Canonical bundle of a complex manifold - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold of complex dimension $n$ . Its canonical bundle is the top exterior power

K_X=\bigwedge\nolimits^n T^{*1,0}X=\Omega_X^n

of the holomorphic cotangent bundle. It is a holomorphic line bundle whose fiber at $x$ consists of alternating complex $n$ -linear forms on $T_x^{1,0}X$ . In holomorphic coordinates $z^1,\ldots,z^n$ , the form $dz^1\wedge\cdots\wedge dz^n$ is a local holomorphic frame. Thus local sections are holomorphic top-degree forms, and the bundle is determined without choosing a metric or orientation.

Coordinate transformations

Under a holomorphic coordinate change $w=w(z)$ , the standard local frames satisfy

dw^1\wedge\cdots\wedge dw^n =\det\left(\frac{\partial w}{\partial z}\right) dz^1\wedge\cdots\wedge dz^n.

The Jacobian determinant is nowhere zero on a coordinate overlap, so these factors are precisely the transition functions of $K_X$ . This also explains why top forms, rather than their coefficient functions alone, are coordinate-independent objects.

Sections and triviality

A holomorphic section of $K_X$ is a holomorphic $n$ -form. A nowhere-vanishing global holomorphic $n$ -form determines an isomorphism $K_X\cong X\times\mathbb C$ ; conversely, any holomorphic trivialization supplies such a form. The form is additional data, however: triviality asserts existence and does not choose a preferred trivializing section.

Examples and scope

On $\mathbb C^n$ , the form $dz^1\wedge\cdots\wedge dz^n$ trivializes the canonical bundle. For the Riemann sphere, the coordinate change $w=1/z$ gives $dw=-z^{-2}dz$ ; its canonical bundle is therefore nontrivial and has degree $-2$ .

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §2.2, holomorphic cotangent bundles, exterior powers, and the canonical bundle.
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley, 1978. DOI record. Relevant: Chapter 0, holomorphic vector bundles and differential forms.