COMPLEX GEOMETRY Course notes

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4.4 Derived functors Given a functor F : C −→ C ′ between abelian categories such that C has enough injectives. Definition 4.4.1. For every object M ∈ C, there exists an object R i F (M) for every i ≥ 0 in C ′ , unique up to isomorphisms, satisfying the following conditions: (1) R 0 F (M) = F (M), (2) Every short exact sequence 0 −→ A −→ B −→ C −→ 0 gives rise to a long exact sequence in C ′ 0 −→ F (A) −→ F (B) −→ F (C) −→ R 1 F (A) −→ · · · · · · −→ R i F (A) −→ R i F (B) −→ R i F (C) −→ R i+1 F (A) −→ · · · Remark 4.4.1. (1) Let M 0 , i : A −→ M 0 be an acyclic resolution of A (i.e., R j+1 F (M k ) = 0 for every j, k ≥ 0). Then R j F (A) = H j (F (M 0 )) := CoKer(d i−1 F(M i−1 ) −→ F(M i )) (2) For a sheaf F over X there exists an acyclic resolution (called the Godement resolution for F). Definition 4.4.2. A fine sheaf F os a sheaf of A-modules where the sheaf of algebra A admits a partition of unity: for every open covering {U i }, there exists f i ∈ F(U i ) such that ∑ i f i = 1 (this sum is locally finite). Proposition 4.4.1. H i (X, F) = 0 for every i ≥ 0 for such F. Corollary 4.4.1. (1) Let X be a real (complex) manifold, then H ∗ (X; R) = Ker(d) Im(d) = H∗ dR(X; R), where the same equality holds if we replace R by C. (2) Given a holomorphic vector bundle E −→ X and E its sheaf of holomorphic sections. Then H ∗ (X, E) = Ker(∂ : A0,q (E) −→ A 0,q+1 (E)) Im(∂ : A 0,q−1 (E) −→ A 0,q (E)) Corollary 4.4.2 (Grothendieck Vanishing Theorem). Given a holomorphic vector bundle E −→ X, we have H q (X, E) = 0 for q > n = dim C X. Ker(d) = F −→ F 0 d −→ F 1 −→ · · · −→ F k −→ · · · Consider a Čech resolution where the map F −→ F 0 is given by σ ↦→ σ| U∩Ui and F k = ⊕ |I|=k+1 (j I ) ∗ (F| UI ). 60
Theorem 4.4.1. If H q>0 (U I , F) = 0 for every I ⊆ N then H q (X, F) = Ȟq (U, F) := H q (Γ(F) = F(X), d X ), e.g., if X is a finite dimensional manifold C ∞ , then there exists a good cover (U I contractible for every I) and so Ȟ q (U, Z) = H q cell (X, Z) where U is in an open cover and H q cell (X, Z) de<strong>notes</strong> the cellular cohomology, which is computing via nerves of the open covering. 61
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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
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 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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