Kahler ManifoldEdit
Kahler manifolds occupy a central place in modern geometry because they fuse complex structure, Riemannian geometry, and symplectic form into a single, highly constrained framework. On a Kahler manifold, the complex structure J, the Riemannian metric g, and the associated two-form ω(X,Y) = g(JX, Y) work together so that ω is not only compatible with the metric and the complex structure but also closed. This triad of conditions yields strong consequences for analysis, topology, and algebraic geometry, and it furnishes a natural bridge between seemingly disparate areas of mathematics.
The theory traces its power to a simple observation: when these three structures align, local complex coordinates can be used to express geometric data in a way that respects both holomorphic and symplectic aspects. The resulting geometry serves as a proving ground for deep results in differential geometry, complex geometry, and mathematical physics. Notable examples include complex projective spaces with their canonical metrics, complex tori with flat metrics, and, in higher-dimensional settings, Calabi-Yau and hyperkähler manifolds that play pivotal roles in string theory and algebraic geometry.
Definitions and basic properties
A Kahler manifold is a complex manifold M equipped with a Riemannian metric g that is Hermitian with respect to the complex structure J, meaning g(JX, JY) = g(X, Y) for all tangent vectors X, Y. The corresponding Kahler form is ω(X, Y) = g(JX, Y). The defining condition is that ω be closed: dω = 0.
Equivalently, the Levi-Civita connection ∇ of g preserves the complex structure, i.e., ∇J = 0. In particular, the parallel transport defined by the metric respects the holomorphic structure, and the geometry is highly rigid.
Locally, a Kahler form admits a potential: there exists a real-valued function φ (the Kahler potential) such that ω = i ∂∂̄ φ in local holomorphic coordinates. This local potential form is a distinctive feature of Kahler geometry and underpins many analytic techniques.
For compact Kahler manifolds, the cohomology class [ω] in the real second cohomology group lies in the (1,1)-part of de Rham cohomology; when [ω] is integral, the geometry interacts particularly cleanly with line bundles and projectivity, as explained in the Kodaira embedding framework.
Important related notions include the Kahler class, which is the cohomology class of ω in H^2(M, R) that contains all cohomologous Kahler forms within a fixed complex structure, and the notion of holonomy: Kahler structures force restricted holonomy, leading to special cases such as Calabi-Yau and hyperkähler geometries.
Geometry, curvature, and special metrics
The Ricci form ρ of a Kahler metric is a closed real (1,1)-form representing the first Chern class c1(M). In local coordinates, ρ is given by a trace of the curvature of the holomorphic tangent bundle and plays a central role in canonical metrics.
A central problem in Kahler geometry is the existence of special metrics, notably Kahler-Einstein metrics, which satisfy Ricci(g) = λg for some constant λ. The sign of c1(M) governs the typical scenarios: negative or zero first Chern class tends to admit canonical metrics in each Kähler class, while positive c1(M) (the Fano case) requires deeper stability conditions and is tied to algebro-geometric notions of stability.
The Aubin–Yau theorem provides foundational existence results: on a compact Kahler manifold with c1(M) < 0, there exists a unique Kahler-Einstein metric in each Kähler class; when c1(M) = 0, there exists a unique Ricci-flat Kahler metric in each Kähler class (the Calabi–Yau case). The c1(M) > 0 situation is more subtle and connects with notions of K-stability and algebraic stability.
Local and global holomorphic sections of line bundles interact with the metric structure through the Chern class. In particular, the genus of the underlying complex manifold and the positivity of line bundles influence the range of possible Kahler forms.
Cohomology, Hodge theory, and deformations
A remarkable feature of Kahler geometry is the Hodge decomposition: for a Kahler manifold, the de Rham cohomology groups decompose into the Dolbeault groups H^{p,q}(M) with complex conjugation exchanging (p,q) and (q,p). This powerful structure allows one to study topology using holomorphic data.
The wedge product with the Kahler form induces hard Lefschetz-type theorems and Lefschetz sl(2) symmetries that constrain the topology of M. In particular, the Hodge numbers h^{p,q} are topological invariants of the complex structure in the Kahler setting and satisfy symmetry relations.
The cohomology class [ω], when integral, gives rise to ample line bundles and ties the differential-geometric picture to projective algebraic geometry via the Kodaira embedding theorem: a compact Kahler manifold with an integral Kahler class is projective, meaning it can be embedded into complex projective space as a closed algebraic subvariety.
Deformation theory studies how complex structures on M vary in families. For Kahler manifolds, deformations of the complex structure interact with the space of Kahler forms, leading to moduli spaces that reflect both analytic and algebro-geometric data.
Construction, examples, and consequences
Complex projective spaces CP^n carry the Fubini–Study metric, a canonical Kahler metric that serves as a fundamental model in both differential and algebraic geometry. A related construction uses ample line bundles and their associated metrics to produce Kahler forms with prescribed cohomology classes.
Complex tori and more general abelian varieties provide flat Kahler metrics, illustrating the basic noncompact-to-compact spectrum within Kahler geometry.
Calabi–Yau manifolds (Ricci-flat Kahler manifolds) arise as a key class in both geometry and theoretical physics. Their existence, proven via the Calabi conjecture, underpins mirror symmetry and compactifications in string theory. Hyperkähler manifolds, which carry a triple of complex structures satisfying quaternionic relations and a Ricci-flat metric, form another distinguished family with rich holonomy properties.
The relationship between Kahler geometry and algebraic geometry is deep: the existence of a Kahler metric in a given class interacts with the positivity of line bundles, stability notions, and the possibility of algebraic realization. This interplay is central to modern approaches to understanding complex manifolds and their moduli.
Algebraic aspects and projectivity
The Kodaira embedding theorem provides a bridge from differential geometry to algebraic geometry: if a compact complex manifold admits a Kahler metric whose Kähler class is integral (the class arises from the first Chern class of an ample line bundle), then the manifold is projective. This result links the differential-geometric condition of a closed Kahler form to the algebraic-geometric condition of being embeddable into projective space.
The study of c1(M) and canonical metrics informs the classification of complex manifolds. Fano manifolds (with c1(M) > 0) are the algebraic analogs of manifolds admitting positively curved Kahler metrics, while Calabi-Yau manifolds (c1(M) = 0) occupy a special place in both geometry and physics due to their Ricci-flat Kahler metrics.
Stability conditions, such as K-stability, have emerged as precise algebro-geometric criteria predicting when a Fano manifold admits a Kahler–Einstein metric. This “Yau–Tian–Donaldson” program connects geometric analysis with geometric invariant theory and moduli theory.