Kahler ClassEdit
The Kahler class sits at the heart of a rich synthesis of ideas in geometry. On a complex manifold, a Kahler form provides a metric, a symplectic structure, and compatibility with the complex structure all at once. The cohomology class represented by this form, denoted [ω], is the Kahler class. This invariant ties together local curvature data with global topological information, and it plays a central role in both pure mathematics and its applications to theoretical physics.
Because a Kahler form is closed, its class is stable under small, exact deformations. In practice, that means the Kahler class captures global, topological information about the manifold that persists as you vary the metric within its Kahler family. The class lives in the (1,1) slice of cohomology, reflecting the way the complex structure interacts with the geometry. Its prominence is reinforced by the fact that the space of all Kahler classes—the Kahler cone—records precisely which cohomology classes can be represented by some Kahler form. de Rham cohomology Hodge theory Kähler cone
Definition
Let M be a complex manifold and g a Riemannian metric compatible with its complex structure J. The associated Kahler form ω is the real, closed (1,1)-form defined by ω(X,Y) = g(JX,Y). The pair (M, g) is called a Kähler manifold when such a form ω exists. The Kahler class is the cohomology class [ω] ∈ H^2(M, R); it also lies in the Hodge summand H^{1,1}(M, R) because ω is of type (1,1). In symbols, [ω] ∈ H^{1,1}(M, R) ∩ H^2(M, R). The class completely determines the metric up to exact i∂∂̄-potentials in the usual sense of Kahler geometry. See also Kähler form for the underlying differential-geometric object, and Hodge theory for the decomposition that places [ω] in the (1,1) sector.
Two commonly used viewpoints emerge from this definition. First, the cohomology class emphasizes the topological footprint of the metric: it encodes volumes, intersection numbers, and curvature data endured by the manifold. Second, the potential-theoretic viewpoint shows that any two Kahler forms in the same class differ by i∂∂̄φ for a real-valued function φ, provided the result remains a Kahler form. This duality underlies many existence results for canonical metrics within a fixed class. See Kähler cone for how these classes populate the ambient cohomology space, and i∂∂̄-lemma for the analytic tools often used in this setting.
The Kahler cone and cohomology
The Kahler cone of M is the subset of H^{1,1}(M, R) consisting of all cohomology classes that can be represented by a Kahler form. Symbolically, it is Kähler cone(M) = { [ω] ∈ H^{1,1}(M, R) | ω is a Kahler form }. This cone is open in H^{1,1}(M, R) and reflects the degree of positivity that the manifold can sustain. The interaction between the Kahler class and the complex structure is central: deforming the complex structure changes H^{1,1}(M, R) and hence the form of the cone, even if some numerical invariants stay constant. In algebraic terms, when [ω] is integral (i.e., lies in the image of H^2(M, Z) inside H^2(M, R)), one can often realize ω as the curvature form of a holomorphic line bundle L with c1(L) = [ω], linking complex geometry to projective embeddings via the Kodaira embedding theorem. See Hodge theory, Kähler cone, Chern class, and Kodaira embedding theorem.
Relation to algebraic geometry and metrics
A central theme in this area is the linkage between Kahler classes and algebraic positivity. If [ω] is integral and represented by a Kahler form, then the corresponding line bundle L has positive curvature, and M can often be embedded into projective space as a projective variety. This connection is encapsulated in results like the Kodaira embedding theorem and the broader dictionary between positivity in algebraic geometry and the existence of Kahler metrics with special properties. See Kodaira embedding theorem and ample line bundle for the algebraic side of the story, and Chern class for the curvature–topology bridge.
On the analytic side, a flagship achievement is the Calabi conjecture, proven by Shing-Tung Yau, which asserts that within a fixed Kahler class, one can solve nonlinear partial differential equations to obtain canonical metrics with prescribed Ricci curvature. In particular: - If c1(M) is negative, there exists a unique Kahler-Einstein metric in the given class. See Kähler–Einstein metric and Calabi conjecture. - If c1(M) = 0, there exist Ricci-flat Kahler metrics in each Kahler class, giving the so-called Calabi–Yau metrics. See Calabi conjecture and Calabi–Yau manifold. - For c1(M) > 0 (the Fano case), existence of Kahler–Einstein metrics depends on stability phenomena and remains a nuanced area of research, with deep links to algebraic geometry. See Fano manifold and Kähler–Einstein metric.
These results highlight how the Kahler class governs not just shape and size, but the very curvature and, by extension, the global geometry of the manifold. See Yau's theorem for the definitive existence result in the general Kahler setting.
Examples and implications
Complex projective space CP^n with the Fubini–Study form is a classical example where the Kahler class is proportional to the first Chern class of the hyperplane line bundle O(1), making the geometry highly explicit. The associated class sits in the integral cohomology, tying geometry to a concrete projective embedding.
Complex tori and hyperkähler manifolds admit rich families of Kahler forms whose classes fill out sizable regions of H^{1,1}(M, R). In these spaces, the Kahler class often serves as a coordinate on moduli spaces of metrics, helping to organize deformations of the geometric structure. See Complex torus and Hyperkähler manifold.
Not every complex manifold supports a Kahler metric. There are compact complex manifolds outside the Kahler realm, sometimes called non-Kähler manifolds. The existence or non-existence of Kahler metrics is a foundational question in complex geometry and motivates refined tools to study complex structures that do not admit the classical Kahler condition. See Non-Kähler manifold.
In physics and broader contexts
Kahler geometry is a natural setting for aspects of theoretical physics, especially in supersymmetric theories and in string theory compactifications. The Kahler class plays the role of moduli controlling the sizes of internal dimensions, and the interplay between the topology of M and the family of possible metrics has concrete implications for physical models. See String theory for the broader physics backdrop and Moduli space for the mathematical perspective on parameter spaces of geometric structures.