and complex geometry, a complex manifold is a manifold with a complex structure, that is an atlas of charts to the open unit disc in the complex coordinate...

an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...

especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and...

Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number...

concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions...

phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties ( C...

geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex structures I , J , K...

manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...

simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective...

theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They...

geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly...

hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. Formally, a CR manifold is a differentiable manifold M together with...

differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate...

a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large...

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied...

In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G...

complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms...

geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic...

In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local...

hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic...

applications in two settings—Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed...

F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are...

Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental...

singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and...

of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures;...

more structure, one may view the plane as a 1-dimensional complex manifold, called the complex line. Many fundamental tasks in mathematics, geometry, trigonometry...

that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory...

same if they differ by a phase. Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as Z = ( Z 1 , Z 2 ...

manifold M is a compact complex manifold such that the field of meromorphic functions on each component M has transcendence degree equal the complex dimension...

the name "K3 surface" In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle...