COMPLEX GEOMETRY Course notes

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Construction: Let X be an affine variety over K, and M a module over K[x]. Then X = zeroes of {f 1 , . . . , f l } over C N . We have K[X] := C[z 1 , . . . , z N ]/ 〈f 1 , . . . , f l 〉 Then U ↦→ M ⊗ K[X] O X (U) is an O X -module and this correspondence preserves tensor product, exactness, etc. Call this O X -module ˜M. In part, ˜M is quasi-coherent (and coherent if M is of finite type) and any quasi-coherent O X -module is of this form. Example 4.1.5 (Important). Ideal sheaf I Y Y ↩→ X (i.e., Y is an algebraic subvariety). ↩→ O X corresponding to the subsheaf of O X vanishing on We normally assume that the ground field K is algebraically closed. Then the Nullstellensatz tells us that V (I) := sup(O X /I) ⊆ X is nonempty if and only if I ≠ O X . Hence the ringed space (V (I), O X /I) is identified with the subscheme of X corresponding to I. A ringed space locally isomorphic to subschemes of affine spaces is called al algebraic scheme. Definition 4.1.9 (Associated fibre spaces). Let A be a quasi-coherent sheaf of a O X -algebra of finite type (i.e., locally generated by finitely many sections as O X -algebras). We define a scheme S = Specm X A and a morphism π : S −→ (X, O X ) as follows: Let X be affine. Then set Specm X A := SpecmA(X) := {maximal ideals in A(X)}. Recall that if R = A(X) and M is a maximal ideal, then R/M = K if K is algebraically closed. Let f : S −→ K be a regular function. Then {f = 0} c = a basis of open sets, form the Zariski topology. Let π be the dual to the K-algebra homomorphism K[X] −→ A(X). If D(f) = {x ∈ X / f(x) ≠ 0} for f ∈ O X (X) then by the quasi-coherence of A, we have A(D(f)) = A(X) ⊗ K[X] K[D(f)]. So that SpecmA(D(f)) = π −1 (D(f)). Special case: Given a coherent sheaf of O X -modules F, let A = Sym OX F. Then the associated fibre space is called the vector fibre space, denoted by π : V(F) −→ X. Here, Sym means the symmetric product ⊕ k≥0 (Sym k F). Note that V(F) x = (F x /M x F x ) ∨ Example 4.1.6 (for algebraic geometry). (I) Definition of normal and tangent bundles (cones): – The model for tangent vector space at a point is given by Specm(K[ɛ]/ɛ 2 ). – The Zariski tangent space: Let x ∈ X be a point of al algebraic variety. Then T x X := (M x /M 2 x) ∨ . – The normal bundle of Y ↩→ X (algebraic subvariety): Let I be the ideal sheaf of Y , then I/I 2 = I ⊗ OX O Y . The normal vector bundle (or normal bundle) of Y in X is I/I . It is denoted by N Y |X ( −→ π Y ). 56
– The normal cone of Y in X is defined by C Y |X := Specm Y ( ⊕k≥0 I k /I k+1) N Y |X Y An easiest definition of the tangent bundle to an algebraic variety X is that it is the normal cone to the diagonal X ↩→ ∆ X ×X. These are functorial objects since f : X −→ Y , f ×f : X ×X −→ Y ×Y , so T f : T X −→ T Y . And it coincides with d x f : T x X −→ T f(x) Y , for every x ∈ X. – We say that x ∈ X is a smooth point if C x (X) = T x X, and X is smooth (non-singular) if all points are. (II) The cotangent sheaf to X is defined as the conormal sheaf to the diagonal in X ×X. Its local sections are local forms on T X and such a form d gives a map M x /M 2 x where Ω ′ X = { differential on O X}. ∼ −→ Ω ′ X(x) := Ω ′ X,x/(Ω ′ X,x ⊗ M x ) (III) Blowing up a subscheme: Let I ↩→ O X be an ideal sheaf defining a subscheme Y ↩→ X, A = ⊕ k>0 I k . Then σ : ˜X = proj(A) −→ X is called the blow up of X along Y , where proj(A) = Specm(homogeneous decomposition of A). By functoriality, σ −1 (Y ) is the projection of the algebra A ⊗ OX O Y = ⊕ k≥0 I k /I k+1 , i.e., σ −1 (Y ) −→ Y is the projectivization of the normal cone C Y |N , i.e., 0 = ⊗O X . Definition 4.1.10. A sheaf is torsion free is ⊗O X = 0, i.e., it is supported on a subvariety. Fact 4.1.2. • Any torsion free O X -module F admits a resolution, i.e., a birational morphism σ : Y −→ X such that σ ∗ F is locally free. • (Hironaka) Any variety X (any rational map X −→ Y ) admits a resolution of singularities by repeatedly blow ups along smooth centres (i.e., smooth subvariety) ˜X σ X Y 57
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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
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 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;
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