COMPLEX GEOMETRY Course notes

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Corollary 5.4.1. There exists an orthogonal direct sum decomposition where Im(∆) = Im(d) + Im(d ∗ ). A p (X) = H p (X) ⊕ d(A p−1 (X)) = H p (X) ⊕ d ∗ (A p−1 (X)) Corollary 5.4.2. There is an isomorphism H p (X) ∼ = H p dR (X) given by α ↦→ [α]. Then O ∈ A p (X) =⇒ O = α + dd ∗ γ + d ∗ dγ, where α ∈ H(X). Also, ||d ∗ dγ|| 2 = 〈d ∗ dγ, α〉 = 〈dγ, dα〉 , where dα = 0. 70
5.5 Index Theorem (Heat Equation approach) Let ∆ p : A p (X) −→ A p (X), λ ∈ R ≥0 . Let E p λ denote the λ-eigenspace for ∆p (finite dimensonal). The square root √ ∆ = δ is called the Dirac operator. Lemma 5.5.1. The sequence is exact for λ > 0. 0 −→ E 0 λ d −→ Eλ 1 d −→ · · · −→ Eλ n −→ 0 Proof: ω ∈ E p λ =⇒ ∆p+1 dω = d∆ p ω = λdω =⇒ dω ∈ E p+1 λ . Now ω ∈ E p λ and dω = 0 =⇒ λω = ∆p ω = d ∗ d + dd ∗ ω =⇒ ω = d ( 1 λ d∗ ω ) . Then ∆d ∗ ω = d ∗ ∆ω = λd ∗ ω. Corollary 5.5.1. ∑ p (−1)p dim(E p λ ) = 0. Corollary 5.5.2. Let {λ p i } be the spectrum of ∆p , with terms repeated n times if multi = n. Then where ∑ ′ i is over i where λ(p) i = 0. ∑ (−1) p tre −t∆p = ∑ p p (−1) p e −tλ(p) i = ∑ p (−1) p ′∑ i e −λ(p) i (t) Note that ∑ ′ i e−λ(p) i (t) = dim(Ker(∆ p )). Hence X (X) = ∑ p = ∑ p (−1) p dim(Ker(∆ p )) = ∑ (−1) p tre −t∆(p) , where e −t∆(p) = T t , p (−1) ∑ ∫ p e (p) (x, x, t)dVol(x). i X Proposition 5.5.1. e(x, x, t) ∼ (4πt) −n/2 ∑ ∞ k=0 u k(x, t)t k , where u k (x, t) is explicitly given in terms of components of curvatures. Hence, as t −→ 0, we have ( X ∼ 1 n/2 ∑ ∞ ∫ 4πt k=0 X ) ∞∑ (−1) p tru p k (x, x)dVol(x) t k . p=0 71
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MARCO A. PÉREZ B. Université du Q
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TABLE OF CONTENTS 1 COMPLEX ANALYSI
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Chapter 1 COMPLEX ANALYSIS 1.1 Comp
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Theorem 1.1.3 (Local Structure of f
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and so f (n) (γ) = 0 for every n
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Theorem 1.3.3 (Riemann Extension Th
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(5) C and C ∗ = C − {0}. (6) Th
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2.3 Meromorphic functions and diffe
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This shows that C ∼ = hol P 1 . F
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2.5 Dimension on Riemann surfaces D
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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
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: Theorem 5.4.1. Let (X, g) be a comp
Page 79: BIBLIOGRAPHY [1] Griffiths Phillip;
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