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...

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

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...

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

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

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...

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

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

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

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...

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

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

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...

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

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

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...

Topological manifold Almost complex manifold Almost symplectic manifold Calibrated manifold Complex manifold Contact manifold CR manifold Finsler manifold Hermitian...

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

complex subvarieties of X. A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space...

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

can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X {\displaystyle X} is a complex manifold and if the local trivializations...

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

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...

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

Calabi–Yau manifold Complex manifold, a manifold over the complex numbers Differentiable manifold Einstein manifold Flag manifold Flat manifold G2 manifold Hermitian...

theorem Generalized complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler manifold Symplectic topology...

A complex structure may refer to: Almost complex manifold Complex manifold Linear complex structure Generalized complex structure Complex structure deformation...