COMPLEX GEOMETRY Course notes

4.2 Cohomology of sheaves
Let X be a topological space. We consider sheaves F i together with morphisms d i : F i −→ F i+1 such that
d i+1 ◦d i = 0 for every i. Such set of sheaves and morphisms (F i , d i ) is called a complex of sheaves over X.
It is also called a resolution of a sheaf F if there exists an inclusion ι : F −→ F 0 such that i(F) = Ker(d 0 )
and Ker(d i+1 ) = Im(d i ), for every i.
(1) Čech resolution: Let F be a sheaf over X, {U i } i∈N a covering of X. For every finite subset I ⊆ N
set U I = ∩ i∈I U i , j I : U I ↩→ X and
Define F k := ⊕ |I|=k+i F I and d : F k −→ F k+1 by
F I = (j I ) ∗ (F| UI ) (extension by zero outside U I )
(dσ) j0...j k+1
= ∑ i
(−1) i σ j0 . . . ĵ i . . . j k+1 | U∩UI
where j 0 ≤ j 1 ≤ · · · ≤ j k+1 , σ = (σ I ), σ I ∈ F I (U) and |I| = k + 1. Lastly, we define ι : F −→ F 0 by
ι(σ) i = σ| U∩Ui
for σ ∈ F(U).
Proposition 4.2.1. This is a resolution.
(2) de Rham resolution: Let A k be the sheaf of C ∞ (R or C-valued) differential forms of degree k (on
a real or complex manifold). The d-Poincaré Lemma says that the complex (A k , d) is a resolution of
Ker(d 0 ) = R (resp. C), constant sheaves over X.
(3) Dolbeault resolution: Let E be a holomorphic vector bundle over a complex manifold X and E its
sheaf of holomorphic sections (i.e., E = O X (E)). Let A 0,q be the sheaf of C ∞ -sections of Ω 0,q
X ⊗ E.
Generalizing the d-Poincaré Lemma, we get that (A 0,q (E), ∂) is a resolution of Ker(∂ 0 ) = E = O X (E)
(a coherent O X -module).
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