Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a special kind of space that has three different but compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. These structures allow mathematicians to study the space using tools from complex numbers, geometry, and physics all at once.

The idea of a Kähler manifold was first explored by Jan Arnoldus Schouten and David van Dantzig in 1930, and later formally introduced by Erich Kähler in 1933. The name we use today was settled by André Weil. Kähler geometry is the branch of math that focuses on understanding these special spaces and their properties.

One important fact is that every smooth complex projective variety is a Kähler manifold. This connection makes Kähler manifolds very important in algebraic geometry. For example, Hodge theory, a key part of algebraic geometry, relies on Kähler metrics to prove its results. Kähler manifolds also have special connections like Hermitian Yang–Mills connections and special metrics such as Kähler–Einstein metrics, which help mathematicians explore their shape and structure.

Definitions

A Kähler manifold is a special kind of space in mathematics that has three compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. These structures work together in a way that makes the manifold very symmetric and well-behaved.

From a symplectic viewpoint, a Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure that is compatible with the symplectic form. This means that a certain bilinear form derived from the symplectic form is symmetric and positive definite, making it a Riemannian metric. From a complex viewpoint, a Kähler manifold is a complex manifold with a Hermitian metric whose associated 2-form is closed. This leads to the definition of the Kähler form, which plays a central role in the study of these manifolds.

Kähler potential

A smooth real-valued function on a complex manifold can help describe special geometric shapes called Kähler manifolds. If this function meets certain conditions, it is called a Kähler potential. This potential helps create a Kähler form, which is important for studying these special spaces.

In any small area of a Kähler manifold, there is always a Kähler potential that can describe the local geometry. This makes it easier to study and compare different Kähler metrics within the same class, helping mathematicians understand their properties better.

Kähler manifolds and volume minimizers

For a special kind of space called a compact Kähler manifold, the size of a smaller space inside it can be figured out using a mathematical idea called its homology class. This shows that the shape of these smaller spaces is linked to their basic structure.

There is a formula, known as Wirtinger's formula, that helps calculate the volume of these smaller spaces. It tells us that the volume depends on a special measurement called the Kähler form. Importantly, these volumes are always positive, which means they show a strong positive nature in how these spaces are built. Additionally, each smaller space within a compact Kähler manifold is a minimal submanifold, meaning it has the smallest possible size for its kind.

Kähler identities

Main article: Kähler identities

Because Kähler manifolds have special structures, there are important connections between different mathematical tools used on them. These connections help mathematicians understand the shapes and properties of these special spaces better. They are especially useful when studying complex shapes and their properties.

The Laplacian on a Kähler manifold

A Kähler manifold is a special kind of space in mathematics that has three compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. On such a manifold, we can study how smooth functions and forms behave using something called the Laplacian.

One important idea is that for Kähler manifolds, different versions of the Laplacian turn out to be related. This means that studying these Laplacians helps us understand the shape and properties of the manifold in a deeper way. This relationship also helps connect the study of shapes (topology) with the study of complex structures.

Topology of compact Kähler manifolds

Compact Kähler manifolds have special properties in their topology. One important fact is that certain numbers, called odd Betti numbers, are always even for these manifolds. This is different from other complex manifolds, like the Hopf surface, which does not follow this rule.

There are many important theorems that describe how the shape and structure of compact Kähler manifolds behave, such as the Lefschetz hyperplane theorem and the hard Lefschetz theorem. Scientists are still studying which groups can be the fundamental groups of these special manifolds, and they have found several restrictions on what these groups can be.

Characterizations of complex projective varieties and compact Kähler manifolds

The Kodaira embedding theorem helps us understand special types of Kähler manifolds. It tells us that a compact complex manifold is projective if it has a special kind of form called a Kähler form. This form must meet certain conditions related to its class in a mathematical group.

Kähler manifolds have many interesting properties, and some of these also apply to a broader class called ∂ ∂ ¯-manifolds. In particular, for Kähler manifolds, two different ways of studying their structure give the same results.

Not all Kähler manifolds are projective, especially in higher dimensions. For example, in dimension 4, there are Kähler manifolds that cannot be deformed into projective ones. Researchers have also studied when a compact complex manifold can have a Kähler metric, finding that in some dimensions this depends on certain topological properties.

Kähler–Einstein manifolds

Main article: Kähler–Einstein metric

A Kähler–Einstein manifold is a special kind of Kähler manifold that has constant Ricci curvature. This means the Ricci curvature tensor is the same everywhere, equal to a constant number times the metric tensor. The idea connects to Einstein’s theory of relativity, where space without mass has zero Ricci curvature.

Important work by Shing-Tung Yau showed that certain types of manifolds, like those with ample canonical bundles and Calabi–Yau manifolds, always have Kähler–Einstein metrics. This helps mathematicians classify and understand these shapes better.

Holomorphic sectional curvature

The holomorphic sectional curvature is a way to measure how a special kind of space, called a Hermitian manifold, differs from ordinary Euclidean space. It focuses on the curvature of complex lines within the space. For example, in a special space called CPⁿ, this curvature is always 1. In another space, the open unit ball in Cⁿ, the curvature is -1.

This curvature helps us understand the complex geometry of these spaces. It can tell us important properties, like whether certain maps between spaces behave in a particular way. For instance, if a compact Kähler manifold has positive holomorphic sectional curvature, it has a special property called being "rationally connected."

Examples

Some important examples of Kähler manifolds include:

1. The complex space Cⁿ with the standard Hermitian metric is a Kähler manifold.
2. A compact complex torus Cⁿ/Λ (where Λ is a full lattice) inherits a flat metric and is therefore a compact Kähler manifold.
3. Every Riemannian metric on an oriented 2-manifold is Kähler. In particular, an oriented Riemannian 2-manifold is a Riemann surface in a special way.
4. There is a standard choice of Kähler metric on complex projective space CPⁿ, known as the Fubini–Study metric.
5. The induced metric on a complex submanifold of a Kähler manifold is also Kähler. This includes many important types of manifolds.
6. The open unit ball B in Cⁿ has a complete Kähler metric called the Bergman metric.
7. Every K3 surface is Kähler.