Kähler manifold

Definition

A Kähler manifold is a special type of complex manifold equipped with a Riemannian metric that is both Hermitian and symplectic, meaning it has a compatible complex structure that allows for the integration of geometric and complex analysis. This unique combination allows Kähler manifolds to have rich geometric properties, making them significant in both Riemannian geometry and complex geometry. The interplay between the metric, the symplectic structure, and the complex structure is key to understanding their holonomy properties and behavior under deformation.

5 Must Know Facts For Your Next Test

1. Kähler manifolds are characterized by having both a Riemannian metric and a symplectic form that are compatible with each other, which means they can be expressed through the same underlying complex structure.
2. The holonomy group of a Kähler manifold is contained in the unitary group, indicating that Kähler manifolds are always connected to special geometries like projective spaces.
3. Kähler manifolds are important in both mathematics and theoretical physics, particularly in string theory and mirror symmetry, where their properties facilitate various physical interpretations.
4. Every Kähler manifold can be shown to be a submanifold of some projective space due to its rich geometric structure, allowing for an elegant classification in algebraic geometry.
5. The existence of Kähler metrics on compact manifolds is closely related to the solution of several key problems in differential geometry, including Yau's proof of the Calabi conjecture.

Review Questions

• How do the properties of Kähler manifolds connect to the broader classification of holonomy groups?
  • Kähler manifolds are classified under holonomy groups that are contained within the unitary group, reflecting their special geometrical structure. This connection highlights how Kähler manifolds exemplify certain behaviors related to curvature and symmetry. By understanding this relationship, one can see how Kähler metrics not only relate to complex structures but also influence the holonomy characteristics, leading to insights into their global geometry.
• In what ways does the presence of a Hermitian metric enhance the study of complex structures in Kähler manifolds?
  • The Hermitian metric in Kähler manifolds establishes a connection between Riemannian geometry and complex analysis, allowing for deeper exploration of geometric properties. It enables us to define notions like volume and curvature in a manner compatible with the complex structure. This compatibility aids in examining how complex functions behave within these manifolds and helps draw connections between algebraic geometry and differential geometry.
• Evaluate the implications of Kähler metrics on compact manifolds in relation to classical problems in differential geometry.
  • The existence of Kähler metrics on compact manifolds relates directly to several classical problems, such as those posed by the Calabi conjecture. Yau's proof demonstrated that every compact Kähler manifold admits a Kähler metric with constant scalar curvature. This result not only deepens our understanding of Kähler geometry but also serves as a bridge connecting differential geometry with algebraic topology, offering solutions to longstanding questions about the relationships between curvature properties and topological features.

Related terms

Hermitian Metric: A Hermitian metric is a type of inner product defined on a complex vector space that is conjugate symmetric and positive definite, playing a crucial role in defining Kähler manifolds.

Symplectic Structure: A symplectic structure is a non-degenerate, closed 2-form that provides a geometric framework for the study of Hamiltonian mechanics and forms an integral part of the definition of Kähler manifolds.

Complex Structure: A complex structure on a manifold defines how to differentiate complex functions and provides the underlying framework for understanding the manifold's geometric properties.