1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
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1 Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces Basic configuration Holomorphic functions Complex projective planes Differential forms Poincaré residue Triangulation Ramification Holomorphic functions Divisors Effective and canonical divisors Linear systems Separation of points Separation of tangent vectors Embeddings Elliptic curves Line bundles Construction Sections On Riemann surfaces Note: For a more complete exposition, see the following textbooks, which were used in accompaniment with this course: Algebraic Curves and Riemann Surfaces, Rick Miranda
2 1 Structures Definition A subset U X is termed open if ϕ α (U U α ) C is open α. Definition A set F X is termed closed if X \ F is open. Remark The empty set is defined to be open. Definition A topological space is a set X together with a set S of subspaces of X such that a. S b. X S c. If U 0, U 1, S then j U j S d. If U, V S then U V S Definition A homeomorphism is an injective function between topological spaces that conserves all the topological properties of the given space. Definition A complex chart on a set X is a homeomorphism ϕ : U V for open sets U X, V C. Definition A conformal mapping is a transformation that preserves local angles. conformal wherever it has nonzero derivative. A function is Definition Given a topological space X and a point p (or a set S), a neighborhood of p (or S) is any open set T X containing p (or S). Definition For U open and ϕ : U X a homeomorphism (a chart), the set (U, ϕ), abbreviated to just U, is termed a coordinate neighborhood. Definition A Riemann surface is a set X with the properties: 1. X = α U α where all U α are coordinate neighborhoods 2. For each U α there exists a bijection ϕ α : U α V α for V α C open 3. ϕ β ϕ 1 α is a bijection from ϕ α (U α U β ) to ϕ β (U α U β ) and a conformal mapping for each (α, β) The function ϕ β ϕ 1 α above is termed a transition function. Theorem [Implicit function theorem] Let f : U C for U C be a function on two variables z 1, z 2. Suppose that f z 1 p 0 at p = (p 1, p 2 ) U and f(p) = 0. Then there exist open neighbourhoods U 1, U 2 C and a holomorphic map ϕ : U 1 U 2 such that {(z, ϕ(z)) z U1 } = {(z 1, z 2 ) f(z1, z 2 ) = 0} (U 1 U 2 ). 2 Framework of Riemann surfaces 2.1 Basic configuration Definition A topological space X is termed compact if X = j U j that X = U j1 U jk for U j open sets. = there exist j 1,... j k such Definition The Riemann sphere is C := C 2 { }. It is a compact Riemann surface. Definition A lattice is a set Γ = {n 1 w n k w k n1,..., n k N and w 1,..., w k C \ {0}}. Theorem [Inverse Function Theorem] If a function f : X X has a non-zero derivative at 0, then there exists a neighborhood U X such that f 1 : f(u) U is also smooth. 2
3 Definition A Riemann surface X is not connected if X = U V and U V for U, V open nonempty sets. Definition Define the set Γ = {n 1 w + n 2 z n1, n 2 Z} for some w, z C \ {0} to be a lattice. Note that Γ is a subgroup of C. Define X = C / Γ = {equivalence classes of x X z w z w Γ} to be an elliptic curve. 2.2 Holomorphic functions Definition A function f : X C is termed holomorphic if for each U α the following composition is well-defined: ϕ 1 U α X f V α f Uα (f Uα ) ϕ 1 α C Proposition If X is a compact and connected Riemann surface, then any holomorphic function f : X C is constant. Definition A function f has a pole at p if there exists a coordinate neighborhood U α p with Uα ϕα V α such that f ϕ 1 α has a pole at ϕ α (p). Definition A meromorphic function on X is a function f : X \ S C where S X is a nonempty set without cluster points and f has, at worst, poles at points of S. Remark Any meromorphic function on C is a rational function of z. Theorem If f is meromorphic on C, then ( # of zeros of f ) = ( # of poles of f counting multiplicities ). The same holds if f is meromorphic on X an elliptic curve. 3 Complex projective planes Definition The complex projective plane is a 2-dimensional complex projective (and topological) space described by 3 complex coordinates: CP 2 = {(z 0 : z 1 : z 2 ) z0 0 or z 1 0 or z 2 0} The coordinates (z 0 : z 1 : z 2 ) are termed homogeneous coordinates. They are uniquely defined up to scalar multiplication, i.e. (1 : 2 : 5) = (4 : 8 : 20). Remark The complex plane may be embedded in CP 2 three separate ways: C 2 0 = {(1 : z : w)} C 2 1 = {(z : 1 : w)} C 2 CP 2 2 = {(z : w : 1)} Moreover, we have that C 2 CP 2. Also, we may described this space as CP 2 = {lines in C 3 passing through the origin}. Definition The set {(z 0, z 1, z 2 ) p0 z 0 + p 1 z 1 + p 2 z 2 = 0} with at least one p i 0 is a line in CP 2. 3
4 3.1 Differential forms Definition Let F (z 0, z 1, z 2 ) be a function in three variables, Then Euler s identity is given as df = z 0 F z 0 + z 1 F z 1 + z 2 F z 2 Definition Let U C be open. A holomorphic 1-form on U is an expression ω = f(z)dz for f a holomorphic function on U.Then ω is a holomorphic 1-form in the coordinate z. Definition Let X be a Riemann surface with X = U α and coordinate neighborhoods z α : U α C. Then a holomorphic 1- form on X is a collection of holomorphic 1-forms {ω α }, one for each z α. If U α U β, then f α (z α )dz α = f α (z α ) dzα dz β dz β = f β (z β )dz β. Remark There are no non-zero holomorphic forms on C. Definition Let f : Y X be a holomorphic function of Riemann surfaces. Let ω be a meromorphic form on X. Then the inverse image of ω is ϕ ω. In local coordinates, z = ϕ(ω) with ϕ (f(z)dz) = f(z(ω))ϕ (ω). Definition Given two 1-forms ω = f 1 dz 1 + +f n dz n and η = g 1 dz 1 + +g n dz n on an n-dimensional manifold, define their distributive product by with the following properties: i. dz k dz k = 0 ii. dz k dz l = dz l dz k 3.2 Poincaré residue ω η = (f 1 dz f n dz n ) (g 1 dz g n dz n ) Definition Let X = {(z 1,..., z n ) f(z1,..., z n ) = 0} be a Riemann surface with f : C n C holomorphic. Given a differential form ω = g dz 1 dz 2 dz n, the residue of ω on X is the second expression in the product and is denoted res X (ω). ω = df f (g 1dz g n dz n ) Definition Given a piecewise smooth path γ([a : b]) X for X = U α a Riemann surface and ω a differential 1-form, the integral of ω is γ ω = ω + + [a 0:a 1] ω [a n 1:a n] for a division a = a 0 < a 1 < < a n = b of [a : b] such that [a j, a j+1 ] U β for some β and for all j. Theorem [Cauchy] If γ 1 is homotopic to γ 2, then γ 1 ω = γ 2 ω for any differential 1-form ω. Remark Let X be a Riemann surface, ω a meromorphic differential form on X, p X a pole of ω and γ X a closed path in X that only encircles one pole of ω, namely p. Then res p (ω) = 1 ω 2πi γ 4
5 Proposition Let X be a Riemann surface, ω a meromorphic differential form on X, p X a pole of ω and z be a local coordinate at p such that z(p) = 0. Then ( ) ω = c k z k dz = res p (ω) = c 1 k= N Proposition If ω is a meromorphic form on a compact Riemann surface X, then res p (ω) = 0. p X 3.3 Triangulation Theorem Suppose that X is a compact Riemann surface. Then X may be presented as a disjoint union of a finite number of sets of vertices, edges and faces, such that 1. Each vertex is a point 2. Each edge is homeomorphic to an open line segment 3. Each face is homeomorphic to the interior of a triangle 4. The closure of an edge includes the edge and both vertex endpoints 5. The closure of a face includes that face, all three bordering edges and all three bordering vertices 6. Each edge is a piecewise smooth curve This is termed a triangulation of X. Theorem Each compact Riemann surface is homeomorphic to a sphere with handles. Definition The sphere is a sphere with 0 handles. To attach a handle, remove the interior of two disjoint disks on a surface and attach ends of a cylinder to the disks. The number of handles g of a surface is termed the genus of the surface. Theorem If a surface X has genus g and is triangulated with e edges, v vertices and f faces, then v e + f = 2g 2 Proposition Let f be a meromorphic function on CP n in n variables. Then the zeros and poles of f are (n 1)-dimensional surfaces. 4 Ramification 4.1 Holomorphic functions Definition Let U C be open and connected, and f : U C a holomorphic and non-constant function with a U such that f (a) = 0. Then f is said to ramified at a. Further, if f around a is given by f(a) = b + c k (z a) k + c k+1 (z a) k+1 + for k the smallest index such that c k 0, then k is termed the ramification index of f at a. Proposition With respect to the above conditions, the ramification index of f at a is k if and only if there exist punctured neighborhoods of a where f is k-to-1. Definition Let f : X Y be a non-constant, holomorphic map of Riemann surfaces with a X. Then f is ramified at a with index n if and only if it is ramified in some local coordinates at a with index n. Denote the set of ramification points by R = {a X f is ramified on X at a}. Proposition Suppose X, Y are compact, connected Riemann surfaces with f : X Y holomorphic and non-constant. Then exactly one of the following hold: i. f(x) = Y ii. there exist a finite number of ramification points of f 5
6 Proposition Let B = f(r) Y. Then f X\f 1 (B) : X \ f 1 (B) Y \ B is a covering. That is, for each y Y \ B, there exists a neighborhood U y with U B = such that f 1 (U) = V 1 V 2 V d with V i X \ f 1 (B) open and f Vi : V i U an isomorphism for each i. Theorem Let X, Y be manifolds with f : X Y a continuous function such that: 1. For each x X, there exist open sets V x with f V : V f(v ) homomorphisms and f(v ) Y 2. If K Y is compact, then f 1 (K) X is also compact Then R is a covering. Corollary If y Y \ B, then #(f 1 (y)) does not depend on y and is termed the degree of f. Proposition Let f : X Y be a holomorphic, non-constant map of compact Riemann surfaces with deg(f) = d. For y Y, we have f 1 (y) = {x 1,..., x m } for 1 m d. Then m i=1 ( ) ramification = d = deg(f) index of f at x i Note that if f is not ramified at some x i, then its ramification index is 1. Theorem [Riemann-Hurwitz] Let f : X Y be a non-constant holomorphic mapping of compact Riemann surfaces X, Y. Suppose that R = {x 1,..., x n } with e i = (ramification index of f at x i ) and deg(f) = d. Then 4.2 Divisors 2 2g(X) = d(2 2g(Y )) r (e i 1) Definition Given a compact Riemann surface X, a divisor D is a finite formal linear combination of points of X with integer coefficients i=1 D = j n j p j for n j Z, p j X Definition Let f : X Y be a meromorphic function. Then a principal divisor is denoted by (f) = p X ord p (f) p Definition The degree of a divisor D = j n jp j is deg(d) = j n j Z. Proposition If D is a principal divisor, then deg(d) = 0. Proposition For functions f, g over identical spaces, (f g) = (f)+(g). Moreover, the sum of principal divisors is a principal divisor. Definition Two divisors D 1, D 2 are termed equivalent if D 1 D 2 is principal, and is denoted D 1 D 2. This is an equivalence relation. Remark If D 1 D 2, then deg(d 1 ) = deg(d 2 ). Proposition On the Riemann sphere, every divisor of degree 0 is principal. Proposition On the Riemann sphere, given a finite set {p 1,..., p n } with principal parts at each p i fixed, there exists a meromorphic function with poles at each p i and principal parts as given, and no other poles. 6
7 4.3 Effective and canonical divisors Definition Let D be a divisor. Then L(D) = {f f is meromorphic on X and (f) + D 0}. We write D 0 n j 0 for all j where D = j n jp j. Such a divisor D is termed effective. Proposition If D 0, then dim(l(d)) deg(d) + 1. Definition If ω is a meromorphic form on X, then the following divisor is termed canonical. (ω) = ord p (ω) p p X Proposition Given a compact Riemann surface X, any two canonical divisors are equivalent. They then belong to a common canonical class. Proposition If X is a compact Riemann surface of genus g, then deg(canonical class) = 2g 2. Proposition If D 1 D 2, then L(D 1 ) = L(D 2 ). Proposition If K is a canonical divisor, then L(K) ( space of holomorphic forms on X ). Corollary The space of holomorphic forms on X is always finite-dimensional. Further on, for any divisor D, denote dim(l(d)) = l(d). Theorem For K a canonical divisor, l(k) = g. Theorem [Riemann, Roch] For K a canonical divisor, l(d) = deg(d) + 1 g + l(k D) Proposition Suppose that f is a meromorphic function on a Riemann surface X. If a X is a zero or pole of f, then df/f has a simple pole at a and res a (df/f) = ord a (f). Proposition Let Ω(X) = {ω ω is a holomorphic form on X}. Then dim(ω(x)) = g(x). Proposition If g(x) = 0, then X C. Definition Let X be a Riemann surface defined by X = {(z, w) w 2 = (z a 1 )(z a 2 ) (z a n )}. For n 4, X is termed a hyperelliptic curve. Proposition For X a Riemann surface, if g(x) > 0 and p X, then l(p) = 1. Moreover, l((n + 1) p) l(n p) for n Z 0. Proposition If X is a compact Riemann surface of genus g with deg(d) > 2g 2, then l(d) = deg(d) + 1 g. 5 Linear systems 5.1 Separation of points Definition For D a divisor on X with l(d) > 0, the complete linear system defined by D is D = {D D 0, D D} ( )/ Definition D = P (L(D)) = L(D) \ {0} C 7
8 Remark D 1 D 2 = D 1 = D 2 2. If D is effective, then D D Corollary The space of ordered n-tuples of points of S 2 is homeomorphic to CP n. Definition Let D be a complete linear system. A point p X is termed a basepoint of D if D p for all D D. Definition If X is a Riemann Surface with genus > 1 and K X is its canonical class, then K X is termed a canonical linear system. Proposition If g 1 for a Riemann surface X, then K X has no basepoints. Theorem A divisor D has no basepoints l(d p) = l(d) 1 for all p X. Definition Define the following mapping for a Riemann surface X and a divisor D: ϕ D : X CP n = D by x {D D D, D x} We note that if D has no basepoints, then this mapping is well-defined. Moreover, if D has no basepoints, then it defines a holomorphic mapping into projective space. Proposition If L(D) = f 0,..., f n, then ϕ D : x (f 0 (x) : : f n (x)) Proposition If D has no basepoints, then ϕ D is injective for all p q, l(d p q) = l(d) 2. Definition Let X = C and D be a divisor of X with deg(d) = n > 1. Note that D does not have basepoints, and ϕ D : X CP 2 is injective. Then ϕ D (X) = X n is termed a rational normal curve, or Veronese curve. 5.2 Separation of tangent vectors Definition Let ϕ : X C n be a map of a Riemann surface X C with ϕ : z (ϕ 1 (z),..., ϕ n (z)). Then ϕ (z) = (ϕ 1(z),..., ϕ n(z)) is termed degenerate at p X if ϕ 1(p) = = ϕ n(p) = 0. Proposition The derivative of ϕ D is non-degenerate at p X l(d 2p) = l(d) 2. Theorem If D is a complete linear system without basepoints and l(d p q) = l(d) 2 for all p, q X, then ϕ D : X P dim( D ) is an embedding, and ϕ D (X) is a smooth curve. Proposition Suppose that X is a compact Riemann surface, D is a divisor of X with deg(d) 0. Then D has no basepoints and ϕ D embeds X as a smooth curve. Proposition ϕ D is not an embedding there exists a D such that deg(d) = 2 and l(d) = 2. In this case X is a hyperelliptic curve. Corollary K X does not define an embedding X is a hyperelliptic curve. Remark If g > 2, then a Riemann surface of genus g is not hyperelliptic. Moreover, if X is not hyperelliptic, then ϕ KX : X CP g 1 is an embedding, and deg(ϕ KX (X)) = 2g 2. Here ϕ Kx is termed a canonical curve. Proposition A Riemann surface of genus 1 is isomorphic to an elliptic curve. Definition Define the Weierstrass p-function to be : X R, given by (z) = 1 z 2 + ( 1 (z γ) 2 1 ) γ 2 γ Γ\{0} Note that this function converges on X = C/Γ an elliptic curve. 8
9 Theorem [Properties of the -function] 1. (z) = ( z) 2. (z) = ( z) 3. (z) = 2 1 (z γ) 3 γ Γ 4. (z + α) = (α) for all α Γ 5. (z + α) = (α) for all α Γ Theorem Let X = C/Γ for Γ = ω 1, ω 2 a lattice. Then (z) has three distinct zeros on X. The value of at these points is denoted: e 1 = ( ω1 ) 2 e 2 = ( ω2 ) 2 ( ) ω1 + ω 2 e 3 = 2 Moreover, ( (z)) 2 = 4( (z) e 1 )( (z) e 2 )( (z) e 3 ) 5.3 Embeddings Remark ( Let X be the ) space of homogeneous polynomials of degree d in m variables. d + m 1 Then dim(x) =. d Proposition Let X be a compact Riemann surface. For n > 3, X may be embedded in CP n 1. Proposition Let X = C/Γ be an elliptic curve for Γ a lattice. Then X is isomorphic to the Riemann surface Y = {(x, y) y 2 = 4(x e 1 )(x e 2 )(x e 3 )} for e i as above. Remark A Riemann surface of genus 1 is isomorphic to a cubic curve. Any Riemann surface of genus 2 is a hyperelliptic curve. Proposition If D 0, then l(d) = deg(d). Note the following new notation on divisors of a Riemann surface X. The sets below are groups with the group operation of addition. Div(X) = {all divisors on X} Div 0 (X) = {D Div(X) deg(d) = 0} Principal(X) = {D Div(X) D is principal} Pic 0 (X) = Div 0 (X)/Principal(X), the Picard group 5.4 Elliptic curves Theorem [Abel, Jacobi] Let +, denote operations for points of a divisor. Let, denote operations on group elements in C/Γ. Suppose that D = m 1 a m k a k is a divisor on X = C/Γ with m j Z, a j X, and deg(d) = m m k = 0. Then D is principal m 1 a 1 m k a k = 0. Corollary If X = C/Γ, then Pic 0 (X) X. Theorem Pic 0 (X) = C g /Γ for Γ C g a lattice of rank 2g. Definition Let k Z 2 and Γ C a lattice. Then G k := γ Γ\{0} 1 is termed an Eisenstein series. γ2k 9
10 Remark By rewriting in terms of Eisenstein series, the following conclusions are reached: ( (z)) 2 = 4 3 (z) 60G 2 (z) + 140G 3 e 1 + e 2 + e 3 = 0 e 1 e 2 + e 1 e 3 + e 2 e 3 = 15G 2 e 1 e 2 e 3 = 20G 3 Definition Let C CP n be a smooth curve of degree d, and L a line in CP n. If L C = {q} and the divisor L C = d q, then q is termed an inflection point of C. Proposition Let C be as above, and q 1, q 2 two inflection points of C. Then the line that passes through q 1 and q 2 also passes through a distinct third inflection point of C. Proposition Let Γ be a lattice on C and f : X X C a holomorphic map with a fixed point f(0) = 0. Then there exists λ C such that f(z) = λz (mod Γ). Proposition Let X 1 = C/Γ 1 and X 2 = C/Γ 2 be two elliptic curves. Then X 1 X 2 there exists non-zero λ C such that λγ 1 = Γ 2. 6 Line bundles 6.1 Construction Definition Suppose that X is a complex manifold with U X open. A line bundle on X is both: 1. A complex manifold T, the total space of the bundle 2. A holomorphic mapping π : T X, the projection that satisfies the following conditions: a. Φ is an isomorphism such that π Φ = π b. For each y U, Φ(π 1 (y)) = (π ) 1 (y) c. For each x X, π 1 (x) C and 0 π 1 (x) This may be envisioned as the following graph that commutes: π T X i i π 1 (U) U π Φ π U C where: π = π π 1 (U) π : (x, λ) x is also a projection i : U X is an inclusion Moreover, the function Φ is termed the local trivialization. Definition Consider two domains U, V X with local trivializations over them: Φ U : π 1 (U) U C Φ V : π 1 (V ) V C Then for all x U V, both functions are defined, and they differ by a linear automorphism given by Φ π 1 (x) = g UV Φ V (x) The functions of the type g UV are termed transition functions of the line bundle. 10
11 Proposition [the Cocycle condition] Let U, V, W X and x U V W. Then g UW (x) = g UV (x)g V W (x). Proposition Suppose that X is a complex manifold and U j = X is an open covering with, for each i j a holomorphic function g ij : U i U j C \ {0} such that on U i U j U k with k / {i, j} Then there exists a line bundle T transition functions. g ij g jk = g ik g ij g ji = 1 π X such that T has trivializations over each U j for which the g ij are Remark The line bundle as constructe.d above is usually denoted O X ( 1). 6.2 Sections Definition A section of a line bundle O X ( 1) is a holomorphic mapping s : X T such that π s = id X. Note that if x X, then s(x) π 1 (x). If s 1, s 2 are sections, then we may define addition of sections by (s 1 + s 2 ) : x s 1 (x) + s 2 (x). Definition Let s : X T be a section with x U X and Φ U : π 1 (U) U C the local trivialization. Then there exists a holomorphic function s U : U C such that Moreover, if x U V for V X, then Φ U (s(x)) = (x, s U (x)) s U (x) = g UV (x)s V (x) Proposition There is a 1-1 correspondence between sections of a given line bundle and collections of the holomorphic s U : U C with s U = g UV s V. Remark Every section of O CP n( 1) is identically zero. Definition Suppose that L, M are line bundles on X with transition functions on U j g ij : U i U j C \ {0} h ij : U i U j C \ {0} for L for M Then the tensor product L M is the line bundle with transition functions g ij h ij over U i U j Definition Using the same notation as above, (O CP n( 1)) = O CP n(1). 6.3 On Riemann surfaces Proposition (Sections of O X (D)) L(D). Proposition Any line bundle on a compact Riemann surface X is of the form O X (D) for some divisor D. Definition (The space of sections of a line bundle L on X) = H 0 (X, L). Here H 0 denotes the 0th cohomology group. Proposition Suppose that S 1, S 2 H 0 (X, O X (D)). Then (S 1 ) (S 2 ). Remark Let L 1 = O X (D 1 ) and L 2 = O X (D 2 ). Then L 1 L 2 = O X (D 1 + D 2 ). 11
12 Proposition Let D be a divisor on a Riemann surface X. Then O X (D) is trivial D is principal. Proposition Let D 1, D 2 be divisors on a Riemann surface X. Then the following are equivalent: i. D 1 D 2 ii. O X (D 1 ) O X (D 2 ) iii. O X (D 1 ) O X (D 2 ) 1 = O X (D 1 D 2 ) is trivial Corollary Line bundles are equivalence classes of divisors, with the canonical line bundle having sections of holomorphic forms. 12
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