This book consists of the lecture notes from a year course in the elementary theory of compact Riemann surfaces, from a modern point of view. There are two noteworthy aspects to the author's approach to the subject. First, the exposition is heavily sheaf-theoretic. The concepts of sheaves and Čech cohomology with values in a sheaf are used whenever possible. The sheaf theory used is developed from the beginning, and the prospective reader need not have any background in it. Second, the main analytic tool is the Serre duality theorem instead of the more usual theory of harmonic integrals. Serre's theorem states that H1(M,O(ξ))≅H0(M,O1,0(ξ−1)), where M is a compact Riemann surface, ξ is a holomorphic complex line bundle, O(ξ) is the sheaf of germs of holomorphic cross-sections of ξ, and O1,0(ξ−1) is the sheaf of germs of ξ−1-valued holomorphic 1-forms of type (1,0). The book is so arranged that the reader can skip the proof of Serre's theorem (§ 6) without losing the thread of the book, because the ideas in the proof do not recur. The author essentially uses Serre's own proof [Comment. Math. Helv. 29 (1955), 9–26; MR0067489], that is, he constructs distributions and sheaves of germs of distributions, which are naturally dual to the sets of forms used to compute H1(M,O(ξ)) by Dolbeault's theorem. The proof is elegant and illustrates many of the techniques used in applying distribution theory to linear partial differential equations, in the easy special case of the operator ∂/∂z¯¯¯. All the relevant distribution theory is developed from the beginning. These notes are well written and quite readable. The heavily sheaf-theoretic approach may make this book seem too formal and algebraic for a classically oriented mathematician, but this may also make the book the ideal introduction to this beautiful subject (and its various generalizations) for the reader raised with the contemporary bias toward abstraction and formalization. This approach also makes the theories of one and of several complex variables look like part of the same subject. The prerequisites for the reader are familiarity with the standard course material of real and complex analysis, as well as with the topology of compact oriented surfaces. The author includes an appendix on the latter subject which is an adequate guide to the literature for the reader deficient in this topic. § 1 presents the basic definitions and examples of Riemann surfaces, and § 2 presents the same thing for sheaves. § 3 contains a lucid exposition of Čech cohomology with values in a sheaf, including the long exact sequence, fine resolutions, a version of the Leray isomorphism theorem, and a special case of Dolbeault's theorem. § 4 discusses the relation between divisors, their linear equivalence classes, and holomorphic line bundles, where a holomorphic line bundle is viewed as an element of H1(M,O∗), where M is a Riemann surface and O∗ is the sheaf of germs of non-vanishing holomorphic functions. The last part of § 4 establishes the finite-dimensionality of the spaces Hq(M,O(ξ)), q=0,1, ξ∈H1(M,O∗), by introducing square-integrable cochains and some elementary functional analysis. In § 5 some familiarity with differential forms is assumed. The de Rham and Dolbeault sequences are discussed and the Serre duality theorem is stated. § 6 consists of the proof of the Serre duality theorem. In § 7 in some sense the theory that is peculiar to Riemann surfaces begins. In 7a the Chern class of a holomorphic complex line bundle is introduced, first sheaf-theoretically, then as an integral, and finally as the ``order'' of a meromorphic cross-section provided one exists. 7b contains the proofs of the existence of meromorphic cross-sections of holomorphic line bundles, and of the Riemann-Roch formula. In the remainder of § 7, Weierstrass gaps are defined, the Weierstrass gap theorem is proved, and the Weierstrass points and their weights are discussed. At this point it might prove useful to skip to 10a, where meromorphic functions are used to represent a given surface as a branched covering of the Riemann sphere. § 8 introduces both the Picard and Jacobi varieties. The Picard variety is first defined as the holomorphic line bundles of zero Chern class. In 8a this is represented as quotients of various cohomology groups. In 8b lattice subgroups of Cq, and tori are discussed, and the Picard variety is seen to be a torus. Then the period matrix of abelian differentials, the Jacobi variety, and the relationship between the Picard and Jacobi varieties are introduced. In 8c the cup product structure on M is reflected in the behavior of abelian differentials, this leading to Riemann's bilinear relations, and the isomorphism between the Picard and Jacobi variety. § 8 ends with a proof of Abel's theorem on the existence of meromorphic functions with prescribed zeros and poles. Now it is again useful to skip to § 10. 10b presents a brief introduction into the connection between the field of meromorphic functions on a Riemann surface, algebraic function fields, and algebraic plane curves. In 10c the author defines the principal mapping and the associated principal curve, and discusses some elementary properties of this map. 10c closes with a definition of the Jacobi map. This completes what one would call the elementary part of the book. § 9 is a special topic, which comprises nearly a fourth of the book, and deals with relatively new material. In this section the author studies the affine and projective structures subordinate to a given complex analytic structure. The author defines these structures first in terms of a pseudo-group property, but then quickly shows that a projective (affine) structure is a choice of an equivalence class of complex analytic atlases A on M, A=(Uα,φα)α∈A such that φβ∘φβ−1:φα(Uα∩Uβ)→φβ(Uα∩Uβ) is a linear fractional (affine) transformation for each α and β. In 9a the author defines and studies the elementary cohomological analysis of these structures, which is applied to show the existence of projective structures subordinate to any given complex analytic structure on a compact surface of genus >1, and to show that only the torus admits affine structures. 9b contains a discussion of the Čech cohomology set H1(M,G), where G is an arbitrary (not necessarily commutative) group. In the case G=PL(2,C), the complex projective linear group of rank 2, the author associates to each projective structure on M an element of H1(M,PL(2,C)), called the coordinate cohomology class, and then shows that the projective structures subordinate to a given complex structure are uniquely determined by their coordinate cohomology classes. In the remainder of 9b an isomorphism between H1(M,G) and Hom(π1(M)1G)/(inner automorphisms of G) (i.e., for φ,Ψ∈Hom(π1(M),G), φ∼Ψ if and only if there exists g∈G such that φ=gΨg−1) is proved. In 9c the elements of H1(M,PL(2,C)) which are coordinate cohomology classes of projective structures are determined. In 9d the Eichler sequence and the Eichler cohomology groups of a Riemann surface with a fixed projective structure are introduced and studied. In 9e the author introduces the universal covering surface M~ of a surface with a fixed projective structure, and then lifts these structures to the covering space. This determines a map ρ:M~→D, where D is an open connected subset of the Riemann sphere. It also determines a homomorphism ρ∗:π1(M)→PL(2,C) by associating, to each T~∈π1(M) considered as a deck transformation, a map T=ρ∗(T~)∈PL(2,C), T:D→D, such that ρ(T~p~)=ρ∗(T~)ρ(p~) for all p~∈M~. The pair (ρ,ρ∗) is called a geometric realization of the given projective structure on M. By introducing the forced equivalence relation among geometric realization, one then shows that projective structures and equivalence classes of geometric realizations are in a natural one-to-one correspondence. § 9 ends with a nice reduction of this abstract problem to this rather concrete discrete problem.
Reviewer: Feldman, E. A. [form MathSciNet]