Sheaf Cohomology 11: Canonical Bundles, Chern Classes, and Curvature

Sheaf cohomology / 11

Canonical bundles, Chern classes, and curvature

Curvature gives differential-form representatives of Chern classes; the canonical bundle records how holomorphic one-forms transform.

Bridge from differential forms: curvature is a cohomology representative

The normalized Chern curvature form of a Hermitian holomorphic line bundle is a closed real 2-form. Changing the metric changes that representative by an exact form, so the de Rham class remains fixed. Thus Chern classes connect the differential-forms quotient

\[{\text{closed forms}\over\text{exact forms}}\]

to holomorphic line bundles and divisors.

Example: same class, different metric

If two Hermitian metrics differ by \(e^{-\psi}\), their normalized curvature representatives differ by a form proportional to \(\partial\bar\partial\psi\) up to the conventional constant. In the complexified de Rham complex this is \(d\)-exact, and the corresponding real representative is exact as well. Therefore the Chern class is unchanged.

Example: Gauss-Bonnet as Chern-Weil

For a Riemann surface, the normalized Ricci form represents \(c_1(T_X)\) once a sign convention is fixed. Integrating it recovers the same topological number that appears in Gauss-Bonnet.

Chapter 13 of the PDF brings together local complex-coordinate formulas, curvature, Ricci form, Chern classes, and canonical divisors. The formulas carry sign conventions, so this article fixes one convention: if a Hermitian metric on a line bundle is locally \(h=e^{-\varphi}\), take the Chern curvature form to be \(F_\nabla=\partial\bar\partial\varphi\) and normalize the real Chern form as \(\frac{i}{2\pi}F_\nabla\). Authors using the opposite curvature sign must reverse \(F_\nabla\) consistently.

Canonical bundle

For a complex manifold \(X\) of dimension \(n\), the canonical bundle is

\[K_X=\Omega_X^n,\]

the line bundle of holomorphic top forms. On a Riemann surface, it is simply the bundle of holomorphic 1-forms.

If \(z_i\) and \(z_j\) are holomorphic coordinates on a Riemann surface, then

\[dz_j={dz_j\over dz_i}\,dz_i.\]

Thus the transition functions of \(K_X\) are Jacobians of coordinate changes.

Example 1: \(K_{\mathbb{CP}^1}\cong\mathcal O(-2)\)

On \(\mathbb{CP}^1\), use coordinates \(z\) and \(w=1/z\). Then

\[dw=-z^{-2}\,dz, \qquad dz=-z^2\,dw.\]

Thus the frame transition in the convention of Article 01 is \(z^2\) up to a nonzero constant, which is the transition of \(\mathcal O(-2)\) because \(\mathcal O(n)\) uses \(z^{-n}\). Hence

\[K_{\mathbb{CP}^1}\cong\mathcal O(-2).\]

Equivalently, the meromorphic form \(dz\) has a double pole at infinity and no finite poles.

Example 2: the elliptic curve

For \(E=\mathbb C/\Lambda\), the form \(dz\) is invariant under translations \(z\mapsto z+\lambda\). It descends to a nowhere-zero holomorphic 1-form on \(E\). Therefore

\(K_E\cong\mathcal O_E, \qquad \deg K_E=0.\)

Chern forms of Hermitian line bundles

Let \(L\) be a holomorphic line bundle with local holomorphic frame \(e\) and Hermitian metric

\[|e|_h^2=e^{-\varphi}.\]

The Chern connection has curvature locally of type \((1,1)\). With the convention above, the first Chern class is represented in de Rham cohomology by

\[c_1(L,h)=\left[\frac{i}{2\pi}F_\nabla\right].\]

Changing the metric changes the representative by an exact form, so the cohomology class is independent of the metric.

Example 3: Fubini-Study curvature of \(\mathcal O(1)\)

On \(U_0\subset\mathbb{CP}^1\), the Fubini-Study metric on \(\mathcal O(1)\) has local potential

\[\varphi(z)=\log(1+|z|^2).\]

The Chern form is

\[\omega_{\mathrm{FS}} =\frac{i}{2\pi}\partial\bar\partial\log(1+|z|^2),\]

and

\[\int_{\mathbb{CP}^1}\omega_{\mathrm{FS}}=1.\]

This realizes \(c_1(\mathcal O(1))\) as a curvature form.

Example 4: tensor products

If \(L\) and \(M\) have metrics with local weights \(\varphi_L\) and \(\varphi_M\), then \(L\otimes M\) has weight \(\varphi_L+\varphi_M\). Therefore

\[c_1(L\otimes M)=c_1(L)+c_1(M).\]

This is the curvature-form version of multiplying transition functions.

Ricci form and the canonical bundle

For a Hermitian metric on a Riemann surface written locally as

\[ds^2=\lambda(z)^2 |dz|^2,\]

the Ricci form represents the curvature of the anticanonical direction, equivalently the negative of the canonical-bundle curvature under the convention below:

\[\operatorname{Ric}=-i\,\partial\bar\partial\log \lambda^2.\]

The Chern class relation is

\[\left[\frac{1}{2\pi}\operatorname{Ric}\right] =c_1(T_X)=-c_1(K_X).\]

Thus Gauss-Bonnet and the degree of the canonical bundle are the same invariant in analytic and algebraic clothing:

\[\deg K_X=2g-2.\]

Example 5: genus two surface

If \(X\) has genus \(2\), then \(\deg K_X=2\). A nonzero holomorphic 1-form has a zero divisor of total degree two. Hyperelliptic models make this visible: holomorphic differentials vanish at ramification patterns whose total order is forced by the canonical degree.

Example 6: flat torus has zero Ricci form

For the flat metric on \(E=\mathbb C/\Lambda\), \(\lambda\) is constant, so

\[\partial\bar\partial\log\lambda^2=0.\]

The Ricci form vanishes and \(c_1(T_E)=0\). This matches the triviality of \(K_E\).

Canonical divisors

A nonzero meromorphic 1-form \(\omega\) on a compact Riemann surface has a divisor

\[(\omega)=\sum_p \operatorname{ord}_p(\omega)p.\]

Any divisor obtained from a nonzero meromorphic 1-form is a canonical divisor. Its linear equivalence class is the class of \(K_X\), and its degree is

\[\deg(\omega)=2g-2.\]

The same object can therefore be read in three ways: as transition functions of \(K_X\), as a divisor of a meromorphic differential, or as a Chern class represented by curvature.