Kahler FormEdit
The Kähler form is a central object in complex differential geometry, arising from a harmonious pairing of a complex structure with a compatible Riemannian metric. Named after Karl Kähler, it sits at the crossroads of complex geometry, symplectic geometry, and algebraic geometry, and it governs much of the geometry and topology of complex manifolds that carry a suitable metric. Given a complex manifold (M, J) equipped with a Hermitian metric g, one forms a real 2-form ω by ω(X,Y) = g(JX,Y). This 2-form encodes both the metric and the complex structure in a single geometric object. When ω is closed (dω = 0), the metric is called Kähler, and the geometry becomes especially rigid and tractable: the complex structure is parallel with respect to the Levi-Civita connection, and the manifold satisfies a powerful suite of tools from Hodge theory and algebraic geometry.
In many texts, the Kähler form is viewed as the bridge between the analytic and algebraic viewpoints on a complex manifold. Its existence imposes strong topological constraints and yields a rich set of invariants. The study of the Kähler form is thus a cornerstone of modern geometry, with deep consequences for the shape of the manifold, the behavior of differential operators, and the structure of cohomology.
Definition and basic properties
Construction and compatibility: Let M be a complex manifold with almost complex structure J, and let g be a Riemannian metric that is Hermitian with respect to J (i.e., g(JX, JY) = g(X, Y) for all tangent vectors X, Y). The associated 2-form ω, defined by ω(X, Y) = g(JX, Y), is real, of type (1,1) with respect to J, and is positive in the sense that ω(v, Jv) > 0 for all nonzero tangent vectors v. When viewed in local holomorphic coordinates, ω can be written as ω = i g_{α \bar β} dz^α ∧ d\bar z^{\bar β}.
Closedness and the Kähler condition: The form ω need not be closed in general. When dω = 0, the Hermitian metric g is called a Kähler metric, and the pair (g, J) enjoys special rigidity: the Levi-Civita connection preserves J (i.e., ∇J = 0), and ω is parallel (∇ω = 0). In a Kähler manifold, the geometry is simultaneously complex, Riemannian, and symplectic.
Type and potentials: The Kähler form is a (1,1)-form, reflecting its compatibility with the complex structure. Locally, if ω is closed, there exists a real-valued function φ, called a Kähler potential, such that ω = i ∂∂̄ φ in a neighborhood. Globally, a global potential need not exist, but the local potential is a powerful tool for computations and for understanding the local geometry.
Cohomology class: Since ω is closed, it defines a de Rham cohomology class [ω] ∈ H^2(M, R). Because ω is of type (1,1), [ω] lies in the Hodge subspace H^{1,1}(M) ∩ H^2(M, R). The class [ω] is often referred to as the Kähler class of the metric and plays a central role in questions about deformations, polarizations, and embeddings in algebraic geometry.
Positivity and polarization: The positivity of ω on tangent vectors makes (M, ω) a symplectic manifold in the sense of differential geometry, while its (1,1) type ties it to the complex structure. In the algebro-geometric setting, ω encodes a polarization: a choice of ample line bundle whose first Chern class coincides with [ω].
Kähler potential and local expressions
Local coordinates: In a neighborhood with holomorphic coordinates z^1, …, z^n, a Kähler metric can be described locally by the Hermitian matrix (g_{α \bar β}) of coefficients, with ω = i ∑ g_{α \bar β} dz^α ∧ d\bar z^{\bar β}.
Kähler potential: If ω is closed, there exists φ (defined locally) with ω = i ∂∂̄ φ. The potential φ is not unique; adding a purely holomorphic plus anti-holomorphic function modifies φ by a pluriharmonic term, leaving ω unchanged. This local representation is a practical computational tool and underlies many problems in complex Monge–Ampère equations.
Relation to the metric: The components g_{α \bar β} are given by g_{α \bar β} = ∂^2 φ / ∂ z^α ∂ \bar z^{\bar β} in a local potential φ, reflecting how the metric is encoded in a single scalar function locally. This scalar function-based viewpoint is especially important in complex differential geometry and in the study of canonical metrics.
Global aspects, cohomology, and projectivity
The Kähler class: The cohomology class [ω] is central to questions about deformation and moduli. Deformations of the complex structure that preserve the Kähler condition can often be studied by tracking how [ω] varies within H^{1,1}(M, R).
Polarizations and projectivity: On a compact complex manifold, if [ω] is an integral class (i.e., lies in the image of H^2(M, Z) under the natural inclusion into real cohomology) and represents a positive form, then by the Kodaira embedding theorem the manifold is projective. In algebraic geometry terms, a polarization in the form of an ample line bundle corresponds to an integral Kähler class.
Hodge theory and Lefschetz theory: Kähler geometry brings powerful cohomological tools. The Hodge decomposition, the Lefschetz decomposition, and the hard Lefschetz theorem all apply to compact Kähler manifolds, leading to deep restrictions on the topology and to a rich structure of harmonic forms.
Canonical metrics: The Kähler framework supports the study of canonical metrics, such as Kähler–Einstein metrics, where the Ricci form is proportional to ω. The existence of such metrics on a given complex manifold relates to the sign of the first Chern class c1(M) and to stability notions in algebraic geometry (e.g., K-stability in the Fano case). The Calabi conjecture, solved by Shing-Tung Yau, guarantees the existence of a unique Ricci-flat Kähler metric in a given Kähler class on manifolds with c1(M) = 0, and related results provide a broad toolkit for geometric analysis.
Examples and canonical metrics
Complex projective space: The projective space CP^n carries the Fubini–Study metric, whose associated Kähler form is canonical and yields a standard polarization. This form generates the positive curvature needed for many embedding theorems and for the study of projective geometry.
Complex tori and abelian varieties: Complex tori equipped with flat metrics furnish Kähler forms that are parallel and yield simple, globally defined structures. In the algebraic setting, these spaces often come endowed with natural polarizations, linking the analytic and algebraic pictures.
K3 surfaces and hyperkähler manifolds: Compact, simply connected complex surfaces like K3 surfaces carry rich families of Kähler metrics. In higher dimensions, hyperkähler manifolds provide fertile ground for exploring multiple compatible complex structures with common Kähler forms.
Calabi–Yau manifolds and Fano manifolds: Calabi–Yau manifolds admit Ricci-flat Kähler metrics by the Calabi conjecture, a fact with deep implications for string theory and algebraic geometry. Fano manifolds admit Kähler–Einstein metrics in many cases, with existence linked to stability conditions.
Existence, obstructions, and non-Kähler geometry
Existence on complex manifolds: While many natural geometric objects carry Kähler forms, not every complex manifold admits a Kähler metric. The obstruction is often topological or cohomological; a classic necessary condition is that the first Betti number b1 is even for compact complex manifolds. When these conditions fail, the manifold is non-Kähler.
Non-Kähler examples and features: Hopf surfaces and many non-Kähler complex surfaces (and more generally some non-algebraic complex manifolds) lack a global Kähler form. In such cases, the powerful machinery of Kähler geometry (Hodge theory, Lefschetz theory) does not apply in full strength, and alternative geometric tools must be used.
Nonexistence results and obstructions: The study of non-Kähler geometry is an active area of differential geometry. Researchers investigate what features of a complex manifold force the absence of a Kähler metric and how one can still extract meaningful geometric and topological information in the absence of a Kähler structure.