Holomorphic tangent bundle

Definition

Let $X$ be a complex manifold of complex dimension $n$ . Its holomorphic tangent bundle $T^{1,0}X\to X$ is the rank- $n$ holomorphic vector bundle obtained by gluing the coordinate bundles with the complex Jacobian matrices of holomorphic coordinate changes. Its fiber at $x$ is the complex tangent space $T_x^{1,0}X$ . Under the type decomposition, it is the $i$ -eigenbundle of the complexified almost-complex structure and is a direct summand of the complexification of the underlying real tangent bundle.

Local frames and sections

Holomorphic coordinates $z^1,\ldots,z^n$ give a local holomorphic frame

\frac{\partial}{\partial z^1},\ldots,\frac{\partial}{\partial z^n}.

A holomorphic section of $T^{1,0}X$ is a holomorphic vector field, locally $\sum_j a^j(z)\partial/\partial z^j$ with holomorphic coefficients. These sections form the holomorphic tangent sheaf.

Duality and functoriality

The holomorphic dual of $T^{1,0}X$ is the holomorphic cotangent bundle $\Omega_X^1$ . A holomorphic map $f:X\to Y$ has a complex-linear differential

df:T^{1,0}X\longrightarrow f^*T^{1,0}Y,

which is a holomorphic vector-bundle morphism. This construction is compatible with composition.

Conventions and scope

Some authors write $T_X$ for the holomorphic bundle or its sheaf of holomorphic sections. It is not the same object as the underlying real tangent bundle $TX$ , whose real rank is $2n$ , nor as the full complexification $TX\otimes_{\mathbb R}\mathbb C=T^{1,0}X\oplus T^{0,1}X$ . The identification with an eigenbundle uses the integrable complex structure Huybrechts, §2.2.

References

D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §2.2, holomorphic tangent and cotangent bundles.
R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: Chapter I, §2, complex tangent bundles.

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