COMPLEX GEOMETRY Course notes

More documents

Recommendations

Info

This implies 4π n/2 ∫ ∞∑ { (−1)tru p 0 k (x, x)dVol(x) = X (X) X p=0 if k ≠ n/2, k = n/2. Theorem 5.5.1 (Gauss-Bonet). Let n = dim(X) be even. Then ∫ X (X) = ω, where ω is given in a local frame by ∑ ω = c n (signσ)(signτ)R σ(1)σ(2)τ(1)τ(2) · · · R σ(n−1)σ(n)τ(n−1)τ(n) and c n = (−1)n/2 (8π) n/2 ( . n 2 )! σ,τ X For n = 2, ω = 1 8π (R 1212 −R 1221 −R 2112 −R 2121 )dA = − 1 2π RdA 1212 = K 2π dA where K is the Gaussian curvature. 72
BIBLIOGRAPHY [1] Griffiths Phillip; Harris Joseph. Principles of Algebraic Geometry. Pure & Applied Mathematics. John Wiley and Sons, Inc. New York (1978). [2] Voisin, Claire. Hodge Theory I. [3] Voisin, Claire. Hodge Theory II. [4] Arapura, Danu. Algebraic Geometry over the Complex Numbers. [5] Rosemberg, Steven. The Laplacian on a Riemannian Manifold. [6] Yu, Yan-Lin. The Index Theorem and the Heat Equation Method. 73
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 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: 5.5 Index Theorem (Heat Equation ap
show all

COMPLEX GEOMETRY Course notes

Create successful ePaper yourself

Delete template?

Save as template?