Let $X$ be a compact Riemann surface. Recall that a holomorphic line bundle on $X$ is a complex line bundle (i.e. the fibers are complex one-dimensional vector spaces) $\pi: L \to X$ whose trivialization maps $\phi_i:\pi^{-1}(U_i)\to U_i \times \mathbb{C}$ are biholomorphic or, equivalently, whose transition functions $\tau_{ij}: U_i \cap U_j \to \mathbb{C}^\ast$ are holomorphic.

We will henceforth adopt the convention that “line bundle” refers to a holomorphic line bundle, while “complex line bundle” refers to a not-necessarily-holomorphic complex line bundle.

Define the following sheaves on $X$:
$\mathscr{O}$: holomorphic functions
$\mathscr{O}^\ast$: nowhere-vanishing holomorphic functions
$\mathscr{M}$: meromorphic functions
$\mathscr{M}^\ast$: nonzero meromorphic functions
$\Omega$: holomorphic 1-forms
$\mathbb{Z}$: locally constant integer-valued functions
$\mathbb{C}$: locally constant complex-valued functions
$\mathscr{E}$: smooth complex functions
$\mathscr{E}^k$: smooth complex $k$-forms
$\mathscr{E}^{p,q}$: smooth complex $p, q$-forms

Some of the questions I am interested in:

  • What are all the line bundles on $X$?

  • What information about $X$ can be recovered from a given line bundle? From all line bundles?

  • Which line bundles admit (holomorphic) sections?

The last question assumes significance because, while the trivial line bundle has no interesting sections (the only holomorphic functions on $X$ are constants), there are bundles with interesting sections. In fact, meromorphic functions correspond to sections of certain bundles, and once we know these bundles have sections, we know that $X$ has meromorphic functions. With enough meromorphic functions, we can construct an embedding of $X$ into some $\mathbb{C}\mathbb{P}^n$, where every analytic subvariety is algebraic, and obtain the equivalence

\[\text{\{Riemann surfaces\}} \longleftrightarrow \text{\{nonsingular projective curves\}}\]

The Picard group

The isomorphism classes of line bundles on $X$ form a group under the tensor product operation (with the trivial bundle being the identity and inverses obtained by taking duals), called the Picard group $\text{Pic}(X)$. Transition functions of line bundles satisfy the cocycle condition $\tau_{ij}\tau_{jk}=\tau_{ik}$ and, in fact, this is a sufficient condition for any collection of holomorphic sections $(\tau_{ij}) \in \mathscr{O}^\ast(U_i \cap U_j)$ to define a line bundle. Thus we obtain a surjective map

\[Z^1(X, \mathscr{O}^\ast) \to \text{Pic}(X)\]

from 1-cocycles of the sheaf $\mathscr{O}^\ast$ to the line bundles they define. The kernel of this map (i.e. the cocycles that define the trivial bundle) is precisely the coboundaries: if the transition functions $\tau_{ij}$ can be written as $\tau_{ij} = f_if_j^{-1}$ for $(f_i) \in \mathscr{O}^\ast(U_i)$, then $f_i$ agrees with $f_j\tau_{ij}$ on intersections, thus giving a global nowhere-vanishing section and hence triviality. Conversely, if $\tau_{ij}$ define a bundle isomorphic to the trivial bundle, then this isomorphism can be used to obtain a global section $s$ which restricts to $s_i \in \mathscr{O}^\ast(U_i)$ satisfying $s_is_j^{-1} = \tau_{ij}$. It follows that

\[\text{Pic}(X) \simeq H^1(X, \mathscr{O}^\ast).\]

Divisors

Global sections of the quotient sheaf $\mathscr{M}^\ast/\mathscr{O}^\ast$ are called divisors. Such a global section consists of nonzero meromorphic functions $f_i \in \mathscr{M}^\ast(U_i)$ on an open cover ${U_i}$ such that $f_if_j^{-1} \in \mathscr{O}^\ast(U_i \cap U_j)$. Given $p \in X$, this last condition ensures that there is at most one $f_i$ with a zero or pole at $p$, and compactness ensures there are only finitely many such poles or zeroes. Thus we can associate to a divisor a finite combination $\sum_k a_kp_k$ where $a_k \in \mathbb{Z}$ is the order of zero or pole at $p_k$. Conversely, given such a finite combination, we can construct a divisor with the prescribed poles and zeroes. If the free abelian group generated by points of $X$ is denoted by Div$(X)$, we have just shown that

\[\text{Div}(X) = \mathscr{M}^\ast(X)/\mathscr{O}^\ast(X).\]

The short exact sequence of sheaves

\[0 \to \mathscr{O}^\ast \to \mathscr{M}^\ast \to \mathscr{M}^\ast/\mathscr{O}^\ast \to 0\]

gives the long exact sequence

\[0 \to \mathscr{O}^\ast(X) \to \mathscr{M}^\ast(X) \to \mathscr{M}^\ast(X)/\mathscr{O}^\ast(X) \to H^1(X, \mathscr{O}^\ast) \to H^1(X, \mathscr{M}^\ast) \to \ldots,\]

part of which can be rewritten as

\[\mathscr{M}^\ast(X) \xrightarrow{\alpha} \text{Div}(X) \xrightarrow{\beta} \text{Pic}(X) \to H^1(X, \mathscr{M}^\ast).\]

The map $\beta$ assigns to the divisor $f_i \in \mathscr{M}^\ast(U_i)$ the line bundle whose transition functions are $f_if_j^{-1} \in \mathscr{O}^\ast(U_i \cap U_j)$. Its kernel (which is the image of $\alpha$) consists of those divisors that arise from global meromorphic functions, the principal divisors PDiv$(X)$. The quotient Div$(X)$/PDiv$(X)$ is called the divisor class group Cl$(X)$. The kernel of the map $H^1(X, \mathscr{O}^\ast) \to H^1(X, \mathscr{M}^\ast)$ consists of those line bundles whose transition functions $\tau_{ij}$ are of the form $s_is_j^{-1}$ for $s_i \in \mathscr{M}^\ast(U_i)$, which can be glued together to form a global meromorphic section. Thus the image of $\beta$ consists of those bundles which admit a nontrivial global meromorphic section. It is a consequence of Riemann-Roch that all line bundles on a compact Riemann surface do. Therefore

\[\text{Cl(X)} \simeq \text{Pic}(X).\]

In particular, the line bundle corresponding to the divisor of a section of the line bundle $L$ is isomorphic to $L$.

The Picard variety

The exponential map $exp: f \mapsto e^{2\pi i f}$ gives a short exact sequence

\[0 \to \mathbb{Z}\to \mathscr{O}\xrightarrow{exp} \mathscr{O}^\ast \to 0\]

from which we get the long exact sequence

\[0 \to \mathbb{Z}\to \mathscr{O}(X) \to \mathscr{O}^\ast(X) \xrightarrow{\delta_1} H^1(X, \mathbb{Z}) \to H^1(X, \mathscr{O}) \to H^1(X, \mathscr{O}^\ast) \xrightarrow{\delta_2} H^2(X, \mathbb{Z}) \to 0.\]

Since $\mathscr{O}^\ast(X)$ on a compact Riemann surface is just $\mathbb{C}^\ast$, the first connecting morphism $\delta_1$ is the zero map. Denoting by Pic$^0(X)$ the quotient $H^1(X, \mathscr{O})/H^1(X, \mathbb{Z})$, we have the exact sequence

\[0 \to \text{Pic}^0(X) \to \text{Pic}(X) \xrightarrow{\delta_2} H^2(X, \mathbb{Z}) \to 0.\]

The connecting map $\delta_2$ is the map that assigns to a line bundle $L$ its first Chern class $c_1(L)$ (recall that $c_1(L\otimes L’) = c_1(L) + c_1(L’)$). Under the identifications $\text{Cl(X)} \simeq \text{Pic}(X)$ and $H^2(X, \mathbb{Z}) \simeq \mathbb{Z}$, it is also the degree map that assigns to a divisor $\sum_k a_k p_k$ the integer $\sum_k a_k$.

Pic$^0(X)$ is the kernel of $\delta_2$ and so consists of line bundles with trivial first Chern class. Since Chern classes are a complete invariant for complex line bundles, Pic$^0(X)$ thus consists of those line bundles on $X$ which are trivial as complex line bundles. Stated differently, Pic$^0(X)$ represents the various non-isomorphic holomorphic structures that can be assigned to the trivial complex line bundle.

It is a fact that, if $X$ has genus $g$, then $H^1(X, \mathscr{O}) \simeq \mathbb{C}^g$, $H^1(X, \mathbb{Z}) \simeq \mathbb{Z}^{2g}$ and Pic$^0(X)$ is a complex torus of real dimension $2g$. This is called the Picard variety of $X$, and is a topological invariant.

Forms

Holomorphic (respectively, meromorphic) 1-forms on $X$ are holomorphic (respectively, meromorphic) sections of the holomorphic cotangent bundle $T^\ast X$ (which coincides in this case with the canonical bundle $K_X$), whose sheaf of sections is $\Omega$. Using the short exact sequences (both of which are Dolbeault resolutions)

\[0 \to \mathscr{O}\to \mathscr{E}\xrightarrow{\bar{\partial}} \mathscr{E}^{0,1} \to 0\] \[0 \to \Omega \to \mathscr{E}^{1,0}\xrightarrow{\bar{\partial}} \mathscr{E}^2 \to 0\]

and the fact that the higher cohomology groups of $\mathscr{E}, \mathscr{E}^k$ and $\mathscr{E}^{p, q}$ vanish, we obtain the isomorphisms

\[H^1(X, \mathscr{O}) \simeq\frac{\mathscr{E}^{0,1}(X)}{\bar{\partial}\mathscr{E}(X)}\qquad\left(=: H^{0,1}(X)\right)\] \[H^1(X, \Omega) \simeq\frac{\mathscr{E}^2(X)}{\bar{\partial}\mathscr{E}^{1, 0}(X)} \qquad\left(=: H^{1,1}(X)\right).\]

These are Dolbeault isomorphisms. The third Dolbeault isomorphism, $H^{1,0}(X) \simeq \Omega(X)$, is immediate from the definition. Now the short exact sequence (every holomorphic 1-form on a Riemann surface is closed, hence locally exact)

\[0 \to \mathbb{C}\to \mathscr{O}\xrightarrow{d} \Omega \to 0\]

gives the long exact sequence

\[0 \to \mathbb{C}\to \mathscr{O}(X) \xrightarrow{d_0} \Omega(X) \to H^1(X, \mathbb{C}) \to H^1(X, \mathscr{O}) \xrightarrow{d_1} H^1(X, \Omega) \xrightarrow{Res} H^2(X, \mathbb{C}) \to 0.\]

The map $d_0$ is zero because $\mathscr{O}(X)$ is just the constants. In light of the Dolbeault isomorphisms, the map $d_1$ is the map $d: H^{0,1}(X) \to H^{1,1}(X)$. The de Rham isomorphism tells us that

\[H^2(X, \mathbb{C}) \simeq H^2_{dR}(X; \mathbb{C}) = \mathbb{C}\] \[H^1(X, \mathbb{C}) \simeq H^1_{dR}(X; \mathbb{C}) = \mathbb{C}^{2g}\]

The map $Res$ is the map

\[\begin{aligned} Res:H^{1,1}(X) &\to& \mathbb{C}\\ \omega &\mapsto& \int_X \omega \end{aligned}\]

and is well-defined by Stokes’ theorem.

Let $L \to X$ be a line bundle, and denote also by $L$ the sheaf of sections of $L$ (in particular, $K_X$ and $\Omega$ will be used interchangeably). Sections of $L$ can be paired with those of $K_X \otimes L^\ast$ to obtain sections of $K_X$. This pairing descends to cohomology and can be composed with the map $Res$ to obtain a bilinear pairing

\[\langle , \rangle: H^1(X, L) \times H^0(X, K_X \otimes L^\ast) \to H^1(X, \Omega) \xrightarrow{Res} \mathbb{C}.\]

That this pairing is nondegenerate is the content of Serre duality. In particular,

\[\Omega(X) \simeq H^1(X, \mathscr{O}) \simeq \mathbb{C}^g\] \[H^1(X, \Omega) \simeq \mathscr{O}(X) = \mathbb{C}\]

and $Res$ is injective while $d_1$ is zero. We have the exact sequence

\[0 \to \Omega(X) \to H^1(X, \mathbb{C}) \to H^1(X, \mathscr{O}) \to 0,\]

a splitting of which is provided by the Hodge decomposition

\[H^1(X, \mathbb{C}) = H^{1,0}(X) \oplus H^{0,1}(X),\]

whose summands are isomorphic by Serre duality. A consequence of the Hodge theory is that this isomorphism is explicitly given by conjugation: $H^{1,0}(X) \simeq \overline{H^{0,1}(X)}$.

Finally, the one-dimensional vector space

\[H^{1,1}(X) \simeq H^1(X, \Omega) \simeq H^2_{dR}(X)\]

is where the cohomology classes of volume (and therefore symplectic and Kahler) forms reside.

Riemann-Roch

The main result in the study of (line bundles on) Riemann surfaces is the Riemann-Roch theorem, which states that, for a line bundle $L$,

\[\text{dim}H^0(X, L) - \text{dim}H^1(X, L) = 1 - g + \text{deg}(L),\]

where $\deg(L)$ is the degree of the divisor corresponding to $L$. In particular, we have the inequality (which is apparently the result Riemann originally proved)

\[\text{dim}H^0(X, L) \geq 1 - g + \text{deg}(L).\]

This is a sufficient condition (a necessary condition is $\deg(L) \geq 0$) for a line bundle to have a section: if it has sufficiently large positive degree, it must have sections. (In fact, it turns out that $L$ is ample if and only if deg$(L) > 0$.) This is an example of the “positivity” in algebraic geometry.

Using the definition of the Euler characteristic $\chi(L)$ of a vector bundle, and the equality deg$(L) = c_1(L)$, Riemann-Roch can be restated as

\[c(L) - \chi(L) = g,\]

where $c(L)$ is the total Chern class. So the two quantities on the left may vary across line bundles on $X$, but their difference is fixed. I am trying to figure out a nice way to interpret this observation.

We can also use Serre duality to rewrite Riemann-Roch as

\[\text{dim}H^0(X, L) - \text{dim}H^0(X, K_X \otimes L^\ast) = 1 - g + \text{deg}(L).\]

Using $L = K_X$, we conclude that deg$(K_X) = 2g-2$.