In differential geometry, the **holomorphic tangent bundle** is a concept that arises in the context of complex manifolds, which are spaces that locally resemble complex Euclidean space and have a complex structure. ### Basic Definitions: 1. **Tangent Bundle**: For a smooth manifold \(M\), the tangent bundle \(TM\) is the collection of all tangent spaces at every point in \(M\).
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The holomorphic tangent bundle of a complex manifold is locally spanned in holomorphic coordinates by . Under a holomorphic coordinate change , the chain rule giveswhose coefficient matrix is holomorphic; these frames therefore define a holomorphic vector bundle.