COMPLEX GEOMETRY Course notes

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In part, if n is even then d ∗ = − ∗ d∗. Similarly, Fact 5.1.2. ∂ ∗ = − ∗ ∂∗ and ∂ ∗ = − ∗ ∂∗ are formal adjoint of ∂ and ∂ with respect to the L 2 metric in A k C . Proof: (∂α, β) = ∫ X ∂α ∧ ∗β = − ∫ X (−1)|α| α ∧ ∂ ∗ β = − ∫ X (−1)|α| α ∧ ∗ ∗ −1 ∂ ∗ β = (α, ∂ ∗ β). More generally, if (E, h) is a Hermitian vector bundle then there exists a C-anti linear isomorphism of vector bundles given by h : Ω 0,q X ⊗ E = Ω0,q (E) −→ (Ω 0,q ⊗ E) ∨ ∼ = Ω n,n−q ⊗ E ∨ , where ∆ 2n X = Ωn,n = RdVol(x). So it gives am antilinear isomorphism ∗ E : Ω 0,q (E) ∼ −→ Ω n,n−q (E ∨ ) = K X ⊗ Ω 0,n−q (E ∨ ) called the Hodge Star, where K X = Ω n,0 = ∆ n C T X ∨ bundle of a complex manifold X. (holomorphic line bundle) is called the canonical Fact 5.1.3. ∂ ∗ X = (−1) q ∗ −1 E ◦∂ K X ⊗E ∨ : A0,q (E) −→ A 0,q−1 (E) is the formal adjoint of ∂ E . Fact 5.1.4. (d ∗ ) 2 = (∂ ∗ E) 2 = (∂ ∗ ) 2 = 0. Definition 5.1.2. Let (X, g) be a Riemannian manifold, ∆ := dd ∗ + d ∗ d = (d + d ∗ ) 2 Definition 5.1.3. Let (X, h) be a Hermitian manifold, ∆ ∂ := ∂∂ ∗ + ∂ ∗ ∂ = (∂ + ∂ ∗ ) 2 , ∆ ∂ := ∂∂ ∗ + ∂ ∗ ∂ = (∂ + ∂ ∗ ) 2 . If further E −→ X is a holomorphic vector bundle with a Hermit metric, we write ∆ E for ∆ ∂E = (∂ E +∂ ∗ E) 2 . From construction, 〈α, ∆ d α〉 = ||dα|| 2 + ||d ∗ α|| 2 and analogously for the Hermitian case. Corollary 5.1.1. Ker(∆ d ) = Ker(d) ∩ Ker(d ∗ ). Definition 5.1.4. An element of Ker(∆ d ) is called harmonic, i.e., it is killed by d and d ∗ . 64
Theorem 5.1.1 (Main Theorem of the <strong>Course</strong>). Let { ( ⊕k Ω (F, φ) = ( k X , ∆ d) ) ⊕q Ω 0,q (E), ∆ ∂E for the Riemannian case, for the Hermitian case. where φ : F −→ F is an automorphism, and F = ⊕ k Ω k X or F = ⊕ qΩ 0,q (E). 65
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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
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
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: Chapter 5 HARMONIC FORMS 5.1 Harmon
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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