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On a Proof of Torelli's Theorem

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On a Proof of Torelli's Theorem Contents 0.1 Assumed Knowledge . 5 0.2 Outline . 5 Chapter 1 Sheaves and sheaf cohomology 7 1.1 Sheaves . 7 1.2 Sheaves associated with functions . 12 1.3 Sheaf cohomology . 14 1.4 Vanishing theorems . 22 n n 1.5 Cohomology of C and P ........................... 24 Chapter 2 Riemann surfaces 27 2.1 Properties of Riemann surfaces . 28 2.2 Examples of Riemann surfaces . 29 2.3 Cohomology of Riemann surfaces . 31 2.4 The Riemann-Hurwitz formula . 32 2.5 Hyperellipticity . 33 Chapter 3 The classical theorems of Abel and Jacobi 35 3.1 Divisors . 35 3.2 The Abel-Jacobi map and the Jacobian variety . 38 3.3 Line bundles . 41 3.4 Pic(S)...................................... 45 Chapter 4 Linear systems and the Riemann-Roch theorem 51 4.1 The Riemann-Roch theorem . 57 4.2 Application and Examples . 59 Chapter 5 Complex tori 65 5.1 Cohomology of complex tori . 66 5.2 Line bundles on complex tori . 67 1 5.3 Theta functions . 72 Chapter 6 The Jacobian Variety 77 6.1 Motivation: Abelian integrals . 77 6.2 Properties of the Jacobian variety . 79 6.3 Riemann’s theorem . 83 Chapter 7 The Torelli theorem 91 7.1 Proof of the Torelli theorem . 91 Chapter 8 Concluding remarks 98 8.1 The Schottky problem . 98 Chapter 9 Background material 100 9.1 Group cohomology . 100 9.2 Major theorems . 101 2 Acknowledgements I would like to extend my gratitude to all those who helped in the production of this thesis, and to those who supported me throughout this year. I thank my supervisor, Dr Daniel Chan, for all of his guidance and assistance. Also to my family and friends, especially Lorraine, who put up with me during all this time. Special thanks go to Prof. Tony Dooley for lending me his copy of [Sha74]. Finally, I would like to thank God, who makes all things possible. 3 Introduction In this thesis, the principal objects of study are Riemann surfaces, and the aim will be to expound the classical Torelli theorem relating Riemann surfaces to their Jacobians, which are central to their study. Riemann’s original definition in his doctoral thesis [Rie51] amounts to saying that a 1 Riemann surface is an n-sheeted branched cover of P . At that time, Riemann surfaces were merely a convenient way to represent multi-valued functions. Klein took up the subject after Riemann and studied Riemann surfaces via differential geometry as objects in their own right. Weyl formalised Klein’s ideas in his famous monograph Die Idee der Riemannschen Fl¨ache [Wey23]. Today we define a Riemann surface as a (compact) connected one dimensional complex manifold. It is interesting to note that the definition of a complex manifold did not appear in the literature until mid 40’s. The phrase komplexe analytische Mannigfaltigkeit 1 first appeared in Teichm¨uller’s [Tei44], and the English version appears in Chern’s [Che46] in 1946. For more on the history of Riemann surfaces, see Remmert’s delightful recount in [Rem98]. The Jacobian of a Riemann surface S is a complex torus, and in fact, is an abelian variety. Its definition is intrinsic to S, and captures much of its information. Torelli’s theorem states that given a Jacobian of a Riemann surface and an additional piece of data, called the principal polarisation, one can recover the Riemann surface up to isomorphism. The proof which we present follows Andreotti’s [And58]. It is interesting to note that Marten published a new proof of the Torelli theorem [Mar63], which uses combinatorial tech- niques together with Abel’s theorem and the Riemann-Roch theorem. Torelli’s original publication on Jacobians can be found here [Tor13]. In [Mum75], Mumford speaks of the “amazing synthesis” of algebra, analysis, and geome- try that is at the heart of the geometry of algebraic curves. This trichotomy is evident in that complex algebraic curves are in a one to one correspondence with Riemann surfaces, each emphasising different methods used to explore the geometry of these objects. The 1This is German for complex analytic manifold. 4 amazing synthesis goes much further; to quote Mumford again [Mum95], algebraic geom- etry is not an “elementary subject” but draws from, and contributes to, many disparate disciplines in mathematics. So the difficulty for any initiate of algebraic geometry lies in the tremendous amount of background which has to be covered, as well as the depth and breadth of the ideas in algebraic geometry which itself has enjoyed a long and illustri- ous history. This is the vindication for the long list of topics assumed. For a history of algebraic geometry, we refer the reader to Dieudonn´e’sarticle, [Die72]. 0.1 Assumed Knowledge We assume knowledge of very basic differential geometry and complex manifolds, an elementary treatment of complex manifolds can be found in chapter 7 of [Che00]. To give an idea of the depth of knowledge assumed, concepts such as K¨ahler manifolds, Hermitian metrics, differentials, tangent and cotangent bundles will be used without comment. Also assumed is a basic understanding of algebraic geometry, where the relevant back- ground can be found in the notes of a course on algebraic geometry taught by Daniel Chan at UNSW in 2004. The course was based on [Sha74], and the notes, edited by the author, can be found at [CC04]. We will not list the concepts assumed, and explicit references to these notes will be made in the thesis. Another very good source of information for the subset of algebraic geometry associated to the thesis material is [Mum95]. The basic concepts algebraic topology, homological algebra, and category theory are also assumed. Again to give some idea of what is assumed, the following concepts will be used without digression; categories, functors, cochain complexes, exact sequences, Poincar´e duality, the Euler characteristic, the Mayer-Vietoris sequence, and simplicial, de Rham, and Dolbeault cohomology. For an exposition of these concepts we refer the reader [Hat02] for algebraic topology, and [Osb00] for homological algebra. 0.2 Outline This section provides an outline for the development of the material. The chapters should be read in sequence to maintain coherence. In chapter 1, we begin with sheaf theory and their cohomology. The reason for beginning with this technical topic is that the chapters which follow employ extensively the language and techniques of sheaves and sheaf cohomology. The definition of coherent sheaves can be found in [Uen01]. Chapters IX and X of [Mir95] contains a very clear exposition on sheaves. Pages 11-18 of [EH00] contain a basic introduction to sheaf theory. 5 From chapter 2 onwards, the main reference will be [GH78], which is a very comprehensive treatment of algebraic geometry from an analytic perspective. In chapter 2, we introduce Riemann surfaces, and derive some of their selected properties which will be used in proving the Torelli theorem. The differences between hyperelliptic and non-hyperelliptic Riemann surfaces are discussed. Kirwan’s book [Kir92] is elemen- tary in its treatment; [Cle80] contain many interesting examples, but assumes previous knowledge in many areas. The topic of Riemann surfaces are thoroughly developed in both [Mir95] and [FK92]. Chapter 3 and 4 develop more advanced material concerning Riemann surfaces. The Abel and Jacobi theorems are first discussed, then the concepts of divisors, line bundles are introduced. Linear systems are explored in chapter 4, with an emphasis on linear systems on Riemann surfaces, culminating in the Riemann-Roch theorem, which is a formula for the dimension of a linear system on a Riemann surface. Complex Tori are discussed at length in chapter 5, in anticipation to the discussion of the Jacobian variety in chapter 6. The line bundles on complex tori are classified, and the theta functions are obtained from global sections of such line bundles. The references for this chapter are [Pol03] and [GH78]. The next chapter on the Jacobian variety applies the results for complex tori, culminating in Riemann’s theorem. The penultimate chapter gives the proof of the Torelli theorem, using much of material developed above. Finally, we end with some concluding remarks regarding the Torelli theorem. 6 Chapter 1 Sheaves and sheaf cohomology Algebraic geometry was transformed by Serre in the 1950’s by his introduction of sheaf theoretic techniques [Ser55]. The main reason for using sheaves in this thesis is to access sheaf cohomology, which is, as we shall see, a powerful and concise technique. We seek to emphasis this approach as much as possible. This chapter contains the elementary definitions and theorems of sheaves and their cohomology. For a full treatment of sheaves in the context of algebraic geometry, see [Uen01]. 1.1 Sheaves The machinery of sheaves allows one to organise local information and extract global properties of a topological space, X. A sheaf associates algebraic data to each open set of X, and does so functorially. Let X denote a topological space; we first define a presheaf over X. Definition 1.1 A presheaf F of abelian groups over X is a contravariant functor from the category of open sets of X, where the morphisms are given by the inclusion maps, to the category of abelian groups Ab, where the morphisms are given by the group homomor- phisms. That is, for every pair of open sets V and U such that V ⊂ U, we have the restric- tion homomorphism ρU,V : F(U) −→ F(V ). Moreover these restriction homomorphisms satisfy 1. ρU,U = idU for all U, and 2. that the following diagram commutes for all open sets W ⊂ V ⊂ U ρU,V ρV,W F(U) / F(V ) 3/ F(W ) ρU,W For concision, we will often write ρU,V (σ) = σ|V for σ ∈ F(U).
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