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Manifold
differential geometry and complex geometry, a complex manifold or a complex analytic manifold is a manifold with a complex structure, that is an atlas
Complex_manifold
Smooth manifold
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
Almost_complex_manifold
Manifold with Riemannian, complex and symplectic structure
especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and
Kähler_manifold
Topological space that locally resembles Euclidean space
manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,
Manifold
Riemannian manifold with SU(n) holonomy
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
Calabi–Yau_manifold
Type of Riemannian manifold
geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex structures I , J , K
Hyperkähler_manifold
Type of mathematical functions
phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties and
Function of several complex variables
Function_of_several_complex_variables
Theorem about complex manifolds
for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a compact complex manifold
Holomorphic Lefschetz fixed-point formula
Holomorphic_Lefschetz_fixed-point_formula
Branch of mathematics
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
Differential_geometry
Model of the extended complex plane plus a point at infinity
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
Riemann_sphere
Study of complex manifolds and several complex variables
concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions
Complex_geometry
Term in mathematics
theory of several complex variables and complex manifolds, a Stein manifold is a closed, complex submanifold of the vector space of n complex dimensions. More
Stein_manifold
Mathematical manifold theory
applications in two settings—Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed
Hodge_theory
Concept in differential geometry
geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly
Hermitian_manifold
Special tangential structure
for such situations. Orientable manifolds with a spinc structure are called spinc manifolds. C stands for the complex numbers, which are denoted C {\displaystyle
Spinc_structure
Differential form on a manifold which is permitted to have complex coefficients
complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms
Complex_differential_form
One-dimensional complex manifold
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied
Riemann_surface
Differentiable manifold
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
CR_manifold
an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a nilmanifold
Iwasawa_manifold
Manifold upon which it is possible to perform calculus
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow
Differentiable_manifold
Unsolved problem in geometry
complex subvarieties of X. A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space
Hodge_conjecture
Complex three dimensional gauge theory
Chern–Simons theory is a gauge theory on a three-dimensional complex manifold. It is a complex analogue of Chern–Simons theory, named after Shiing-Shen Chern
Six-dimensional holomorphic Chern–Simons theory
Six-dimensional_holomorphic_Chern–Simons_theory
the complex hyperbolic space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex
Complex_hyperbolic_space
Tool to track locally defined data attached to the open sets of a topological space
singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and
Sheaf_(mathematics)
Type of geometry in mathematics
Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental
Ricci-flat_manifold
Property of a differential manifold that includes complex structures
geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic
Generalized_complex_structure
2D surface which extends indefinitely
more structure, one may view the plane as a 1-dimensional complex manifold, called the complex line. Many fundamental tasks in mathematics, geometry, trigonometry
Plane_(mathematics)
Type of smooth complex surface of kodaira dimension 0
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
K3_surface
Study of the topology of a complex manifold
of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces
Picard–Lefschetz_theory
Type of metric in Riemannian geometry
Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein
Kähler–Einstein_metric
Topics referred to by the same term
Calabi–Yau manifold Complex manifold, a manifold over the complex numbers Differentiable manifold Einstein manifold Flag manifold Flat manifold G2 manifold Hermitian
Manifold_(disambiguation)
Kind of complex manifold
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian
Complex_torus
In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of
Holomorphic_tangent_bundle
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
Complex_dimension
projective space, RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold Grassmann manifold Stiefel manifold Lie groups provide
List_of_manifolds
Riemannian metrics, complex manifolds
about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957). It was proved by Shing-Tung
Calabi_conjecture
Mathematics concept
complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex
Linear_complex_structure
Complex-differentiable (mathematical) function
a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate
Holomorphic_function
Complex vector bundle on a complex manifold
holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π :
Holomorphic_vector_bundle
Type of manifold in differential geometry
In differential geometry, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate differential 2-form,
Symplectic_manifold
Type of mathematical space
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
Generalized_flag_variety
Stability conditions for triangulated cateogires
conditions on the triangulated category carries the structure of a complex manifold, thus furnishing an invariant of the category that is topological in
Bridgeland stability condition
Bridgeland_stability_condition
J-holomorphic curve) is a smooth map, from a Riemann surface into an almost complex manifold, that satisfies the Cauchy–Riemann equations. Introduced in 1985 by
Pseudoholomorphic_curve
Index of articles associated with the same name
space over the complex numbers, with no distinguishable point of origin Complex analytic space, a generalization of a complex manifold, with singularities
Complex_space
Mathematical space
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four,
4-manifold
Algebraic variety
For an algebraic manifold, the ground field will be the real numbers or complex numbers; in the case of the real numbers, the manifold of real points is
Algebraic_manifold
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
Complex_vector_bundle
Configuration space
configuration space of n distinct labeled points in a compact complex manifold is a compact complex manifold that contains the configuration space as an open dense
Fulton–MacPherson compactification
Fulton–MacPherson_compactification
Concept in differential geometry
In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the
Spin_structure
Compact complex manifold in algebraic geometry
In mathematics, a Moishezon manifold M is a compact complex manifold such that the field of meromorphic functions on each component M has transcendence
Moishezon_manifold
Chinese-American mathematician (born 1949)
complex manifold has positive first Chern class. A proposal of Calabi's suggested that Kähler–Einstein metrics exist on any compact Kähler manifolds with
Shing-Tung_Yau
Mathematical theorem
vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to Grauert and Riemenschneider (1970). The Grauert–Riemenschneider
Grauert–Riemenschneider vanishing theorem
Grauert–Riemenschneider_vanishing_theorem
in the field of complex geometry, a holomorphic curve in a complex manifold M is a non-constant holomorphic map f from the complex plane to M. Nevanlinna
Holomorphic_curve
cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge
Hodge–de Rham spectral sequence
Hodge–de_Rham_spectral_sequence
Mathematical concept
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
Complex_projective_space
Branch of mathematics studying functions of a complex variable
natural way to maps between Riemannian or semi-Riemannian manifolds. One of the central tools in complex analysis is the line integral. The line integral around
Complex_analysis
Pseudometric of complex manifolds
mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by
Kobayashi_metric
Topics referred to by the same term
A complex structure may refer to: Almost complex manifold Complex manifold Linear complex structure Generalized complex structure Complex structure deformation
Complex_structure
Vector bundles theorem
Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Algebraic variety in a projective space
that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory
Projective_variety
Counterexample in algebraic geometry
geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka (1960, 1962). Hironaka's
Hironaka's_example
Generalization of a complex manifold that allows the use of singularities
differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold that allows the presence
Complex_analytic_variety
Two-dimensional manifold
In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere
Surface_(topology)
Parametrizes complex structures on a surface
of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features
Teichmüller_space
theorem Generalized complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler manifold Symplectic topology
List of differential geometry topics
List_of_differential_geometry_topics
theorem (complex analysis) Bôcher's theorem (complex analysis) Borel–Carathéodory theorem (complex analysis) Branching theorem (complex manifold) Carathéodory's
List_of_theorems
geometry, a complex manifold is called Fujiki class C {\displaystyle {\mathcal {C}}} if it is bimeromorphic to a compact Kähler manifold. This notion
Fujiki_class_C
Analyzes the topology of a manifold by studying differentiable functions on that manifold
complex manifolds is Picard–Lefschetz theory. To illustrate, consider a mountainous landscape surface M {\displaystyle M} (more generally, a manifold)
Morse_theory
Mathematical function on a space that is invariant under the action of some group
is a complex manifold and the group is a discrete group. In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic
Automorphic_function
Mathematical term
Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H p , q ( M ,
Dolbeault_cohomology
Branch of mathematics
from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one
Topology
Mathematical concept
the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations
Infinity
Theorem in complex geometry
\omega )} is a compact Kähler manifold and α ∈ Ω p , q ( X ) {\displaystyle \alpha \in \Omega ^{p,q}(X)} is a complex differential form of bidegree (p
Ddbar_lemma
bundle on a complex manifold, ∂ ¯ : L ↦ L ⊗ Λ 0 , 1 ( M ) {\displaystyle {\bar {\partial }}:\;L\mapsto L\otimes \Lambda ^{0,1}(M)} its complex structure
Positive_form
Algebra used in 2D conformal field theories and string theory
sophisticated examples such as affine W-algebras and the chiral de Rham complex on a complex manifold arise in geometric representation theory and mathematical physics
Vertex_operator_algebra
Mathematical function that preserves angles
maps between Riemannian or semi-Riemannian manifolds. If U {\displaystyle U} is an open subset of the complex plane C {\displaystyle \mathbb {C} } , then
Conformal_map
Manifold equipped with a quaternionic structure
define integrable almost complex structures. If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic
Hypercomplex_manifold
Algebro-geometric stability condition
K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced
K-stability
26-dimensional string theory
the given topological surface, and is in fact a finite-dimensional complex manifold. The fundamental problem of perturbative bosonic strings therefore
Bosonic_string_theory
in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking
Positive_current
Smooth manifold with an inner product on each tangent space
In differential geometry, a Riemannian manifold (or Riemann space) is a geometric space on which many geometric notions such as distance, angles, length
Riemannian_manifold
Theorem about complex manifolds
holomorphic vector field of a compact complex manifold. If v is a holomorphic vector field on a compact complex manifold M, then ∑ v ( p ) = 0 P ( A p ) det
Bott_residue_formula
Type of geometric transformation
category, by endowing the symplectic manifold with a compatible almost complex structure and proceeding with a complex blow-up. This makes sense on a purely
Blowing_up
Japanese mathematician (1915–1997)
known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers
Kunihiko_Kodaira
Mathematical result in differential geometry
(1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions)
Atiyah–Singer_index_theorem
Seven-dimensional Riemannian manifold
In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G
G2_manifold
Type of function in mathematics
locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point
Analytic_function
scalar curvature Kähler metric (cscK metric) is a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is a Kähler–Einstein
Constant scalar curvature Kähler metric
Constant_scalar_curvature_Kähler_metric
geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected)
Affine_manifold
metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both
Bergman_metric
In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is
Calabi–Eckmann_manifold
Characteristic classes of vector bundles
space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space
Chern_class
Vector bundle of cotangent spaces at every point in a manifold
generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties
Cotangent_bundle
Mathematical symbol used for partial derivatives and other concepts
boundary operator in a chain complex, and the conjugate of the Dolbeault operator on smooth differential forms over a complex manifold. It should be distinguished
Partial_differential
Affine space over the complex numbers
implies that the structure sheaf of a complex-analytic space (e.g., a complex manifold) is coherent. Every complex affine space is a domain of holomorphy
Complex_affine_space
Indian mathematician (1937–2015)
University of Chicago who worked on real and complex manifolds and who solved the Levi problem for complex manifolds. He attended Loyola College in Madras,
Raghavan_Narasimhan
American mathematician (born 1938)
known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He is a major developer in particular
Phillip_Griffiths
Mathematical structure in differential geometry
Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in
Poisson_manifold
Generalizes the Kodaira vanishing theorem for ample vector bundle
states the following Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here H p , q (
Le_Potier's_vanishing_theorem
Italian-born American mathematician (1923–2023)
1950. His doctoral dissertation, titled "Isometric complex analytic imbedding of Kähler manifolds", was done under the supervision of Salomon Bochner
Eugenio_Calabi
COMPLEX MANIFOLD
COMPLEX MANIFOLD
Boy/Male
Tamil
Complete
Boy/Male
Tamil
Complete
Girl/Female
Tamil
Complete
Girl/Female
Hindu, Indian
Complex
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
Shesha Harani | ஷேஷ ஹரணீÂ
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Girl/Female
Tamil
Complete
Boy/Male
Indian
Complete
Boy/Male
Indian
Complete
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Complete
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Girl/Female
Bengali, Indian
Good Complex
Girl/Female
Muslim
Complex, Zigzag, Curling
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Boy/Male
Tamil
Complete
COMPLEX MANIFOLD
COMPLEX MANIFOLD
Male
Norse
Old Norse name derived from the word óðr, ÓÃINN means "poetry, song" and "eager, frenzied, raging." In mythology, this is the name of the chief god of the Aesir. Equated with Anglo-Saxon Woden.
Boy/Male
French, German, Teutonic
Noble Eagle; Noble; Courageous; Noble Strength; Brave
Boy/Male
British, English
Ring
Boy/Male
Biblical American Hebrew
Laughter.
Girl/Female
English
Girl/Female
Muslim
Dear
Boy/Male
Muslim
Following, Next
Biblical
Same as Kenah
Boy/Male
Muslim
Attached, Intent
Boy/Male
American, Anglo, Australian, British, Christian, English, Hebrew, Indian, Jamaican
Son of the Red Earth; In the Bible God Created Adam-the First Man-out of the Red Earth and Breathed Life into Him; Child of Adam; Son of Adam
COMPLEX MANIFOLD
COMPLEX MANIFOLD
COMPLEX MANIFOLD
COMPLEX MANIFOLD
COMPLEX MANIFOLD
a.
Intricate; entangled; complicated; complex.
imp. & p. p.
of Comply
a.
See Couple-close.
a.
That which joins or links two things together; a bond or tie; a coupler.
n.
A complex; an aggregate of parts; a complication.
a.
Repeatedly compound; made up of complex constituents.
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
imp. & p. p.
of Couple
imp. & p. p.
of Compile
a.
Complex, complicated.
adv.
In a complex manner; not simply.
pl.
of Couple-close
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.
n.
One who compiles; esp., one who makes books by compilation.
a.
Finished; ended; concluded; completed; as, the edifice is complete.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.
a.
Not complex; uncompounded; simple.
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.