COMPLEX GEOMETRY Course notes

3.3 Metrics and connections
Let E −→ X be a C ∞ -vector bundle on X, and let A i (E) be the vector space of C ∞ E-valued forms on X.
Definition 3.3.1. A real (complex) connection on E is a real (resp. complex) linear map
∇ : A 0 (E) −→ A 1 (E)
satisfying the Leibniz rule:
∇(fσ) = df ⊗ σ + f∇σ.
For a vector field ψ and σ ∈ A 0 (E) we write
∇ ψ σ = (∇σ)(ψ) ∈ A 0 (E).
In the case where E is a holomorphic vector bundle, we have the operation
∂ E : A 0 (E) −→ A 0,1 (E)
which defines holomorphic sections of E via Ker(∂ E ). It satisfies the ∂-Leibniz rule instead:
∂(fσ) = ∂f ⊗ σ + f∂σ
but it is not a complex connection.
Proposition 3.3.1 (For a Riemannian manifold). If (M, g) is a R-manifold then there exists a unique
connection ∇ on T M called the Levi-Civita connection satisfying:
(1) d(g(ψ 1 , ψ 2 )) = g(ψ 1 , ∇ψ 2 ) + g(∇ψ 1 , ψ 2 ), i.e., g is ∇-invariant.
(2) ∇ ψ1 ψ 2 − ∇ ψ2 ψ 1 = [ψ 1 , ψ 2 ], i.e., g is torsion free of ∇.
Theorem 3.3.1 (and definition). Let E −→ X be a holomorphic vector bundle with a Hermitian metric.
There exists a unique complex connection ∇ on E, called the Chern connection satisfying:
(1) d(h(σ, τ)) = h(∇σ, τ) + h(σ, ∇τ), i.e., ∇ is invariant under (or compatible with) h.
(2) Let ∇ 0,1 be its composition with A 1 (E) −→ A 0,1 (E). Then ∇ 0,1 = ∂ E .
Theorem 3.3.2. The following statements for a complex Hermitian manifold (X, h) are equivalent:
(1) h is Kähler.
(2) J is flat for the Levi-Civita connection.
(3) Chern connection = Levi-Civita connection.
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