COMPLEX GEOMETRY Course notes

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2.9 Topology of Riemann surfaces One known fact about Riemann surfaces is that any Riemann surface is orientable. Recall that a manifold is orientable if its transition functions have positive Jacobian . Let Z be a Riemann surface, consider two intersecting charts U α and U β , and let z ∈ U α ∩ U β . Then we have df dz (z) = f ′ (z) as an R-matrix. We write f = u + iv, then df = α + iβ. β α ◦ β −1 holomorphic α C By the Cauchy-Riemann equations, we have ( α β df = −β α ) and so Jac(α ◦ β −1 ) = det(D(α ◦ β −1 )) = |df| 2 = det(df) = α 2 + β 2 > 0. It is also known that every Riemann surface is obtained by attaching handles to CP 1 = S 2 . # handles = # holes = : topological genus = g ∞ 0 Theorem 2.9.1. Any Riemann surface is triangularizable. 24
This fact is easy to prove for Compact Riemann surfaces, and shows that any such is obtained by attaching handles to S 2 = CP 1 = C ∪ {∞}. We denote Z g for a Riemann surface Z of genus g. It is known that the first homotopy group of Z g is given by π 1 (Z g ) = Z a1 ∗ Z b1 ∗ · · · ∗ Z ag ∗ Z bg / 〈 a 1 b 1 a −1 1 b−1 1 · · · a g b g a −1 g Notice that every hole gives two generators. b −1 g 〉 b 1 b g a g a 1 a 1 b 1 a −1 1 b −1 1 b −1 g Let A i denote the space of C ∞ -complex valued i-forms on a Riemann surface, then A = A 1,0 ⊕ A 0,1 , ω ′ = fdx + gdy = hdz + hdz, where hdz ∈ A 1,0 and hdz ∈ A 0,1 . We also have differential operators making the following diagram commutes d A 1 d C ∞ − functions = A 0 ∂ A 0,1 ∂ A 2 ∂ A 1,0 ∂ where d = ∂ + ∂. Definition 2.9.1. An i-form ω ∈ A i is called closed if dω = 0. It is called exact if ω = dα. Since d ◦ d = 0, we have Im(d) ⊆ Ker(d), and so we define the de Rham cohomology groups of Z and the quotient HdR i (Z) = {closed i-forms}/{exact i-forms} 25
Page 1: MARCO A. PÉREZ B. Université du Q
Page 5 and 6: TABLE OF CONTENTS 1 COMPLEX ANALYSI
Page 7 and 8: Chapter 1 COMPLEX ANALYSIS 1.1 Comp
Page 9 and 10: Theorem 1.1.3 (Local Structure of f
Page 11 and 12: and so f (n) (γ) = 0 for every n
Page 13: Theorem 1.3.3 (Riemann Extension Th
Page 16 and 17: (5) C and C ∗ = C − {0}. (6) Th
Page 18 and 19: 2.3 Meromorphic functions and diffe
Page 20 and 21: This shows that C ∼ = hol P 1 . F
Page 22 and 23: 2.5 Dimension on Riemann surfaces D
Page 24 and 25: 2.6 Covering spaces Recall that an
Page 26 and 27: 2.7 The Riemann surface of an algeb
Page 28 and 29: Theorem 2.8.2. Let P (T ) = T n + c
Page 32 and 33: Note that HdR 0 (Z) = C if Z is con
Page 34 and 35: 1 b g a g a 1 Since S is oriented w
Page 36 and 37: Recall that Ω denotes the space of
Page 38 and 39: 2.12 Harmonic differentials and Hod
Page 40 and 41: 2.13 Analysis on the Hilberts space
Page 42 and 43: 2.14 Review Theorem 2.14.1 (Orthogo
Page 44 and 45: Claim 2.15.3. For every α ∈ N 2
Page 46 and 47: 2.17 Projective model Theorem 2.17.
Page 49 and 50: Chapter 3 COMPLEX MANIFOLDS 3.1 Com
Page 51 and 52: where α ′ 1 involves only the co
Page 53 and 54: Corollary 3.2.2. If a symplectic st
Page 55 and 56: 3.4 Review A complex structure on a
Page 57: Similarly for a holomorphic vector
Page 60 and 61: Lemma 4.1.1. If F is a presheaf ove
Page 62 and 63: Construction: Let X be an affine va
Page 64 and 65: 4.2 Cohomology of sheaves Let X be
Page 66 and 67: 4.4 Derived functors Given a functo
Page 69 and 70: Chapter 5 HARMONIC FORMS 5.1 Harmon
Page 71 and 72: Theorem 5.1.1 (Main Theorem of the
Page 73 and 74: 5.3 Review Theorem 5.3.1. Let X be
Page 75 and 76: Theorem 5.4.1. Let (X, g) be a comp
Page 77 and 78: 5.5 Index Theorem (Heat Equation ap
Page 79: BIBLIOGRAPHY [1] Griffiths Phillip;

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