COMPLEX GEOMETRY Course notes

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1 b g a g a 1 Since S is oriented we have that there exists an intersection pairing in H 1 (any two loops, or elements in H 1 , are homotopic to loops which are transversal). Note that a and b are the same element in H 1 if and only if a + (−b) = a − b = ∂U, for some open set U ⊆ S, i.e., a is homologous to b. Here −b de<strong>notes</strong> reverse orientation. So if a is homotopic to b then they are homologous. Consider the following picture: chart −→ b y C p −→ a x We have −→ a ∧ p −→ b = kx ∧p y, and denote and (a, b) p = sign(k) (a, b) = ∑ p∈a∩b (a, b) p Since the previous sum does not depend on the choice of the prepresentative, we have a well defined map ( , ) : H 1 × H 1 −→ Z If S is a Riemann surface of genus g, we have in terms of a basis that (a i , b j ) = (b i , b j ) = 0, for every i, j, (a i , b j ) = −(b j , a i ) = δ ij . 28
Then we have that each (a i , b j ) is a skew-linear pairing which is unimodular (det = 1). a i b i a i 0 1 b i −1 0 det 1 Hence H 1 (S; Z) = Hom(H 1 (S; Z), Z) =: H 1 (S; Z). Let ω ∈ M ′ (S) be a closed meromorphic differential. Recall that ω is exact if and only if ω = df for some meromorphic function f ∈ M(S). Can we choose a function f ∈ M(S) such that f = ∫ z ω? As an exercise, p think if it is possible that Res p (ω) = 0 for every p ∈ S. Consider the period homomorphism Π ω : H 1 (S; Z) −→ C ∫ [γ] ↦→ ω, γ is a loop. γ This map is well defined. For if γ = γ ′ in H 1 then γ − γ ′ = ∂u, and by the Stokes Theorem we have ∫ γ ω = ∫ γ ′ ω. Also, Π ω is a C-linear map. By the de Rham Theorem, we have an isomorphism HdR 1 (Z) ∼ = H 1 (S; Z) ⊗ C if Z is compact. isomorphism is given by [ω] ∈ HdR(Z) 1 ↦→ Π ω Such an Corollary 2.10.1. dim C HdR 1 (S) = 2g. Proof: We have 2 − 2g = χ(S) = ∑ (−1) i h i dR(S), where h i dR (S) = dim(H1 dR (S)), = h 0 dR(S) − h 1 dR(S) + h 2 dR(S) = 1 − dim C HdR(S) 1 + 1. Theorem 2.10.1. The de Rham ismomorphism carries wedge product of forms defined by isomorphically to the (cup) product on H 1 (S). HdR 1 × HdR 1 −→ C ∫ ([ω 1 ], [ω 2 ]) ↦→ ω 1 ∧ ω 2 S 29
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 30 and 31: 2.9 Topology of Riemann surfaces On
Page 32 and 33: Note that HdR 0 (Z) = C if Z is con
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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