COMPLEX GEOMETRY Course notes

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1.2 Analyticity A function f : U −→ C is said to be real analytic on U if for every z 0 = (x 0 , y 0 ) ∈ U there exists a neighbourhood V of z 0 such that f(z) = ∑ ∞ α,β=0 a α,β(x − x 0 ) α (y − y 0 ) β for every z ∈ V. Similarly, f is said to be complex analytic on U if for every z 0 ∈ U there exists a neighbourhood V of z 0 such that f(z) = ∑ ∞ n=0 a n(z − z 0 ) n for every z ∈ V. In both cases the equality means normal convergence in U, i.e., uniform convergence on compacts in U. Theorem 1.2.1. f ∈ O(U) if and only if f is complex analytic on U. Proof: We know that On the other hand, f(z) = 1 ∫ f(w) 2πi ∂D r(z 0) w − z dw. ( 1 w − z = 1 1 w − z 1 − z−z0 w−w 0 ) 1 = w − z 0 ∞ ∑ n=0 ( z − z0 w − z 0 ) n . It follows that where and |w − z 0 | = r on ∂D r (z 0 ). f(z) = 1 ∫ 2πi ∂D r(z 0) f(z) ∞∑ (z − z 0 ) n ∞ (w − z 0 ) n+1 dw = ∑ a n (z − z 0 ) n n=0 n=0 a n = 1 ∫ f(w) 2πi ∂D r(z 0) (w − z 0 ) n+1 dw = 1 n! f (n) (z 0 ) Theorem 1.2.2. If f ∈ O(U) is non-constant, where U is connected, then f −1 (0) is discrete in U. Proof: Suppose f −1 (0) is not discrete. Let γ be an isolated point in f −1 (0) and consider the Taylor expansion of f about γ, ∞∑ f (n) f(z) = (z − γ) n n! n=0 for every z ∈ D r (γ), where r is the radius of convergence of the series. Since γ is not isolated in f −1 (0), there exists z ∈ f −1 (0) ∩ D r (γ). We have 0 = ∞∑ n=0 f (n) (z − γ) n n! 4
and so f (n) (γ) = 0 for every n ≥ 0. It follows that f is constant on an neighbourhood of γ, getting a contradiction. Corollary 1.2.1 (Analytic Continuation or Identity Theorem). If f = g on a non-discrete subset of U and f, g ∈ O(U), then f ≡ g on U. Lemma 1.2.1 (Schwartz). If f ∈ O(D) and f ≤ M on ∂D, |f(z)| ≤ M|z| on ∂D r for every r ∈ (−ɛ, 1), then f(z) = Nz; where |N| < M. Another version: If |f(z)| ≤ M on D and f(0) = 0, then f(z) = Nz with |N| < M. Here, D de<strong>notes</strong> the Poncaré disk, i.e., the disk {z ∈ C : |z| 2 < 1} endowed with the metric |z − w| 2 δ(z, w) = 2 (1 − |z| 2 )(1 − |w| 2 ) . 5
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: Theorem 1.1.3 (Local Structure of f
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 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
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Lemma 4.1.1. If F is a presheaf ove
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Construction: Let X be an affine va
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4.2 Cohomology of sheaves Let X be
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4.4 Derived functors Given a functo
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Chapter 5 HARMONIC FORMS 5.1 Harmon
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Theorem 5.1.1 (Main Theorem of the
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5.3 Review Theorem 5.3.1. Let X be
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Theorem 5.4.1. Let (X, g) be a comp
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5.5 Index Theorem (Heat Equation ap
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BIBLIOGRAPHY [1] Griffiths Phillip;
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