Complex manifolds

Complex manifolds are a type of manifold that is equipped with a complex structure, allowing for the use of complex numbers in their local charts. More formally, a complex manifold is a differentiable manifold that has an atlas of charts (local coordinate systems) where the transition functions between charts are holomorphic (i.e., complex differentiable).

The Bergman metric is a Riemannian metric used in the context of several complex variables, particularly on domains in complex manifolds. It is defined on a domain \( \Omega \subset \mathbb{C}^n \) and serves as a way to measure distances in a way that reflects the complex structure of the domain.

The Bismut connection, named after Jean-Michel Bismut, is a concept from differential geometry and the theory of connections on vector bundles. It is particularly significant in the context of studying geometric structures and their associated differential operators, especially in relation to heat kernels and the analysis of elliptic operators.

The Bott residue formula is a result in the field of differential topology and algebraic topology that relates the topology of smooth manifolds with specific types of mappings. It is particularly associated with the study of smooth maps between manifolds and the properties of their critical points. The formula is named after Raoul Bott, and it generalizes the classical concept of residues from complex analysis into the realm of differential forms and manifolds.

A CR manifold (Cauchy-Riemann manifold) is a differential manifold equipped with a special kind of geometric structure that generalizes the concept of complex structures to a more general setting. Specifically, a CR manifold has a certain kind of foliation by complex subspaces that leads to the definition of CR structures. To define a CR manifold, consider the following components: 1. **Smooth Manifold**: Start with a smooth manifold \( M \) of real dimension \( 2n \).

The Calabi conjecture is a significant result in differential geometry, particularly in the study of Kähler manifolds. Formulated by Eugenio Calabi in the 1950s, the conjecture addresses the existence of Kähler metrics with special properties on certain compact complex manifolds. Specifically, the conjecture states that for a given compact Kähler manifold with a vanishing first Chern class, there exists a unique Kähler metric in each Kähler class that is Ricci-flat.

A Calabi–Eckmann manifold is a type of complex manifold that is constructed as a special case of a more general theory involving complex and symplectic geometry. Specifically, Calabi–Eckmann manifolds are a class of compact Kähler manifolds that serve as examples of non-Kähler, simply-connected manifolds with rich geometric structures.

Complex affine space is a mathematical structure used primarily in algebraic geometry and complex analysis. It can be understood as a generalization of affine space, but specifically over the field of complex numbers. ### Definition An \( n \)-dimensional complex affine space, denoted as \( \mathbb{C}^n \), consists of all ordered \( n \)-tuples of complex numbers.

A complex differential form is a mathematical object used in the field of complex analysis and differential geometry. It extends the notion of differential forms to the context of complex manifolds, allowing for the study of functions, integrals, and geometry in complex spaces. ### Key Concepts: 1. **Differential Forms**: In real analysis, differential forms are antisymmetric tensor fields that can be integrated over manifolds.

A **complex torus** is a type of mathematical structure that arises in the field of complex geometry and algebraic geometry. Specifically, a complex torus is defined as a quotient of a complex vector space by a discrete subgroup of complex numbers that forms a lattice.

A **constant scalar curvature Kähler (cscK) metric** is a special type of Kähler metric that arises in the field of differential geometry, particularly in the study of Kähler manifolds. To understand this concept, it's helpful to break down the components involved: 1. **Kähler Manifold**: A Kähler manifold is a complex manifold \( (M, J) \) equipped with a Kähler metric \( g \).

The \( \bar{\partial} \)-lemma, often referred to as the \( \overline{\partial} \)-lemma, is a fundamental result in complex analysis, particularly in the context of several complex variables and complex geometry. It provides conditions under which a \( \overline{\partial} \)-closed form can be expressed as the \( \overline{\partial} \) of another form.

The Hitchin functional, named after mathematician Nigel Hitchin, is an important concept in differential geometry and mathematical physics, particularly in the study of moduli spaces of Higgs bundles. In essence, the Hitchin functional is a specific type of energy functional defined on a space of Higgs bundles.

The Holomorphic Lefschetz fixed-point formula is an important result in complex geometry and algebraic geometry that relates fixed points of holomorphic maps to topological invariants of the underlying space. It is an extension of the classical Lefschetz fixed-point theorem which applies to smooth (differentiable) maps. ### Key Concepts 1.

A holomorphic vector bundle is a specific type of vector bundle in the context of complex geometry. In mathematics, a vector bundle is a topological construction that associates a vector space to each point of a base space, which can be a manifold. When we add the structure of complex numbers and holomorphic functions, we arrive at the concept of a holomorphic vector bundle. Here's a more detailed description: 1. **Base Space**: Consider a complex manifold \(X\).

A Hopf manifold is a specific type of complex manifold that can be defined through the quotient of a complex vector space by the action of a group. More specifically, Hopf manifolds are obtained from the complex projective space \(\mathbb{C}P^n\) by removing a point and then taking the quotient by a specific action of a group.

The Kähler quotient is a construction in differential geometry and algebraic geometry that allows one to form a new space from a symplectic manifold by quotienting out by a group action. Specifically, it is commonly associated with Kähler manifolds, where the underlying structure combines a symplectic structure and a Riemannian metric that is compatible with the complex structure.

The Lelong number is a concept from complex analysis, particularly in the study of plurisubharmonic functions, and is named after the mathematician Pierre Lelong. It provides a measure of the "growth" or "behavior" of a plurisubharmonic function near a point in complex space.

In the context of Jordan algebras, a **mutation** refers to a particular process or operation that alters the elements of the algebra in a structured way. Jordan algebras are a class of non-associative algebras that arise in various areas of mathematics, particularly in the study of symmetries, physics, and quantum mechanics.

Nonabelian Hodge correspondence is a mathematical framework that establishes a deep connection between certain geometric structures on a Riemann surface (or, more generally, on algebraic varieties) and particular types of representations of the fundamental group of these surfaces. This correspondence generalizes classical results in Hodge theory that relate complex geometry to the algebraic topology of varieties.

The term "period domain" can refer to different concepts depending on the context. Here are two primary interpretations: 1. **Mathematics and Complex Analysis**: In complex analysis, the period domain refers to a certain subset of the complex space associated with abelian varieties or more generally, with algebraic varieties. It often relates to the study of periods of differential forms and can involve analyzing how certain structures or functions behave under transformations defined by these periods.

"Positive current" typically refers to the direction of electric current flow in a circuit. In conventional terms, current is said to flow from the positive terminal to the negative terminal of a power source, like a battery. This definition dates back to the early studies of electricity, before the discovery of electrons and their actual movement, which flows from negative to positive.

A **quadratic differential** is a mathematical concept that arises primarily in the fields of complex analysis and differential geometry, often used to study the properties of Riemann surfaces and their associated geometric structures. In a more formal description, a quadratic differential on a Riemann surface can be seen as a section of the tensor product of the cotangent bundle with itself, specifically a differential form of type (2,0).

A Siegel domain is a type of domain used in the field of several complex variables and complex geometry. It is named after Carl Ludwig Siegel, who made significant contributions to the theory of complex multi-dimensional spaces. More formally, a Siegel domain is defined as a specific type of domain in complex Euclidean space \(\mathbb{C}^n\) that can be described as a product of a complex vector space and a strictly convex set in that space.

Siu's semicontinuity theorem is a result in the field of complex geometry, particularly concerning the behavior of the plurisubharmonic pluri-Laplacian energies of complex manifolds. One of the main contexts in which Siu's theorem is applied involves the study of the canonical metrics on complex manifolds and the stability of certain geometrical properties under deformations.

A **Stein manifold** is a concept from complex geometry which refers to a particular class of complex manifolds that generalize certain properties of complex affine varieties. Stein manifolds are considered the complex-analytic counterpart of affine algebraic varieties.

Twistor space is a mathematical construction that arises in the context of theoretical physics, particularly in the study of certain fundamental aspects of spacetime and quantum field theory. Introduced by Roger Penrose in the 1960s, twistor theory provides a framework for understanding the relationships between geometrical and physical entities in a novel way, combining aspects of geometry with concepts in physics.