1. Degree of Tangent Bundle of a compact Riemann surface
Let X be a compact complex manifold. If L is a line bundle on X with a curvature form Θ, the first Chern class of L is defined to be the de Rham cohomology class
c1(L) =
√
−1 2π Θ
∈ HdR2 (X).
Let X be a compact Riemann surface and L be a complex line bundle on X. The degree of L is defined to be
deg L = hc1(L), [X]i,
where h·, ·i : HdR2 (X) × H2(X) → C is the natural pairing defined by the integration. If Θ is a curvature form of L, then
deg L =
√−1 2π
Z
X
Θ.
When X is a compact Riemann surface of genus g, its holomorphic tangent bundle T0X is a complex line bundle over X. Assume that ds2 = h2dz ⊗ dz is a Hermitian metric on X, where h is a positive real valued smooth function on X. Its corresponding curvature form Θ is given by
Θ = −1
2(∆ log h)dz ∧ dz.
Lemma 1.1. Suppose ds2 = h2dz ⊗ dz is a hermitian metric on a Riemann surface X. The Gaussian curvature of the corresponding Riemannian connection is given by
K = −∆ log h h2 .
Proof. The proof will be given in another note.
Denote θ = dz/h. Then Θ = 1
2Kθ ∧ θ, where K is the Gaussian curvature. On the other hand, the volume form dµ on X is dµ =
√−1
2 θ ∧ θ. Then
√−1
2π Θ = 1 2πKdµ.
The degree of the holomorphic tangent bundle T0X of X is deg T0X = 1
2π Z
X
Kdµ.
By Gauss-Bonnet theorem, deg T0X = χ(X), the Euler characteristic of X. Since χ(X) = 2 − 2g, the degree of T0X is
deg T0X = 2 − 2g.
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