COMPLEX GEOMETRY Course notes

More documents

Recommendations

Info

Similarly, if Ω X,R := T ∨ X,R , then with the identification Ω X,R ⊗ C = Ω X ⊕ Ω X Ω X ←→ Ω 1,0 X ←→ dz i, Ω X ←→ Ω 0,1 X ←→ dz i. Definition 3.1.2. An almost complex structure is called integrable if it comes from a complex structure. The only spheres with an almost complex structure are S 2 and S 6 . The 6-sphere S 6 has an almost complex structure via the octonions, by taking the multiplication structure in the multiplicative structure in the sphere in the purely imaginary part of octonions. Question: Does it exist complex structures on S 2 ? Theorem 3.1.2 (Newlande - Niremberg). J is integrable if and only if [T 1,0 X , T 1,0 X ] ⊆ T 1,0 X . Proposition 3.1.1 (Poincaré Lemma). Let α be a closed differential from a differentiable manifold with deg(α) > 0. Then locally α = dβ, for some form β. Proposition 3.1.2 (∂-Poincaré Lemma). Let α be a ∂-closed differential from a differentiable manifold with (p, q) = deg(α) and q > 0. Then locally α = ∂β, for some form β. Proof: First we show that we can reduce the problem to the case p = 0 and q = 1. In general, we know α = ∑ α I,J dz I ∧ dz J ∂α = ∑ dα I,J ∧ dz I ∧ dz J = 0 Then α I = ∑ α I,J ∂z J is ∂-closed and of type (0, q), and so α I = ∂β if and only if α = (−1) p ∂( ∑ ∂z I ∧β I ). Henceforth assume that α is of type (0, q), i.e., α = ∑ α J dz J . Apply the induction on the largest integer k such that k ∈ J and α J ≠ 0. Necessarily k ≥ q and k = q implies α = fdz 1 ∧ · · · ∧ dz q . In the latter case ∂α = 0 if and only if f is holomorphic in the variables z i with i > q. We may know apply the following result: Proposition: There exists a differentiable function g holomorphic in the variables z i , with i > q, such that ∂g ∂z q = f and hence α = (−1)q−1 ∂(gdz 1 ∧ · · · ∧ dz q−1 ). Now assume the ∂-Poincaré Lemma proved for k − 1 > q. Write α = α 1 + α 2 dz k , α 2 = ∑ α 2,J dz J where |J| = q − 1 and J ⊆ {1, . . . , k − 1}. So ∂ = 0 implies α 2,J is holomorphic in variables z l , with l > k. Hence by the previous proposition, we have α 2,J = ∂β 2,J ∂z k , where β 2,J is holomorphic in z l , l > k. Then ∂β = ∂(β 2,J dz J ) = (−1) q−1 α 2 ∧ dz k + α ′ 1 44
where α ′ 1 involves only the coordinate z l for l < k. Thus, α = α ′′ 1 + ∂β where α ′′ 1 for l < k. Since β is holomorphic in the z l for l > k, we have ∂α ′′ 1 = 0, q = deg(α ′′ 1) < k. We conclude by induction. Proof of the previous proposition: We restrict to the case p = 0 and q = 1. Let α = fdz. As statement is local, we may assume that supp(f) is compact. Define g = 1 ∫ 2πi C f(z) dζ ∧ dζ := lim ζ − z ɛ→0 ∫ 1 2πi C\D(z,ɛ) f(z) dz ∧ dz. ɛ − z This limit exists since f is bounded and 1 ζ−z in integrable on D. We want to show that ∂g = α = fdz. ∫ Now g = g ɛ where g ɛ = 1 f(γ+z) 2πi C\D(0,ɛ) γ dγ ∧ dγ. Then ( ∫ ) ∂g ɛ (z) = 1 dγ ∧ dγ ∂ z f(γ + z) dz 2πi C\D(0,ɛ) γ implies ∂ ∂z g(z) = ∂ zg(z) = 1 2πi ∫ 1 = lim ɛ→0 2πi ∫ C C\D(z,ɛ) ∂ ∧ dγ f(γ + z)dγ ∂z γ ∂ ∂ζ f(ζ)dζ ∧ dζ ζ − z . ( ) On C\D(z, ɛ), we have ∂f dζ∧dζ (z) ∂ζ ζ−z = −d ɛ f(ζ) dζ ζ−z . By the Stokes Theorem, we get Hence ∂g = fdz. ∫ 1 ∂f dζ ∧ dζ 2πi C\D(z,ɛ) ∂ζ ζ − z = 1 ∫ f(ζ) −→ f(z) as ɛ −→ 0. 2πi ∂D(z,ɛ) ζ − z 45
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 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: Chapter 3 COMPLEX MANIFOLDS 3.1 Com
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;

COMPLEX GEOMETRY Course notes

Create successful ePaper yourself

Delete template?

Save as template?