Complex abelian varieties Por Sebasti´anTorres Tesis presentada a la Facultad de Matem´aticas de la Pontificia Universidad Cat´olicade Chile para optar al grado de Mag´ısteren Matem´aticas Profesora gu´ıa: Rub´ıRodr´ıguez- Universidad de la Frontera Comisi´oninformante: Anita Rojas - Universidad de Chile Gonzalo Riera - Pontificia Universidad Cat´olicade Chile Agosto de 2015 Santiago, Chile Contents 1 Introduction 1 2 Complex manifolds 2 2.1 Vector bundles . 5 2.2 Tangent bundle . 8 2.3 Differential forms . 11 2.4 Integration . 13 3 Riemann surfaces 15 3.1 Divisors on Riemann surfaces . 18 3.2 Group actions on Riemann surfaces . 19 3.3 Jacobian variety . 20 4 Complex Tori 22 4.1 Differential forms . 24 4.2 Theta functions . 26 4.3 Divisors . 29 4.4 Line bundles . 32 5 Complex abelian varieties 36 5.1 Homomorphisms of abelian varieties . 37 6 Representations of finite groups 40 6.1 Decomposition of semi-simple rings . 40 6.2 Rational central idempotents . 46 7 Group representations in abelian varieties 49 7.1 Involutions on ppav's . 51 7.2 Automorphisms of order three on ppav's . 52 i 1 Introduction The present work is devoted to the study of complex abelian varieties and the action of a group on it. In the main part of the text, we describe the theoretic frame regarding abelian varieties, for which we need to talk about complex manifolds, complex tori and topics surrounding those. Most of the results here will not be proved; these contents can be found in classical books. In section 2 we define complex manifolds and the main concepts we will need later, especially vector bundles, sections and differential forms. The next section gives a one-dimensional flavor to the theory of complex manifolds, stating a few classical results in compact Riemann surfaces, without proof. This section is not strictly necessary to understand what follows. In section 4 we describe how the theory of complex tori works. A complex torus is the compact quotient of Cn by some discrete additive subgroup. One of the most important things here is understanding how divisors can be seen as line bundles, for what we need to speak about theta functions, among others. Next we talk about abelian varieties, which are complex tori with a bilinear form satisfying certain properties, and describe how this can be written in terms of matrices. We also mention the important fact that an abelian variety can always be seen as a submanifold of some projective space. To understand how a group acts on an abelian variety, in section 6 we study part of the algebra we need regarding group representations, for in section 7 apply this theory to our case, stating Lange and Recillas' isotypical decomposition. The algebra needed can be found in classical books like [3], while its applications to abelian varieties are studied in specific articles such as [13], [7] or [2], for instance. Finally, we try to work out specific group actions on abelian varieties. In [12], Rodr´ıguezcompletely describes abelian varieties admitting an involution. In the same spirit, the present works makes an attempt to describe the action of an order-three group in an abelian variety. For this we need to study symplectic matrices. Further interesting work would be to work out the way different finite groups act on a complex abelian variety. 1 2 Complex manifolds Roughly speaking, a complex manifold is a space that locally looks like Cn, so that we can define holomorphic functions on it. Here we describe these ideas more precisely. Definition 2.1. A function f from an open set D ⊂ Cn to C is said to be holomorphic if every point a 2 D has a neighborhood in which f can be expressed as a power series 1 X k1 kn f(z) = ck1;:::;kn (z1 − a1) ··· (zn − an) : k1;:::;kn=0 A function between open sets D ⊂ Cn, E ⊂ Cm is said to be holomorphic if every coordinate function is so. Holomorphic functions in several variables quite behave like single-variable holomorphic functions. We state a couple of useful lemmas, whose proofs can be found, for instance, in [14]. The first one is known as Osgood's lemma. Lemma 2.2. Let f : D ! C be a continuous function. If f is holomorphic in each variable separately, then it is holomorphic on D. As a consecuence of Osgood's lemma, we have that a C1 function f : D ! is holomorphic iff @f = ··· = @f = 0. The following is the identity C @z¯1 @z¯n principle for several variables. Lemma 2.3. If f, g are holomorphic functions on a connected open set n D ⊂ C , and U ⊂ D is nonempty open subset satisfying fjU = gjU , then f = g. Definition 2.4. Let X be a topological space. An atlas is a covering of X given by a family of open subsets Ui ⊂ X, together with a set of homeomor- n phisms φi : Ui ! Di, where Di is an open set of C , which satisfy that −1 φi ◦ φj : φj(Ui \ Uj) ! φi(Ui \ Uj) 2 is a biholomorphism for every par (i; j). We call a Hausdorff, second count- able space X with such an atlas a complex manifold. The functions φi are called charts. We may always assume that the family f(Ui; φi)g is maximal with respect to the condition above. We also assume that X is connected, unless otherwise stated. In this case, the number n does not depend on the chart and will be called the dimension of X.A function f : X ! Y between complex manifolds is said to be holomorphic −1 if every composition 'j ◦ f ◦ φi is holomorphic, where φi and 'j are charts for X, Y , respectively. Of course, the condition need only be checked for one chart at each point. Note that C itself has a natural structure of complex manifold. Holomorphic functions between X and C will be referred to simply as holomorphic functions on X. Example 2.5. Consider Cn+1nf0g with the equivalence relation given by: x ∼ y iff x = ty for some t 2 C∗. The quotient is called the projective space n P and is made into a complex manifold by setting Ui = f[x0 : ::: : xn] j xi 6= 0g and n φi : Ui ! C x0 xi−1 xi+1 xn [x0 : ::: : xn] 7! ; ··· ; ; ; ··· ; : x1 xi xi xi Pn is the set of lines in Cn+1. For n = 1, we obtain the Riemann sphere, that can be seen as C plus one point called infinity, 1 = [1 : 0]. We denote it by C^ = C [ f1g = P1. Definition 2.6. A meromorphic function on X is a holomorphic function f : X ! C^ which is not identically 1. The space of such functions is denoted M(X). Let f; g be meromorphic functions on X, with g not identically zero. If f(z) = [f1(z): f2(z)] and g(z) = [g1(z): g2(z)], we define the quotient 3 f=g simply as [f1g2 : f2g1]. It is not difficult to check that this indeed is well defined and gives a meromorphic function, so this makes M(X) into a field. A holomorphic function h on X can also be regarded as a meromorphic function by identifying it with [h : 1]. Next we will come to an alternative definition of complex manifold using the language of sheaves. Definition 2.7. Let X be a topological space. A geometric structure O is an assignment O(U) for every open subset U ⊂ X, where O(U) is a ring of continuous functions defined on U, which satisfies the following conditions: (1) The constants are in O(U) for every open subset U. (2) If V ⊂ U are open sets, then fjV 2 O(V ) for every f 2 O(U). (3) If U is a collection of open subsets and f 2 O(U ) satisfy f j = i i i i Ui\Uj f j 8i; j, then there exists a unique f 2 O(U) such that f = j Ui\Uj i fj 8i, where U = S U . Ui i We call such a pair (X; O) a geometric space. Note that every open subset U ⊂ Cn has a natural structure of geometric space by letting O(V ) be the set of holomorphic functions on V . If X is a geometric space and U ⊂ X is open, U inherits a structure of geometric space that we denote (U; OjU ), given by OjU (V ) = O(V ) for V ⊂ U an open set. Definition 2.8. Let (X; OX ), (Y; OY ) be geometric spaces. A morphism between them is a continuous function f : X ! Y such that for every U ⊂ Y −1 open, g ◦ f 2 OX (f (U)). An isomorphism is a morphism with a two sided inverse that is also a morphism. Now we give an alternative definition for a complex manifold. 4 Definition 2.9. Let X be a second countable, Hausdorff topological space with a geometric structure O. We say that (X; O) is a complex manifold if it has a covering by open sets Ui such that every (Ui; O(Ui)) is isomorphic to some open subset of Cn. It is not too difficult to see that both definitions coincide. If we have a geometric space X satisfying the conditions above and fi :(Ui; O(Ui)) ! n (Di; ODi ) are isomorphisms with open subsets of C , then the mappings fi : Ui ! Di give coordinate charts for X, and the compatibility conditions are easily checked to be satisfied. Conversely, if we have a complex manifold with respect to the first definition, we can give it a complex structure putting O(U) as the holomorphic functions in U for every open subset U ⊂ X, and the coordinate charts give isomorphisms of geometric spaces.