Complex Analytic and Differential Geometry 2025, 18 - Hermitian and Kaehler Manifolds

Write the fundamental (1, 1)-form ω associated to a hermitian metric h(z)

A hermitian manifold is a pair (X, ω) where ω is a C^∞ positive definite (1, 1)-form of X. ω is Kaehler, if dω = 0. X is a Kaehler manifold if X carries at least one Kaehler metric. If (X, ω) is compact Kaehler and {ω} its cohomology class in H²(X, ℝ), then {ω}ⁿ ≠ 0. If X is compact Kaehler, then H^2k(X, ℝ) ≠ 0 for 0 ≤ k ≤ n.

For ω positive definite, all points in X have to correspond a holomorphic coordinate system centered at it for h_jk = 2ω_jk with ω_lm = δlm + O(|z|²) for ω to be Kaehler. If so, then the coordinates can be chosen to define ω_lm as a differential form. with coefficients of the Chern curvature tensor associated to the tangent space of X. On Kaehler manifolds:

The space of primitive elements of total degree k is Prim^kT*_X = ⊕_p+q=kPrim^p,q T*_X. For all primitive u, there is a unique decomposition u Σ_r≥(k-n)+ L^ru_r