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Riemann Surfaces Riemann Surfaces R. R. Simha and V. Srinivas Dedicated to Prof. K. R. Parthasarathy on turning 60. This expository article on Riemann surfaces is an outgrowth of a Winter School on Riemann Surfaces held in December, 1995, organised by the Centre of Advanced Study in Mathematics of the University of Bombay, and sponsored by the University Grants Commission and the National Board for Higher Math- ematics. It is a survey of some recent developments in the subject, which could not be included in the Winter School because of lack of time. We have however tried to make the exposition reasonably self-contained by recalling most of the basic definitions and concepts, as well as some key examples. We thank Prof. M. G. Nadkarni, the chief organiser of the Winter School, for providing us this opportunity to take a look at this old but very active subject. We also thank the rest of the faculty as well as the participants of the Winter School for the stimulation they have provided. 1 Introduction, and some examples Riemann surfaces are (Hausdorff) topological spaces on which one can do com- plex analysis of one complex variable. These generalizations of open subsets of the complex plane C occur in a natural way in a large number of areas of mathematics (including complex analysis in C); their study is an active area of current research, though the subject is almost two centuries old. We recall that a Riemann surface is a Hausdorff topological space X, to- gether with a rule X which assigns to each open U X a set X (U) of functions f : U CO (which will be called holomorphic⊂ functions onOU) such that: → (i) (Sheaf property) if U ∈ are open subsets of X, and U = ∈ U , then { i}i I ∪i I i a function f : U C lies in X (U) f Ui : Ui C lies in X (Ui) for all i I; → O ⇔ | → O ∈ (ii) for each x X, there exists an open neighbourhood U of x, and a homeo- morphism ∈f : U V onto an open subset V C, such that for any open W V , a function→ g : W C is holomorphic⊂ on W (in the usual sense) ⊂g f (g−1(W )). → ⇔ ◦ ∈ OX 1 The above definition is easily seen to be equivalent to the more usual one, according to which one is given a family of pairs (U ,f ), i I (called coordinate i i ∈ charts), each consisting of an open set Ui and a homeomorphism fi of Ui onto an open subset of C such that (i)′ U = X; ∪i i ′ −1 (ii) fj fi : fi(Ui Uj ) fj (Ui Uj ) C is holomorphic whenever Ui Uj is nonempty.◦ ∩ → ∩ ⊂ ∩ A holomorphic map f : (X, X ) (Y, Y ) between Riemann surfaces is a continuous map f : X Y Osuch→ that forO all open subsets V Y , and → −1 ⊂ g Y (V ), we have that g f X (f (V )); f is said to be biholomorphic if f ∈is O a homeomorphism of X◦onto∈ OY , and f and f −1 are holomorphic. However, it is easy to show that a holomorphic homeomorphism of one Riemann surface onto another is a biholomorphic map. Clearly, C and its open subsets are Riemann surfaces in a natural way; if U, V C are open sets, then a holomorphic map f : U V is just a holomorphic⊂ function on U with f(U) V . More generally, an→ open subset of a Riemann surface is a Riemann surface⊂ in a similar way. Most of the properties of holomorphic functions on open subsets in C carry over to holomorphic maps between Riemann surfaces. For example, if f,g : X Y are holomorphic maps and X is connected, then either f g, or x →X f(x)= g(x) is a discrete1 subset of X. Similarly, the open mapping≡ theorem{ ∈ | carries over} in the obvious way. The theory of Riemann surfaces is concerned with the study of holomorphic maps between Riemann surfaces. 1.1 Examples of Riemann surfaces 1 Example 1.1: The extended complex plane C (or PC). This is the first example of a “new” Riemann surface, i.e., one which is not (biholomorphic to) an open subset of C. As a topological space, C = C is just the one-point compactification of C; in particular, C is compact,∪ and {∞} in fact homeomorphic to the two-sphere S2. The Riemann surface structure can be defined, e.g., by two coordinate charts (C, Id) and (C 0 , w), where Id denotes the identity function, and −{ } 0 if x = C w(x)= 1 ∞ ∈ if x C 0 . ( x ∈ −{ } A meromorphic function on a Riemann surface X can be defined to be a holomorphic map f : X C such that f on any connected component of X. When X is an open→ subset of C, this6≡ ∞ reduces to the usual definition; 1A closed subset all of whose points are isolated. 2 further, when X = C, one can show that meromorphic functions are precisely the rational functions. The poles of a rational function f are the points in f −1( ). The∞ Riemann surface C can be naturally identified with the 1-dimensional 1 complex projective space PC (see Example 1.5). Example 1.2: The complex tori. These are compact (connected) Riemann surfaces homeomorphic to the two- 1 1 torus S S , and arise as follows. Choose ω1,ω2 C which are linearly independent× over R (e.g., ω = 1, ω C R). Let Γ∈ = Γ(ω ,ω ) denote the 1 2 ∈ \ 1 2 additive subgroup of C generated by ω1 and ω2. Consider the quotient space E defined by the equivalence relation z z z z Γ. Since the map Γ 1 ∼ 2 ⇔ 1 − 2 ∈ R2 C, → (s,t) sω + tω 7→ 1 2 is an R-linear isomorphism carrying Z Z R2 isomorphically onto Γ, the ⊕ ⊂ quotient space EΓ is homeomorphic to R2/Z Z = R/Z R/Z = S1 S1; ⊕ × × in particular it is a compact Hausdorff topological group. The Riemann surface structure on E is the one which makes the quotient map π : C E locally Γ → Γ biholomorphic; thus if U EΓ is an open set, then f : U C is holomorphic f π : π−1(U) C is⊂ holomorphic. And meromorphic→ functions on E are ⇔ ◦ → Γ precisely the doubly periodic meromorphic functions on C admitting ω1 and ω2 as periods. Remark 1.1 Let X be a Riemann surface, and G be a subgroup of the group Aut (X) of biholomorphic maps X X. Suppose that any p, q X → ∈ have neighbourhoods Up,Uq (respectively) such that g G g(Up) Uq = is finite; G is then said to be a discrete subgroup of Aut{ (X∈). In| this case,∩ it6 can∅} be shown that the orbit space Y = X/G (whose points are G-orbits gx g G ), with the quotient topology induced by the natural quotient map{ π :| X∈ }Y , is a Hausdorff space, and has a unique Riemann surface structure making→π a holomorphic map. This result is easy to prove if G is fixed-point free (i.e., if g G is not the identity, then gx = x for all x X). If X is simply connected, ∈ 6 ∈ and G is fixed-point free, then G is isomorphic to the fundamental group π1(Y ) of Y , and X is the universal covering space of Y . The complex tori are examples of this situation, where X = C, and G = Γ, acting by translations. The complex tori were the first (non-trivial) Riemann surfaces studied, and their complex analytic theory is completely understood. For example, if EΓ′ ′ ′ ′ ′ ′ ′ corresponds to Γ = Γ(ω1,ω2), such that τ = ω1/ω2, τ = ω1/ω2 have positive imaginary part, then E is biholomorphic to E ′ there exists Γ Γ ⇔ a b SL (Z) c d ∈ 2 3 such that aτ + b τ ′ = . (1.1) cτ + d Thus, if H is the upper half-plane in C, then isomorphism classes of tori are in bijection with points of the quotient space H/SL 2(Z) for the action of SL 2(Z) on H given by the formula (1.1). More interestingly, every Riemann surface homeomorphic to S1 S1 is biholomorphic to some E . × Γ Example 1.3: Riemann surfaces of algebraic functions. These were the first “abstract” Riemann surfaces constructed by Riemann. The construction begins with an easy observation: if X is a Hausdorff space, and π : X Y is a local homeomorphism2 and if Y is a Riemann surface, then X becomes→ a Riemann surface by requiring that π be locally biholomorphic. ∞ In the case at hand, X will be the set of all convergent power series n=0 an(z b)n which are obtained by analytic continuation (along some path in C) of a− ∞ n P fixed convergent power series n=0 αn(z β) ; Y will be the complex plane C, and π : X Y will be defined by − → P ∞ π( a (z b)n)= b. n − n=0 X ∞ n (Thus, the series n=0 an(z b) is obtained by analytic continuation along some path in C from β to b.) Note− that there is a natural Hausdorff topology on X making π a localP homeomorphism, and thus X becomes a connected Riemann surface. Defining f : X C by → ∞ f( a (z b)n)= a , n − 0 n=0 X we obtain a holomorphic function on X; this single-valued function on X repre- sents the “multivalued” holomorphic function consisting of all analytic contin- ∞ n uations of the given power series n=0 αn(z β) .
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