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Chern connection of a Hermitian holomorphic line bundle

Statement

Let $X$ be a Riemann surface and $E \to X$ a holomorphic line bundle with Hermitian metric $h$ (Holomorphic line bundles and meromorphic sections on a Riemann surface, Hermitian metric and $L^{2}$ pairing on a compact Riemann surface). Identify the complexified cotangent bundle with the Whitney sum $Λ^{1, 0} T^{*} X \oplus Λ^{0, 1} T^{*} X$ , and hence identify the bundle of complex-valued one-forms with values in $E$ as $(Λ^{1, 0} T^{*} X \oplus Λ^{0, 1} T^{*} X) \otimes E$ (Cotangent space and cotangent bundle as a disjoint union, Whitney sums of vector bundles, Whitney sums are smooth vector bundles).

A connection on $E$ is a $C$ -linear map $\nabla : C^{\infty} (X, E) ⟶ C^{\infty} (X, (Λ^{1, 0} \oplus Λ^{0, 1}) T^{*} X \otimes E)$ satisfying $\nabla (f s) = d f \otimes s + f \nabla s$ for $f \in C^{\infty} (X; C)$ and $s \in C^{\infty} (X, E)$ (The exterior derivative of a function is its differential). Its $(1, 0)$ - and $(0, 1)$ -parts $\nabla^{'}$ and $\nabla^{''}$ are the projections to the two summands. Extend $h$ to $E$ -valued one-forms by $⟨ α \otimes u, t ⟩_{h} = α h (u, t)$ and $⟨ s, β \otimes v ⟩_{h} = \overset{ˉ}{β} h (s, v)$ , pairing the bundle factors and conjugating the one-form coefficient in the second argument. The connection is compatible with $h$ when $d (h (s, t)) = ⟨ \nabla s, t ⟩_{h} + ⟨ s, \nabla t ⟩_{h}$ for all smooth sections $s, t$ .

There exists exactly one connection $\nabla_{E}$ such that $\nabla_{E}^{''} = {\overset{ˉ}{\partial}}_{E}$ and $\nabla_{E}$ is compatible with $h$ . It is the Chern connection. In a holomorphic frame $e$ over a holomorphic chart, with $ψ = h (e, e) > 0$ , it is $\nabla_{E} (f e) = (d f + f ψ^{- 1} \partial ψ) \otimes e,$ so $\nabla_{E}^{'} e = ψ^{- 1} \partial ψ \otimes e$ , $\nabla_{E}^{''} e = 0$ , and its connection form is $ω = ψ^{- 1} \partial ψ = \partial \log ψ$ . In another smooth frame $e^{'} = g e$ , $g \in C^{\infty} (U; C^{\times})$ , the full connection form transforms by $ω^{'} = ω + g^{- 1} d g$ .

Facts & Assumptions

Given: A Riemann surface $X$ , a holomorphic line bundle $E \to X$ , and a supplied smooth Hermitian metric $h$ on $E$ .

[F1]

In a holomorphic frame $e$ , a section is $f e$ with smooth coefficient $f$ ; the canonical Dolbeault operator is ${\overset{ˉ}{\partial}}_{E} (f e) = (\overset{ˉ}{\partial} f) \otimes e$ , and holomorphic frame changes are holomorphic nonvanishing functions (Holomorphic line bundles and meromorphic sections on a Riemann surface, Hermitian metric and $L^{2}$ pairing on a compact Riemann surface, Local and global frames of a vector bundle, Smoothness of a section is equivalent to smooth local components).

[F2]

Complex one-forms split into types and $d = \partial + \overset{ˉ}{\partial}$ ; the type projections and their Leibniz rules are coordinate-independent (Bigraded complex forms and the Dolbeault operators, The d, partial and dbar identities, A smooth differential $k$ -form, The wedge product of differential forms).

[F3]

For a smooth function $f$ , its ordinary differential is $d f$ , and the complex chain rule and Wirtinger derivatives give $\partial \log ∣ g ∣^{2} = g^{- 1} \partial g$ for every nonvanishing holomorphic $g$ (The exterior derivative of a function is its differential, The chain rule for complex derivatives, A complex domain is a nonempty connected open subset of $C$ , The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions).

[F4]

The cotangent bundle is the disjoint union of its cotangent fibres, and the Whitney sum of the two type bundles is a smooth vector bundle (Cotangent space and cotangent bundle as a disjoint union, Whitney sums of vector bundles, Whitney sums are smooth vector bundles).

[F5]

A compact set inside an open set admits a smooth cutoff equal to one on a neighborhood of that set and supported in the open set (A manifold bump for a compact set inside an open set).

Proof

technique · direct local construction and uniqueness

1.1F1F2F3F4F5given

First, any connection in the Statement is local. If a global section $s$ vanishes near $p$ , choose a smooth cutoff $χ$ supported in that neighborhood and equal to one near $p$ by [F5]. Then $χ s = 0$ and the Leibniz rule at $p$ gives $\nabla s (p) = 0$ . A local section can be multiplied by a cutoff compactly supported in its domain and extended by zero; near any point where the cutoff is one this defines its connection independently of the extension, by locality. Thus the connection and compatibility identities apply to local frames. In a holomorphic coordinate chart and holomorphic frame $e$ , put $ω_{e} = ψ^{- 1} \partial ψ$ and define $\nabla_{E} (f e) = (d f + f ω_{e}) \otimes e$ . The target one-form bundle is smooth by [F4]. This is $C$ -linear and obeys the Leibniz rule. Its $(0, 1)$ -part is $\overset{ˉ}{\partial} f \otimes e = {\overset{ˉ}{\partial}}_{E} (f e)$ , since $ω_{e}$ has type $(1, 0)$ .

2.1F2F3step 1.1givenalgebra

For $s = f e$ and $t = g e$ , the right side of metric compatibility is $(d f + f ω_{e}) \overset{ˉ}{g} ψ + f \overline{(d g + g ω_{e})} ψ$ . Since $ω_{e} + {\overset{ˉ}{ω}}_{e} = ψ^{- 1} d ψ$ , this equals $d (f \overset{ˉ}{g} ψ) = d (h (s, t))$ . Thus the local connection is compatible with $h$ .

3.1F1F3step 1.1step 2.1algebra

If $e^{'} = g e$ is another holomorphic frame, then $ψ^{'} = ∣ g ∣^{2} ψ$ . Since $\partial \overset{ˉ}{g} = 0$ , $\partial ∣ g ∣^{2} = \overset{ˉ}{g} \partial g$ , whence $ω_{e^{'}} = \partial \log ψ^{'} = ω_{e} + g^{- 1} d g$ . For an arbitrary smooth change of frame, the same connection's form transforms by the full Leibniz rule: $\nabla_{E} (g e) = (d g + g ω_{e}) \otimes e = (ω_{e} + g^{- 1} d g) \otimes e^{'}$ . Thus the local formulas agree on holomorphic overlaps, define a global connection with the two required properties, and give the stated smooth-frame transformation.

4.1F1F2F5step 1.1step 2.1algebra∎

If $\nabla$ and $\tilde{\nabla}$ are two connections with the prescribed $(0, 1)$ -part, their difference is $C^{\infty}$ -linear by the Leibniz rule. Since $E$ has rank one, it is multiplication by an $E$ -endomorphism-valued one-form $η$ , and equality of $(0, 1)$ -parts forces $η$ to have type $(1, 0)$ . Subtracting their metric-compatibility identities gives $(η + \overset{ˉ}{η}) h (s, t) = 0$ for all $s, t$ ; taking a local nonzero frame yields $η + \overset{ˉ}{η} = 0$ . The two terms have distinct types, so each vanishes and $η = 0$ . Hence the connection is unique. No choice principle is used.

Depends on

Used by

The Dolbeault adjoint and Laplacian: local formulas and ellipticity Lemma

Dependency tree · two levels

90 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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