotopy between two maps f; g : X −! Y ; then one says that two topological spaces (X1; τ1) and (X2; τ2) are homotopic equivalent if there are two continuous maps f1 : X1 −! X2 and f2 : X2 −! X1 such that f2 ◦ f1 is homotopic to idX1 and f1 ◦ f2 is homotopic to idX2 . Still, the problem seems very hard to solve. For simplicity's sake, we will consider only compact and connected topological spaces. The next step in the classification is then to enrich the structure of a topolig- ical space. This can be done in different ways: first of all, one only considers topological spaces over which one can, in some way, perform analysis or alge- bra. A first approach is to define the notion of a manifold: this can be done for different purposes. A real manifold will be a topological space M which is Hausdorff, admits a countable basis for the topology, and admits an open cover fUigi2I , where Ui is an open subset of M and for every i 2 I there is a n n homeomorphism 'i : Ui −! R for some fixed n (here R is a topological space with the euclidean topology). This n is called the (topological) real dimension of M. If one considers i; j 2 I, then we have a map −1 n n 'i ◦ 'j : 'j(Ui \ Uj) ⊆ R −! R : Asking for these maps to be differentiable, of class Ck, of class C1, ana- 5 6 Introduction lytic, one gets the notion of differentiable manifold, Ck−differentiable manifold, C1−differentiable manifold, analytic manifold. Moreover, one can introduce the notion of continuous map, Ck−, C1−differentiable map or analytic map be- tween two manifolds M1 and M2, and hence of equivalence: one gets the notions of homeomorphic manifolds and of diffeomorphic ( Ck−, C1−)differentiable or analytic manifolds. In a similar way one can introduce the notion of complex manifold, which is a topological space X which is Hausdorff, admits a countable basis for the topology, and admits an open cover fUigi2I , where Ui is an open subset of X n and for every i 2 I there is a homeomorphism 'i : Ui −! C for some fixed n (here Cn is a topological space with the euclidean topology). This n is called the (topological) complex dimension of X. Again, for every i; j 2 I we have a map −1 n n 'i ◦ 'j : 'j(Ui \ Uj) ⊆ C −! C : Asking for these maps to be holomorphic, one gets the notion of holomorphic manifold, or simply of complex manifold. As in the previous case, one can introduce the notion of holomorphic map between two complex manifolds, and hence of equivalence: two complex manifolds X1 and X2 are equivalent if they are biholomorphic. Using the notion of sheaf, one can still enrich the structure of a manifold, attaching to it a sheaf of rings: if M is a differentiable or analytic manifold, and U ⊆ M is an open subset, then one can define the ring OM (U) := ff : U −! R j f is differentiable/analyticg: This assignement defines a sheaf of rings over M, and the couple (M; OM ) is called a differentiable or analytic space. In a similar way one defines the notion of a complex space (X; OX ). There is a different approach, relying on algebra instead of analysis. To any commutative ring A with unity 1A, we can associate a topological space Spec(A), whose underlying set is the set of prime ideals of A, and whose topology is defined using ideals of A, and is called Zariski topology. Moreover, using localizations of A, one can define a sheaf of rings OSpec(A), and the couple (Spec(A); OSpec(A)) is called an affine scheme. More generally one can define the notion of a scheme: this is simply a couple (X; OX ), where X is a topological space, OX is a sheaf of commutative rings with unity, and admitting an open cover by affine schemes. Again, one can define the notion of morphism between two schemes, and hence of equivalence: two schemes are equivalent if they are isomorphic. As for manifolds, one can define the notion of (algebraic) dimension of a scheme, relying on the notion of Krull dimension of a commutative ring. In any case, the notion of dimension is crucial: if two spaces (or schemes) are equivalent, they have the same dimension. This is a first important invariant, and we can classify these objects at first with respect to their dimension. In dimension 0, there is only the point, so there is no classification needed. In di- mension 1, things are more complicated: in the complex universe, these objects are called Riemann surfaces. These are, in particular, compact connected real manifolds of dimension 2, whose classification started in the second half of the XIX century by Moebius, Klein and others.
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