Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a special kind of space. It has three structures that work together well: a complex structure, a Riemannian structure, and a symplectic structure. These help mathematicians study the space using ideas from complex numbers, geometry, and physics.

The idea of a Kähler manifold was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930. It was later introduced by Erich Kähler in 1933. The name we use today was set by André Weil. Kähler geometry is a part of math that looks at these special spaces and what they are like.

One important fact is that every smooth complex projective variety is a Kähler manifold. This makes Kähler manifolds very useful in algebraic geometry. For example, Hodge theory, a big part of algebraic geometry, uses Kähler metrics to prove its ideas. Kähler manifolds also have special connections like Hermitian Yang–Mills connections and special metrics such as Kähler–Einstein metrics. These help mathematicians learn about the shape and structure of these spaces.

Definitions

A Kähler manifold is a special space in mathematics. It has three structures that work well together: a complex structure, a Riemannian structure, and a symplectic structure. These structures make the manifold very symmetric and organized.

From a symplectic viewpoint, a Kähler manifold is a symplectic manifold with an integrable almost-complex structure. This structure works with the symplectic form. This means a certain bilinear form 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. This metric's associated 2-form is closed. This leads to the Kähler form, which is important in studying 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.

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

Kähler manifolds have special structures. This creates important links between different mathematical tools used on them. These links help mathematicians learn more about the shapes and properties of these special spaces. They are very useful when studying complex shapes and their features.

The Laplacian on a Kähler manifold

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

An important idea is that on Kähler manifolds, different versions of the Laplacian are related. This helps us understand the shape and properties of the manifold better. It also connects the study of shapes (topology) with the study of complex structures.

Topology of compact Kähler manifolds

Compact Kähler manifolds have special shapes and structures. One important fact is that certain numbers, called odd Betti numbers, are always even for these manifolds. This is different from other complex shapes, like the Hopf surface.

There are many important ideas that describe how the shape and structure of compact Kähler manifolds work, such as the Lefschetz hyperplane theorem and the hard Lefschetz theorem. Scientists are still learning more about these special shapes.

Characterizations of complex projective varieties and compact Kähler manifolds

The Kodaira embedding theorem helps us learn about special types of Kähler manifolds. It says that a compact complex manifold is projective if it has a special kind of form called a Kähler form. This form needs to meet certain conditions.

Kähler manifolds have many interesting properties. Some of these properties also apply to a broader class called ∂ ∂ ¯-manifolds. 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, some Kähler manifolds cannot be changed into projective ones. Researchers have looked at when a compact complex manifold can have a Kähler metric. They found 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. It has constant Ricci curvature. This means the Ricci curvature tensor is the same everywhere. It is equal to a constant number times the metric tensor. The idea connects to Einstein’s theory of relativity. In this theory, space without mass has zero Ricci curvature.

Important work by Shing-Tung Yau showed that certain types of manifolds always have Kähler–Einstein metrics. This helps mathematicians classify and understand these shapes better.

Holomorphic sectional curvature

The holomorphic sectional curvature measures how a special space, called a Hermitian manifold, is different from ordinary flat space. It looks at the curvature of complex lines in the space.

For example, in a 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 shape and properties of these spaces. It can show special features, like whether a compact Kähler manifold is "rationally connected."

Examples

Some important examples of Kähler manifolds are:

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) has a flat metric and is a compact Kähler manifold.
3. Every Riemannian metric on an oriented 2-manifold is Kähler. For example, an oriented Riemannian 2-manifold is a special type of Riemann surface.
4. There is a standard Kähler metric on complex projective space CPⁿ, called the Fubini–Study metric.
5. The 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.