Holomorphic cotangent bundle

Definition

Let $X$ be a complex manifold. Its holomorphic cotangent bundle, denoted $T^{*1,0}X$ or $\Omega_X^1$ , is the holomorphic dual of the holomorphic tangent bundle $T^{1,0}X$ . Thus it is a holomorphic vector bundle whose fiber at $x$ is $(T_x^{1,0}X)^*$ . Holomorphic coordinate functions $z^1,\ldots,z^n$ give the local frame $dz^1,\ldots,dz^n$ , and changes of frame are the inverse transposes of the holomorphic Jacobian matrices. Its underlying smooth bundle is the $(1,0)$ -cotangent summand.

Sections and pullback

Holomorphic sections of $\Omega_X^1$ are holomorphic one-forms. Locally they are sums $\sum_j f_j(z)\,dz^j$ with holomorphic coefficients. A holomorphic map $f:X\to Y$ induces a pullback morphism

f^*\Omega_Y^1\longrightarrow\Omega_X^1

by precomposition with its differential.

Exterior powers

The holomorphic bundle of $p$ -forms is $\Omega_X^p=\Lambda^p\Omega_X^1$ . Its local sections are holomorphic $(p,0)$ -forms. If $n=\dim_{\mathbb C}X$ , the top exterior power

K_X=\Lambda^n\Omega_X^1

is the canonical holomorphic line bundle.

Conventions and comparison

The notation $\Omega_X^1$ may mean either the holomorphic vector bundle or its sheaf of holomorphic sections; context distinguishes them. The holomorphic cotangent bundle is not the full complexification of the real cotangent bundle, which splits into $(1,0)$ and $(0,1)$ summands. It is the holomorphic refinement of the corresponding dual vector bundle 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 cotangent bundles and forms.

Search Knowlpedia

Definition

Examples