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Holomorphic line bundles and meromorphic sections on a Riemann surface

Definition

Let $X$ be a Riemann surface with maximal holomorphic atlas $A$ (Riemann surfaces and holomorphic atlases). A smooth complex line bundle $π : E \to X$ is a smooth real rank-two vector bundle whose fibres carry complex vector-space structures and which has local trivializations $θ_{i} : E ∣_{U_{i}} \to U_{i} \times C$ that are complex-linear on each fibre. The transition functions are defined by $θ_{i} \circ θ_{j}^{- 1} (x, v) = (x, g_{i j} (x) v), g_{i j} : U_{i} \cap U_{j} \to C^{\times} .$ A holomorphic line bundle is such a bundle with a trivializing cover by domains of holomorphic charts for which every $g_{i j}$ is holomorphic. These functions obey $g_{i j} g_{j k} = g_{i k}$ on triple overlaps. Conversely, a holomorphic $C^{\times}$ -valued cocycle on a supplied countable open cover constructs a holomorphic line bundle: regard each scalar as its real $2 \times 2$ multiplication matrix, apply the smooth cocycle construction, and note that the resulting transitions commute with multiplication by $i$ and are holomorphic.

Write $e_{i} = θ_{i}^{- 1} (1)$ for the associated local frame. A smooth section has the form $s = f_{i} e_{i}$ locally, with $f_{i} \in C^{\infty} (U_{i}; C)$ and $f_{i} = g_{i j} f_{j}$ on overlaps. It is holomorphic when each $f_{i}$ is holomorphic; $H^{0} (X, E)$ denotes the complex vector space of holomorphic sections. A meromorphic section is a family of meromorphic functions $f_{i}$ with the same transition law. For a nonzero meromorphic section, define its order at $p$ by ${ord}_{p} (s) : = {ord}_{z_{i} (p)} (f_{i})$ in any holomorphic chart and local frame. Its divisor is $(s) : = \sum_{p \in X} {ord}_{p} (s) p$ ; this sum is locally finite and is finite when $X$ is compact.

The canonical bundle is $K : = Λ^{1, 0} T^{*} X$ . If $z_{i}, z_{j}$ are holomorphic coordinates on an overlap, then its transition function in the convention above is $g_{i j} = d z_{j} / d z_{i}$ : a local differential satisfies $f_{i} d z_{i} = f_{j} d z_{j}$ . Hence holomorphic sections of $K$ are precisely holomorphic differentials, and meromorphic sections of $K$ are meromorphic differentials (Meromorphic differentials, orders and residues). The conjugate cocycle $\overline{g_{i j}}$ defines $Λ^{0, 1} T^{*} X = \overline{K}$ , while $Λ^{0, 0} T^{*} X = X \times C$ . Thus the bundles of $E$ -valued $(0, 0)$ - and $(0, 1)$ -forms are $E$ and $Λ^{0, 1} T^{*} X \otimes E$ ; locally the latter has the form $f (z) d \overset{ˉ}{z} \otimes e$ (Bigraded complex forms and the Dolbeault operators, The exterior power bundle of the cotangent bundle).

For a holomorphic line bundle $E$ , the Dolbeault operator is ${\overset{ˉ}{\partial}}_{E} : C^{\infty} (X, E) ⟶ C^{\infty} (X, Λ^{0, 1} T^{*} X \otimes E), {\overset{ˉ}{\partial}}_{E} (f e_{i}) : = (\partial_{\overset{ˉ}{z}} f) d \overset{ˉ}{z} \otimes e_{i}$ in a holomorphic frame $e_{i}$ and coordinate $z$ . It is $C$ -linear, satisfies ${\overset{ˉ}{\partial}}_{E} (f s) = \overset{ˉ}{\partial} f \otimes s + f {\overset{ˉ}{\partial}}_{E} s$ , and ${\overset{ˉ}{\partial}}_{E} s = 0$ exactly when $s$ is holomorphic. It extends coefficientwise to $E$ -valued forms: in a holomorphic frame, ${\overset{ˉ}{\partial}}_{E} (α \otimes e_{i}) = (\overset{ˉ}{\partial} α) \otimes e_{i}$ . This extension satisfies ${\overset{ˉ}{\partial}}_{E}^{2} = 0$ and ${\overset{ˉ}{\partial}}_{E} (β \land s) = \overset{ˉ}{\partial} β \land s + (- 1)^{\deg β} β \land {\overset{ˉ}{\partial}}_{E} s$ for complex-valued forms $β$ . The complex dual $E^{*}$ has inverse holomorphic cocycle $g_{i j}^{- 1}$ and its corresponding Dolbeault operator.

Facts & Assumptions

Given: A connected Riemann surface $X$ with its maximal atlas, a smooth complex line bundle $E \to X$ , holomorphic trivializations and their transition functions, and a meromorphic section when order or divisor is discussed.

[F1]

The charts of $X$ are compatible biholomorphisms; their transitions are smooth and have nonzero derivative. A Riemann surface is a connected smooth two-manifold (Riemann surfaces and holomorphic atlases, Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces, Smooth manifolds and their smooth charts, Holomorphic functions are real analytic and smooth in their two real coordinates).

[F2]

A smooth vector bundle is locally trivial with linear fibre maps; smooth sections have smooth local components in a frame. A smooth cocycle on a supplied countable cover constructs a smooth real vector bundle (Smooth vector bundles, rank, fibres, and trivial bundles, Vector bundle charts and transition functions, Construction of a vector bundle from a smooth cocycle, Local and global frames of a vector bundle, Smooth sections, local sections, and support, Smoothness of a section is equivalent to smooth local components).

[F3]

Complex forms split by type, $d = \partial + \overset{ˉ}{\partial}$ , and the scalar Dolbeault operator obeys the square-zero and graded Leibniz identities (A smooth differential $k$ -form, The wedge product of differential forms, The exterior power bundle of the cotangent bundle, Bigraded complex forms and the Dolbeault operators, The d, partial and dbar identities).

[F4]

A meromorphic function has isolated zeros and poles unless it is identically zero; a nonzero holomorphic function has a finite zero order and a local factorization by that power. The identity theorem applies on each connected chart. The transition law for a meromorphic differential is $h_{j} (w) = h_{i} (z (w)) z^{'} (w)$ , equivalently $h_{i} d z_{i} = h_{j} d z_{j}$ (Meromorphic functions on a plane domain, Isolated singularities: removable, poles, and essential singularities, Identity theorem for holomorphic functions, The order of a zero is the exponent in its local holomorphic factorization, Holomorphic maps and meromorphic functions on Riemann surfaces, Meromorphic differentials, orders and residues).

[F5]

For a smooth complex-valued function, $\partial_{\overset{ˉ}{z}} f = 0$ is equivalent to complex differentiability and hence holomorphy; the chain rule gives the change-of-coordinate formula (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions, The chain rule for complex derivatives, Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F6]

The smooth dual of a vector bundle is defined fibrewise; for a complex line the complex-linear dual transition is the inverse scalar (Dual and Hom vector bundles).

Proof

technique · direct local verification

1.1F1F2given

The fibrewise complex-linear trivializations give smooth transitions in $C^{\times}$ and composition of the maps $θ_{i} \circ θ_{j}^{- 1}$ gives $g_{i j} g_{j k} = g_{i k}$ . For a supplied countable holomorphic cocycle, its real multiplication matrices are smooth $G L (2, R)$ transitions, so [F2] constructs the underlying smooth rank-two bundle; these matrices commute with the standard complex structure, and their holomorphicity makes the local total-space charts holomorphic.

1.2F1F3F4givenalgebra

In a local frame $e_{i}$ , the cotangent line $K$ is spanned by $d z_{i}$ . If $z_{j} = ϕ (z_{i})$ , then $d z_{j} = ϕ^{'} (z_{i}) d z_{i}$ and a differential $ω = f_{i} d z_{i} = f_{j} d z_{j}$ has $f_{i} = (d z_{j} / d z_{i}) f_{j}$ . Since $ϕ^{'}$ is holomorphic and nowhere zero, these are holomorphic line-bundle transitions; a local section of $K$ is holomorphic or meromorphic exactly when its coefficient $f_{i}$ is, so these sections are precisely the corresponding differentials. Conjugation gives the stated transitions for $\overline{K}$ , and the type decomposition gives the local formulas for $E$ -valued forms.

1.3F1F4given

Let $Z$ be the set of points having a neighborhood on which the section is zero. It is open. If $p$ is in its closure, choose a connected chart and frame around $p$ ; the meromorphic coefficient has zeros accumulating at $p$ . A pole at $p$ is impossible because its finite principal part is nonzero on a punctured neighborhood, and otherwise the holomorphic identity theorem makes the coefficient identically zero near $p$ . Thus $Z$ is closed; since $X$ is connected and the section is nonzero, $Z = \emptyset$ . Each local representative consequently has a finite Laurent order at every point. Under a change of frame it is multiplied by a holomorphic unit, and under a coordinate change its argument is composed with a biholomorphism with nonzero derivative; neither operation changes the zero or pole order. Its zeros and poles are therefore locally isolated, so the divisor is locally finite and has finite support on compact $X$ .

1.4F3F5givenalgebra

On an overlap with $z_{j} = ϕ (z_{i})$ and $e_{j} = g_{i j} e_{i}$ , a section has coefficients $f_{i} = g_{i j} f_{j}$ . The chain rule and holomorphy of $g_{i j}$ give $\overset{ˉ}{\partial} f_{i} = g_{i j} \overset{ˉ}{\partial} f_{j}$ as $(0, 1)$ -forms, so the local formula defines a global operator. The same calculation applies to $E$ -valued forms; the scalar graded Leibniz rule gives the displayed rule, and scalar ${\overset{ˉ}{\partial}}^{2} = 0$ gives ${\overset{ˉ}{\partial}}_{E}^{2} = 0$ . By [F5], its kernel on sections is exactly the holomorphic sections.

2.1F2F6step 1.1step 1.2step 1.3step 1.4∎

In the complex-linear dual frame the transition is $g_{i j}^{- 1}$ , which is holomorphic and nonvanishing, so the dual is a holomorphic line bundle and the same local construction defines ${\overset{ˉ}{\partial}}_{E^{*}}$ . The preceding coordinate and frame calculations establish all stated definitions and well-definedness claims. No choice principle is used.

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Sources

Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry (author manuscript, Universite Grenoble Alpes) (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)