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COMPLEX GEOMETRY Course Notes MARCO A. PEREZ´ B. Universite´ du Quebec´ a` Montreal.´ Departement´ de Mathematiques.´ COMPLEX GEOMETRY Course notes December, 2011. These notes are based on a course given by Steven Lu in Fall 2011 at UQAM.` All errors are responsibility of the author. On the cover: a picture of the Riemann Sphere (taken from: http://en.wikipedia.org/wiki/Riemann sphere). i ii TABLE OF CONTENTS 1 COMPLEX ANALYSIS 1 1.1 Complex Analysis in one variable .............................1 1.2 Analyticity ............................................4 1.3 Complex Analysis in several variables ..........................6 2 RIEMANN SURFACES 9 2.1 Complex manifolds, Lie groups and Riemann surfaces ................9 2.2 Holomorphic maps ....................................... 11 2.3 Meromorphic functions and differentials ......................... 12 2.4 Weierstrass P -function on C ................................. 15 2.5 Dimension on Riemann surfaces .............................. 16 2.6 Covering spaces ........................................ 18 2.7 The Riemann surface of an algebraic function ..................... 20 2.8 Review .............................................. 21 2.9 Topology of Riemann surfaces ............................... 24 i 2.10 Product structures on i HdR(Z) ............................. 27 2.11 Questions about (compact)L Riemann surfaces ..................... 31 2.12 Harmonic differentials and Hodge decompositions ................... 32 2.13 Analysis on the Hilberts space of differentials ..................... 34 2.14 Review .............................................. 36 2.15 Proof of Weyl's Lemma ................................... 37 2.16 Riemann Extension Theorem and Dirichlet Principle ................ 39 2.17 Projective model ........................................ 40 2.18 Arithmetic nature ....................................... 41 iii 3 COMPLEX MANIFOLDS 43 3.1 Complex manifolds and forms ............................... 43 3.2 K¨ahlermanifolds ........................................ 46 3.3 Metrics and connections ................................... 48 3.4 Review .............................................. 49 3.5 The Fubini Study metric ................................... 50 4 SHEAF COHOMOLOGY 53 4.1 Sheaves .............................................. 53 4.2 Cohomology of sheaves .................................... 58 4.3 Coherent sheaves ........................................ 59 4.4 Derived functors ........................................ 60 5 HARMONIC FORMS 63 5.1 Harmonic forms on compact manifolds .......................... 63 5.2 Some applications of the Main Theorem ......................... 66 5.3 Review .............................................. 67 5.4 Heat equation approach ................................... 68 5.5 Index Theorem (Heat Equation approach) ....................... 71 BIBLIOGRAPHY 73 iv Chapter 1 COMPLEX ANALYSIS 1.1 Complex Analysis in one variable Let U C = R2 be an open subset of the complex plane. We shall denote an element z C by z = x + iy, ⊆ 2 where i = p 1. An function f : U C is holomorphic on U if it is complex differentiable at all points of , i.e., − −! U df f(z) f(z0) f (z ) = (z ) = lim − 0 0 dz 0 z z0 z z0 ! − exists for every z0 . We shall denote this by f (U). If S C is any subset, we shall say that f is holomorphic on S (2f U (S)) if f is holomorphic on2 a O open neighbourhood⊆ of S. 2 O @f @f 2 If the function f is R-differentiable on U then @x dx + @y makes sense and df(x; y) HomR(Tz=x+iyU; R ). Recall that 2 dz = dx + idy and dz = dx idy − Using these expressions, we can write the differential df as df = 1 @f i @f dz + 1 @f + i @f dz = @f dz + @f dz 2 @x − @y 2 @x @y @z @z Notice the following relations @f @f @f @f @z = @z and @z = @z Recall that f is complex differentiable @f () f is R-differentiable and @z = 0 (Cauchy-Riemann condition) @u =()i @v @z − @z ux uy () df = is a rotation matrix up to a real scalar multiple (ux = vy and uy = vx). vx vy − 1 We shall denote V U if V U is compact (V is precompact in U) and @V is rectifiable, i.e., @V is piecewise smooth. ⊂⊂ ⊆ Theorem 1.1.1 (Cauchy). f (U) if and only if f = 0, for every V U simple connected. 2 O @V ⊂⊂ R Theorem 1.1.2 (Cauchy's Integral Formula). z V U if and only if 0 2 ⊂⊂ 1 f(z) f(z ) = dz: 0 2πi z z Z@V − 0 If V = D(z0) is a disk centered at z0 of radius , then we shall denote the previous integral by 1 2π f(z ) = Avg (f) := f(z + eiθ)dθ: 0 @V 2π 0 Z0 Corollary 1.1.1 (Liouville Theorem). Every holomorphic function on C is constant. Proof: Let V = B(z0), z0 C. We show that f 0(z0) = 0. Using the Cauchy's Integral formula, we have 2 1 f(z) f 0(z ) = 0 2πi (z z )2 Z@V − 0 Notice that f is bounded on @V . Then f(z) M on @V for some M > 0. So we have j j j j ≤ 1 f(z) 1 f(z) f 0(z ) = dz j j dz j 0 j 2π (z z )2 ≤ 2π2 z z 2 Z@V − 0 Z@V j − 0j 1 M = f(z) dz dz 2π 2 j j ≤ 2π2 Z@V Z@V M M = 2π = 2π2 · It follows f 0(z0) 0 as . Hence f 0(z0) = 0 for every z0 C and f is constant in C. j j −! −! 1 2 Corollary 1.1.2 (Riemann Extension Theorem). If f (U z0), bounded near z0 and continuous at z0, then f (U). 2 O − 2 O Proof: If f (U z0) then f (U B(z0)), for some > 0. Then the result follows since the Cauchy's Integral2 O Formula− still holds2 O in this− case. 2 Theorem 1.1.3 (Local Structure of f (U)). If f (U) is non-constant at z U. Let 2 O 2 O 0 2 m = min n > 0 / f (n)(z ) = 0 : f 0 6 g Then there exists a bi-holomorphic function ' : V W from a neighbourhood of V of z0 to a neighbourhood W of 0 with '(z ) = 0 such that −! 2 C 0 f(z) f(z ) = '(z)m; for every z V: − 0 2 m Proof: Note that f(z) f(z0) = (z z0) g(z) with g(z0) = 0 and g (U). Since the quotient f(z) f(z0) − − 6 2 O − is bounded on U z and continuous at z , we have by the Riemann Extension Theorem that z z0 0 0 − − m 1 f(z) f(z0) (z z ) g(z) = − is holomorphic on U. Proceeding this way, we have that g(z) (U). 0 − z z0 − − 2 O We study several cases: If n = 1 then f 0(z ) = 0 and by the Inverse Function Theorem we can choose 0 6 '(z) = f(z) f(z0). Now assume n = 1. Since g(z0) = 0 then g(z) = 0 on a neighbourhood of z0. So we can write−g = hm on a neighbourhood6 V . We have 6 6 f(z) f(z ) = [h(z)(z z )]m − 0 − 0 m with '0(z ) = h(z ) = 0. Hence, up to a local change of coordinates, f is locally of the form z z for 0 0 6 7! some m. Such a number m is called the ramification degree of f at z0. Corollary 1.1.3 (Open Mapping Theorem). If f (U) is non-constant and U is connected, then f is an open mapping. 2 O Corollary 1.1.4. If f (U) and f has a local maximum at z0 U, where U is an open connected set, then f is constant on U2. O j j 2 Proof: Suppose f is not constant. Then by the Open Mapping Theorem, we have that B (f(z )) f(U) 0 ⊆ for some > 0. In this neighbourhood there are some points of modulus greater that 0. Hence f(z0) is not a local maximum. 3 1.2 Analyticity A function f : U C is said to be real analytic on U if for every z0 = (x0; y0) U there exists a −! 2 neighbourhood V of z0 such that α β f(z) = 1 a (x x ) (y y ) for every z V: α,β=0 α,β − 0 − 0 2 Similarly, f is said to be complexP analytic on U if for every z0 U there exists a neighbourhood V of z0 such that 2 n f(z) = 1 a (z z ) for every z V: n=0 n − 0 2 In both cases the equality means normalP convergence in U, i.e., uniform convergence on compacts in U. Theorem 1.2.1. f (U) if and only if f is complex analytic on U. 2 O Proof: We know that 1 f(w) f(z) = dw: 2πi w z Z@Dr (z0) − On the other hand, n 1 1 1 1 1 z z = = 0 : z z0 − w z w z 1 − w z0 w z0 w w0 ! n=0 − − − − − X − It follows that 1 1 (z z )n 1 f(z) = f(z) − 0 dw = a (z z )n 2πi (w z )n+1 n − 0 @Dr (z0) n=0 0 n=0 Z X − X where 1 f(w) 1 a = dw = f (n)(z ) n 2πi (w z )n+1 n! 0 Z@Dr (z0) − 0 and w z = r on @D (z ). j − 0j r 0 1 Theorem 1.2.2. If f (U) is non-constant, where U is connected, then f − (0) is discrete in U. 2 O 1 1 Proof: Suppose f − (0) is not discrete. Let γ be an isolated point in f − (0) and consider the Taylor expansion of f about γ, 1 f (n) f(z) = (z γ)n n! − n=0 X 1 for every z Dr(γ), where r is the radius of convergence of the series. Since γ is not isolated in f − (0), 2 1 there exists z f − (0) D (γ).
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