Sheaf Cohomology 01: Line Bundles and Transition Functions

Sheaf cohomology / 01

Line bundles and transition functions

A holomorphic line bundle is locally trivial, but the mismatch among local trivializations is the invariant object.

Bridge from forms: local formulas versus global objects

Differential forms already showed the pattern: a closed positive-degree form may be locally a derivative without being globally exact. Line bundles use the same local-to-global pattern, but the local objects are frames rather than primitives. A bundle is locally \(U_i\times\mathbb C\); the global object is encoded by how these local frames disagree on overlaps.

Example: local primitive versus local frame

For \(dz/z\) on \(\mathbb C^*\), local branches of \(\log z\) differ by constants on overlaps. For \(\mathcal O(1)\) on \(\mathbb{CP}^1\), local frames differ by multiplication by \(z\) or \(z^{-1}\) on the overlap, depending on convention. Additive mismatch produces de Rham classes; multiplicative mismatch produces line bundles.

Section 6.1 of the PDF introduces holomorphic line bundles through transition functions. This is the correct first model because it makes the later appearance of \(H^1(X,\mathcal O_X^*)\) almost inevitable: a line bundle is a multiplicative 1-cocycle up to a multiplicative change of local frames.

Definition by gluing

Let \(X\) be a complex manifold and let \(\mathfrak U=\{U_i\}\) be an open cover. A holomorphic line bundle over \(X\) is obtained by gluing the products \(U_i\times\mathbb C\) using nowhere-zero holomorphic functions

\[g_{ij}\in \mathcal O_X^*(U_i\cap U_j)\]

such that

\[g_{ii}=1,\qquad g_{ij}g_{ji}=1,\qquad g_{ij}g_{jk}g_{ki}=1 \quad\text{on }U_i\cap U_j\cap U_k.\]

The last equation is the cocycle condition. It says that if one moves from chart \(i\) to chart \(j\) to chart \(k\) and back to \(i\), the fiber coordinate returns unchanged.

Equivalently, define an equivalence relation on the disjoint union \(\coprod_i U_i\times\mathbb C\) by

\[(x,v)_i\sim (x,g_{ij}(x)v)_j.\]

The quotient is the total space of the line bundle.

Local frames and sections

A local frame \(e_i\) over \(U_i\) identifies a section \(s\) with a holomorphic function \(s_i\) by \(s=s_i e_i\). With the frame convention used here, on overlaps

\[e_i=g_{ij}e_j, \qquad s_j=g_{ij}s_i.\]

If one instead lets transition functions convert local coordinate functions rather than frames, all transition functions are inverted. The convention must be fixed once; the invariant statement is compatibility of the local representatives.

Gauge changes

The same bundle may be described by different frames. If \(h_i\in\mathcal O_X^*(U_i)\) and \(e_i'=h_i e_i\), then the transition functions change to

\[g'_{ij}=h_i g_{ij}h_j^{-1}.\]

Since line bundles have scalar transition functions, the product is commutative; the expression is often written as

\[g'_{ij}=h_i h_j^{-1}g_{ij}.\]

Thus isomorphism classes of line bundles are cocycles modulo coboundaries. This is the geometric content behind

\[\operatorname{Pic}(X)\cong H^1(X,\mathcal O_X^*).\]

Example 1: the trivial bundle is a coboundary

If \(g_{ij}=h_i h_j^{-1}\) for some nowhere-zero holomorphic functions \(h_i\), then changing frames by \(e_i'=h_i^{-1}e_i\) makes every transition function equal to \(1\). The bundle is globally \(X\times\mathbb C\). The data may look nontrivial on overlaps, but it disappears after changing the local frames.

Example 2: \(\mathcal O(1)\) on \(\mathbb{CP}^1\)

Let \(U_0=\{Z_0\ne0\}\) with coordinate \(z=Z_1/Z_0\) and \(U_\infty=\{Z_1\ne0\}\) with coordinate \(w=Z_0/Z_1=1/z\). In the section convention used above, \(\mathcal O(1)\) has compatibility

\[s_\infty(w)=z^{-1}s_0(z)=w\,s_0(z).\]

Equivalently, the reciprocal frame convention uses transition function \(z\). The two independent global sections are represented by the homogeneous coordinates \(Z_0,Z_1\): in these local functions they appear as \((s_0,s_\infty)=(1,w)\) and \((z,1)\).

Operations

If \(L\) and \(M\) have transition functions \(g_{ij}\) and \(h_{ij}\), then

\[L\otimes M:\quad g_{ij}h_{ij}, \qquad L^\vee:\quad g_{ij}^{-1}.\]

The trivial bundle is the identity element, and dualization gives inverses in the Picard group. This is why line bundles form an abelian group under tensor product.

Example 3: canonical bundle on a Riemann surface

For a Riemann surface, the canonical bundle \(K_X\) has local frame \(dz_i\) in a holomorphic coordinate \(z_i\). If \(z_j=z_j(z_i)\), then

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

Thus the coordinate-change derivative gives the frame relation. On \(\mathbb{CP}^1\), with \(w=1/z\),

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

With the frame convention of this article, \(e_0=dz\) and \(e_\infty=dw\) satisfy \(e_0=-z^2e_\infty\). Since \(\mathcal O(n)\) has transition \(z^{-n}\) in this convention, this is \(K_{\mathbb{CP}^1}\cong\mathcal O(-2)\).

Example 4: a flat bundle on an elliptic curve

Let \(E=\mathbb C/\Lambda\). A character \(\chi:\Lambda\to\mathbb C^*\) defines a line bundle

\[(\mathbb C\times\mathbb C)/\Lambda, \qquad \lambda\cdot(z,v)=(z+\lambda,\chi(\lambda)v).\]

If \(|\chi(\lambda)|=1\) for all \(\lambda\), this is a unitary flat line bundle. Its degree is zero; for nontrivial unitary \(\chi\) it is not holomorphically trivial. This example separates topological degree from holomorphic isomorphism class.

What the cocycle records

The transition functions do not describe a line bundle as a set of unrelated local products. They measure the obstruction to choosing one global nowhere-zero frame. If the obstruction vanishes, all local frames can be rescaled into a single global frame. If it does not vanish, the bundle is locally simple but globally twisted.