Chern connection
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| Chern connection | |
|---|---|
| Name | Chern connection |
| Field | Differential geometry; Complex geometry |
| Introduced by | Shiing-Shen Chern |
| Year | 1946 |
| Related | Hermitian metric; Holomorphic vector bundle; Levi-Civita connection; Chern classes |
Chern connection The Chern connection is a canonical connection associated to a Hermitian metric on a holomorphic vector bundle or to a Hermitian metric on a complex manifold. It provides a bridge between holomorphic structures, metric geometry, and topological invariants, and plays a central role in results of complex differential geometry and gauge theory. The construction yields a unique connection compatible with both the holomorphic structure and the Hermitian metric, whose curvature encodes Chern classes and influences theorems by Gauss, Kodaira, Donaldson, and Yau.
Definition and basic properties
On a holomorphic vector bundle E over a complex manifold M endowed with a Hermitian metric h, the Chern connection is the unique connection ∇ satisfying two conditions: ∇ is metric-compatible with h and the (0,1)-part of ∇ coincides with the holomorphic structure, i.e., with the Dolbeault operator ∂̄_E. In the case of the holomorphic tangent bundle of a complex manifold with a Hermitian metric (a Hermitian manifold), the Chern connection is similarly characterized and preserves the complex structure. The connection is linear over the sheaf of smooth functions and yields a decomposition of forms into types (p,q) compatible with the complex structure. Important formal properties include that its torsion is of type (2,0)+(0,2) in general, and it reduces to a torsion-free connection precisely when the Hermitian metric is Kähler, linking to classical results by Gauss, Hirzebruch, and Kodaira.
Construction on Hermitian and complex manifolds
Locally one constructs the Chern connection using a holomorphic frame for a bundle E: given a local holomorphic trivialization and the Hermitian matrix h_{i\bar j} of inner products of frame sections, the connection 1-form Θ is given by Θ = h^{-1} ∂h, where ∂ is the (1,0)-part of the exterior derivative on matrix-valued functions. For the tangent bundle of a complex manifold with Hermitian metric g, choose a local holomorphic coordinate system; the Christoffel symbols of type (1,0) are computed from g_{i\bar j} via Γ^k_{ij} = g^{k\bar ℓ} ∂_i g_{j\bar ℓ}. This construction uses existence results from Dolbeault cohomology and techniques familiar from the work of Dolbeault, Serre, and Cartan. The uniqueness follows from linear algebra and compatibility constraints analogous to those used by Levi and Civita in Riemannian geometry.
Curvature and Chern classes
The curvature F_∇ of the Chern connection is a (1,1)-form with values in End(E) when E is holomorphic, a fact crucial for defining Chern classes in de Rham cohomology. The trace of the curvature yields representatives of Chern forms: the first Chern form is (i/2π) Tr(F_∇) and its de Rham class equals the topological first Chern class c_1(E). Higher Chern forms are obtained from symmetric polynomials in F_∇, producing closed differential forms whose de Rham classes coincide with the Chern classes defined via obstruction theory and K-theory as in the work of Chern, Whitney, and Grothendieck. These relationships underpin index theorems by Atiyah and Singer and vanishing theorems by Kodaira and Nakano, and appear in curvature conditions used by Yau in the Calabi conjecture and by Donaldson in gauge-theoretic stability.
Comparison with Levi-Civita and other connections
Unlike the Levi-Civita connection on a Riemannian manifold, which is uniquely torsion-free and metric-compatible, the Chern connection is generally not torsion-free except in the Kähler case; when the Hermitian metric is Kähler, the Chern and Levi-Civita connections coincide on the holomorphic tangent bundle. Other connections of interest include the Bismut connection, which is metric-compatible and has skew-symmetric torsion related to conformal and string-theoretic constructions, and the Kobayashi connection, used in complex Finsler geometry. In gauge-theoretic contexts, the Chern connection is compared with unitary connections used in Yang–Mills theory; for holomorphic bundles with Hermitian metric, the Chern connection is the unique unitary connection whose (0,1)-part is the holomorphic structure, making it the natural object in the Donaldson–Uhlenbeck–Yau correspondence between stability and Hermitian–Yang–Mills connections.
Examples and computations
On a holomorphic line bundle L with local holomorphic trivialization s and Hermitian metric given by ||s||^2 = e^{-φ}, the Chern connection 1-form is ∂φ and the curvature is ∂∂̄φ, a (1,1)-form whose cohomology class is c_1(L). For the complex projective space CP^n with the Fubini–Study metric, the Chern connection on the tangent bundle or on the hyperplane line bundle yields curvature proportional to the Kähler form, producing explicit Chern forms computed classically by Chern and Weil. On complex tori and Calabi–Yau manifolds, specialized metrics make computations tractable: flat metrics give vanishing curvature and trivial Chern classes, while Ricci-flat Kähler metrics from the Calabi conjecture give vanishing first Chern form, connecting to results of Yau and Tian. Explicit matrix computations in local frames are routine in examples studied by Kodaira, Hirzebruch, and Bott.
Applications in complex geometry and gauge theory
The Chern connection is fundamental in proofs of vanishing theorems and in the formulation of stability conditions: it appears in Nakano and Kodaira vanishing, in the Hermitian–Einstein condition that characterizes stable bundles in the Donaldson–Uhlenbeck–Yau theorem, and in the analytical approach to the Calabi conjecture by Yau. In algebraic geometry it links holomorphic vector bundles to topological invariants used in Riemann–Roch theorems by Grothendieck and Hirzebruch and in index formulas by Atiyah and Singer. In gauge theory and mathematical physics, Chern connections model background gauge fields in Seiberg–Witten and Yang–Mills theories and enter constructions of characteristic classes underlying anomalies studied by Witten and Freed. The ubiquity of the connection across work by Chern, Atiyah, Donaldson, Yau, and others makes it a central tool in modern differential and algebraic geometry.