Let M be a compact C∞ manifold. A Riemannian metric g on M is a smooth section of T * M ⊗ T * M defining a positive definite symmetric bilinear form on TxM for each x ∈ M. In local coordinates x1,…, xn, one has a natural local basis \(
\frac{\partial }{{\partial {{x}_{1}}}}, \cdots ,\frac{\partial }{{\partial {{x}_{n}}}}\) for TM, then g is represented by a smooth matrix-valued function {gij}, for TM, then g is represented by a smooth matrix-valued function {gij}, where
In this section, we will introduce a holomorphic invariant for Kähler manifolds, which was first done by Futaki for manifolds with positive first Chern class and then by Calabi and Futaki for general Kähler manifolds. We will take a slightly different approach from the original one taken.
Let (V, ω) be a simply connected symplectic manifold and let G be a group acting on V preserving the symplectic form. Let \(
\mathfrak{g}
\) be the Lie algebra of G which consists of all left-invariant vector fields on G. Then any v ∈ \(
\mathfrak{g}
\) induces a one-parameter subgroup {øt} of G. Since G acts on V, ø t induces a vector field Xv on V. It is well known that there exists a map m, called moment map, m : V→ \(
\mathfrak{g}
\)*, satisfying
m is G-equivariant with respect to the co-adjoint action on \(
\mathfrak{g}
\)*,
for all v ∈ \(
\mathfrak{g}
\) and all u ∈ TV, ω( u, Xv = dm (u)(v).
We have seen that the Ricci curvature represents the first Chern class. In this section, we will consider the converse problem, namely, given a Kähler class [ω] ∈ H2 (M, ℝ) ∩ H1,2 (M, ℂ) on a compact Kähler manifold M and any form Ω representing the first Chern class, can we find a metric ω ∈ [ω] such that Ric(ω) = Ω? This is known as the Calabi conjecture and it was solved by Yau in 1976. We will state it here as a theorem and refer to it as the Calabi-Yau Theorem.
