Sheaf cohomology / 02

Divisors and associated line bundles

A divisor is a finite prescription of zeros and poles; the sheaf $$\mathcal O(D)$$ turns that prescription into holomorphic sections.

Sections 6.2 and 6.5 of the PDF connect divisors with line bundles. This is the bridge from concrete meromorphic functions to the Picard group and ultimately to Riemann-Roch.

Divisors and local orders

On a compact Riemann surface \(X\), a divisor is a finite formal sum

\[D=\sum_{p\in X} n_p p,\qquad n_p\in\mathbb Z.\]

The degree is \(\deg D=\sum_p n_p\). If \(f\) is meromorphic and \(t\) is a local coordinate at \(p\), write

\[f=t^m u,\qquad u(p)\ne0.\]

Then \(\operatorname{ord}_p(f)=m\). A positive order is a zero, a negative order is a pole, and the principal divisor of \(f\) is

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

The sheaf \(\mathcal O(D)\)

Define

\[\mathcal O(D)(U)= \{f\in\mathcal M_X(U): \operatorname{ord}_p(f)+n_p\ge0 \text{ for every }p\in U\}.\]

Thus a section of \(\mathcal O(D)\) is a meromorphic function whose pole at \(p\) has order at most \(n_p\) when \(n_p>0\), and whose zero at \(p\) has order at least \(-n_p\) when \(n_p<0\).

Example 1: one allowed pole on \(\mathbb{CP}^1\)

Let \(D=m\infty\) on \(\mathbb{CP}^1\). Then

\[H^0(\mathbb{CP}^1,\mathcal O(m\infty))\]

is the vector space of rational functions with no finite poles and pole order at most \(m\) at infinity. These are exactly polynomials of degree at most \(m\):

\[a_0+a_1z+\cdots+a_m z^m.\]

Hence \(h^0(\mathbb{CP}^1,\mathcal O(m\infty))=m+1\) for \(m\ge0\).

Example 2: forcing a zero

Let \(D=-p\). Then

\[\mathcal O(-p)(U)=\{f\in\mathcal O(U): f(p)=0\}\]

for neighborhoods \(U\) of \(p\). A global section of \(\mathcal O(-p)\) is a holomorphic function vanishing at \(p\). On a compact connected Riemann surface, every holomorphic function is constant, so

\[H^0(X,\mathcal O(-p))=0.\]

This is the simplest example where a negative divisor kills global sections.

Local construction of the line bundle

Choose an open cover \(\{U_i\}\) so that on each \(U_i\) there is a meromorphic function \(f_i\) with

\[D|_{U_i}=(f_i).\]

On overlaps, \(f_j/f_i\) is nowhere-zero holomorphic. Therefore

\[g_{ij}=f_j/f_i\]

is the transition function in the section convention of Article 01: a section \(h\in\mathcal O(D)(U)\) is represented locally by the holomorphic functions \(h f_i\), and on overlaps \(h f_j=(f_j/f_i)h f_i\). This defines the holomorphic line bundle associated with \(D\), whose sheaf of holomorphic sections is canonically identified with \(\mathcal O(D)\).

Why \(f_j/f_i\) is holomorphic and invertible

Both \(f_i\) and \(f_j\) have the same divisor on the overlap, because both represent \(D\) there. Their quotient has zero order at every point, so it has neither zeros nor poles. Hence \(f_j/f_i\in\mathcal O^*(U_i\cap U_j)\).

Principal divisors give trivial bundles

If \(D=(f)\) globally, take the same meromorphic function on every open set. Then all transition functions are \(1\). More invariantly,

\[\mathcal O((f))\cong\mathcal O_X.\]

In the convention fixed above, the isomorphism sends a local meromorphic section \(h\) of \(\mathcal O((f))\) to the holomorphic function \(hf\). The inverse sends a holomorphic function \(u\) to \(u/f\). This is the reason divisor classes modulo principal divisors map to the Picard group.

Example 3: \(z-a\) on \(\mathbb{CP}^1\)

For \(f(z)=z-a\),

\[(f)=[a]-[\infty].\]

Thus \([a]\) and \([\infty]\) determine isomorphic line bundles:

\[\mathcal O([a])\cong\mathcal O([\infty]).\]

The actual divisors are different, but their difference is principal.

Example 4: elliptic curve degree-one divisors

On an elliptic curve \(E\) with chosen origin, degree-one divisors \([p]\) and \([q]\) need not be linearly equivalent. Their difference \([p]-[q]\) maps under Abel-Jacobi to \(p-q\in E\), so it is principal exactly when \(p=q\). Thus unlike \(\mathbb{CP}^1\), degree alone does not determine the line bundle.

Meromorphic sections

In the convention fixed above, the local equations \(f_i\) glue to a distinguished meromorphic section of \(\mathcal O(D)\) whose divisor is \(D\). Multiplying this section by a meromorphic function changes the divisor by a principal divisor. The invariant statement is:

\[\text{meromorphic sections of }L \quad\leftrightarrow\quad \text{divisors in the linear equivalence class representing }L.\]

Zeros and poles of a meromorphic section are not decoration; they are the divisor data from which the line bundle can be reconstructed.