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The Riemann-Roch Theorem Is a Special Case of the Atiyah-Singer Index Formula

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The Riemann-Roch Theorem Is a Special Case of the Atiyah-Singer Index Formula S.C. Raynor The Riemann-Roch theorem is a special case of the Atiyah-Singer index formula Master thesis defended on 5 March, 2010 Thesis supervisor: dr. M. L¨ubke Mathematisch Instituut, Universiteit Leiden Contents Introduction 5 Chapter 1. Review of Basic Material 9 1. Vector bundles 9 2. Sheaves 18 Chapter 2. The Analytic Index of an Elliptic Complex 27 1. Elliptic differential operators 27 2. Elliptic complexes 30 Chapter 3. The Riemann-Roch Theorem 35 1. Divisors 35 2. The Riemann-Roch Theorem and the analytic index of a divisor 40 3. The Euler characteristic and Hirzebruch-Riemann-Roch 42 Chapter 4. The Topological Index of a Divisor 45 1. De Rham Cohomology 45 2. The genus of a Riemann surface 46 3. The degree of a divisor 48 Chapter 5. Some aspects of algebraic topology and the T-characteristic 57 1. Chern classes 57 2. Multiplicative sequences and the Todd polynomials 62 3. The Todd class and the Chern Character 63 4. The T-characteristic 65 Chapter 6. The Topological Index of the Dolbeault operator 67 1. Elements of topological K-theory 67 2. The difference bundle associated to an elliptic operator 68 3. The Thom Isomorphism 71 4. The Todd genus is a special case of the topological index 76 Appendix: Elliptic complexes and the topological index 81 Bibliography 85 3 Introduction The Atiyah-Singer index formula equates a purely analytical property of an elliptic differential operator P (resp. elliptic complex E) on a compact manifold called the analytic index inda(P ) (resp. inda(E)) with a purely topological prop- erty, the topological index indt(P )(resp. indt(E)) and has been one of the most significant single results in late twentieth century pure mathematics. It was an- nounced by Michael Atiyah and Isadore Singer in 1963, with a sketch of a proof using cohomological methods. Between 1968 and 1971, they published a series of papers1 in which they proved the formula using topological K-theory, as well as filling in the details of the original proof. The history of the Atiyah-Singer index formula reads as a“Who’s Who” in twen- tieth century topology and analysis. The formula can be seen as the culmination of a project of generalisation of index theorems that began in the mid 1800’s with the Riemann-Roch theorem (and the Gauss-Bonnet theorem), and which involved many of the greatest names in topology and analysis of the last 150 years. It is an achievement for which Atiyah and Singer were awarded the Abel Prize in 2004. The significance of their formula reaches beyond the fields of differential topology and functional analysis: it is also fundamental in much contemporary theoretical physics, most notably string theory. For the purpose of this paper however, the only results which we shall consider are the classical Riemann-Roch theorem (1864), the Hirzebruch-Riemann-Roch the- orem (1954), and the Atiyah-Singer index formula (1963). In fact, we will only really look at the latter two in the context of being direct generalisations of the classical Riemann-Roch theorem. The (classical) Riemann-Roch theorem, proved as an equality in 1864, links analytic properties of certain objects called divisors on compact Riemann surfaces, with topological properties of holomorphic line bundles defined in terms of the divisors. Though the terms involved will only be properly defined later in this paper, it is convenient, nonetheless, to state the theorem here. Let X be a compact Riemann surface and D a divisor on X, that is, a function D : X → Z with discrete support. Then the Riemann-Roch theorem states that 0 1 (0.1) h (X, OD) − h (X, OD) = 1 − g + deg(D). 0 Here h (X, OD) is the dimension of the space of meromorphic functions f such that, for all x ∈ X, ord x(f) ≥ −D(x), where ord x(f) = n if f has a zero of order 1 n or a pole of order −n at x, and h (X, OD) is the dimension of another space of meromorphic functions also with only certain prescribed poles and zeroes (we will discuss this in detail in chapter 3). The degree, deg(D), of the divisor D is the sum 1The index of elliptic operators: I-V. (Paper II from 1968 is authored by Atiyah and Segal, rather than Atiyah and Singer.) [AS1, AS2, AS3, AS4, AS5]. 5 6 INTRODUCTION of its values over X. Since X is compact, the support of D is finite and so deg(D) is well-defined. Finally g denotes the genus of the surface X. It is clear that these are all integral values. The left hand side of equation (0.1) can be described in terms which depend on the holomorphic structure of certain line bundles on X, whilst we shall see that the right hand side depends only on the topology of these bundles. There is a natural equivalence relation on the space of divisors of a Riemann surface X and it will be shown that there is a one to one correspondence between equivalence classes of divisors on X and isomorphism classes of line bundles on X. (This will be described in chapter 3.) The Riemann-Roch theorem provides the conditions for the existence of mero- morphic functions with prescribed zeroes and poles on a compact Riemann surface. Its significance did not go unnoticed and its implications were studied by many of the greatest names in topology and analysis (even including Weierstrass). Interest- ingly it was initially regarded fundamentally as a theorem of analysis and not of topology. It was not until 1954, nearly a century after its original discovery, that Hirze- bruch found the first succesful generalisation of the Riemann-Roch theorem to holomorphic vector bundles of any rank on compact complex manifolds of any di- mension.2 This came a few months after J.P Serre’s 1953 discovery of what is now known as Serre duality, which provides a powerful tool for calculation with the Riemann-Roch theorem, but also deep insights into the concepts involved. Serre had applied sheaf theory to the Riemann-Roch theorem and Hirzebruch also used these newly emerging methods of topology to find techniques suitable for the project of generalisation. The so-called Hirzebruch-Riemann-Roch theorem says that the Euler characteristic χ(E) of a holomorphic vector bundle E on a compact complex manifold X is equal to its T-characteristic T (E). We will define these terms in chapters 3 and 4. Of significance here is that, in the case that the X has dimension 1 and E rank 1, if D is the divisor that corresponds to E, then the Euler charac- teristic χ(E) is equal to the left hand side of equation (0.1) and T (E) is equal to the right hand side of (0.1). After Hirzebruch’s theorem, progress to the Atiyah-Singer index formula was very swift indeed. Grothendieck discovered the Grothendieck-Riemann-Roch theo- rem around 19563, and the Atiyah-Singer index formula was published in its com- plete form in 1964. The Atiyah-Singer index formula is a direct generalisation of the Hirzebruch- Riemann-Roch theorem since we can assosciate a certain elliptic complex ∂(E) with any holomorphic vector bundle E on a compact complex manifold X, and it can be shown that χ(E) = inda(∂(E)) and T (E) = indt(∂(E)). In this paper, we will show how the original Riemann-Roch theorem, formu- lated for divisors on compact Riemann surfaces, is a special case of the Hirzebruch- Riemann-Roch Theorem and the Atiyah-Singer index formula. The paper does not set out to prove any of these theorems. One of the most striking features of the 2 These results can be found in [Hi], originally published as Neue topologische Methoden in der algebraischen Geometrie in 1956. 3Grothendieck had originally wished to wait with publishing a proof. With Grothendieck’s permission, a proof was first published by Borel and Serre [BS] in 1958. INTRODUCTION 7 Atiyah-Singer index formula, and a good illustration of the depth and signifcance of the result, is that it admits proofs by many different methods, from the initial cohomology and K-theory proofs, to proofs using the heat equation. We will limit ourselves here to a cohomological formulation of the formula since this is the most natural choice when dealing with the Riemann-Roch theorem. However it is per- haps worth mentioning that the K-theoretic formulation lends itself best to a more general exposition on the Atiyah-Singer index formula. The paper begins with two purely expository chapters. Chapter 1 sets out the basic definitions and notations concerning vector bundles, sheaves and sheaf cohomology which will be used throughout the paper. Most proofs will not be given. In chapter 2, elliptic differential operators, complexes and the analytic index of an elliptic complex will be defined and a number of examples will be given. The substantial part of the paper begins in chapter 3. Divisors on a Riemann surface X are defined and the Riemann-Roch theorem is stated in terms of divisors. By constructing a holomorphic line bundle L = LD on X, associated with the divisor D, it is then shown that the left hand side of the Riemann-Roch equation (0.1) can be interpreted as a special case of the analytic index of an elliptic operator. Finally we show that this also corresponds to the Euler characteristic χ(L) of L on a Riemann surface. In chapter 4, we turn to the right hand side of the Riemann-Roch equation (0.1) and show that this can be described in terms of purely topological properties of the surface X and the bundle L = LD.
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