The Bochner–Kodaira–Nakano identity is a fundamental result in the study of the geometry of complex manifolds, particularly in the context of the study of Hermitian and Kähler metrics. This identity relates the curvature of a Hermitian manifold to the properties of sections of vector bundles over the manifold, and it plays a crucial role in several areas of differential geometry and mathematical physics.
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For the Chern connection of a Hermitian holomorphic vector bundle on a Kähler manifold, the two Dolbeault Laplacians differ by , where is its curvature form of a connection and is the adjoint Lefschetz operator. The curvature term controls vanishing estimates.