# Category: Complex manifolds

Fujita conjecture
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985.
Complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in
Biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose i
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The de
Complex dimension
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 neighborhoods of points (or of non-singular
Stein manifold
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and nam
Genus of a multiplicative sequence
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to sui
Complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in , such that the transition maps are holomorphic. The term complex manif
Holomorphic tangent bundle
In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorph
Bott residue formula
In mathematics, the Bott residue formula, introduced by Bott, describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold.
Siegel domain
In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel. They were introduced by Piatets
Higgs bundle
In mathematics, a Higgs bundle is a pair consisting of a holomorphic vector bundle E and a Higgs field , a holomorphic 1-form taking values in the bundle of endomorphisms of E such that . Such pairs w
Calabi–Eckmann manifold
In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, ho
Mutation (Jordan algebra)
In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the give
Hypercomplex manifold
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions define integrable almost
Iwasawa manifold
In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauc
Exponential sheaf sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of holomorphic functio
Siu's semicontinuity theorem
In complex analysis, the Siu semicontinuity theorem implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong num
Kähler quotient
In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold by a Lie group acting on by preserving the Kähler structure and with moment map (with respect to the Kähler f
Fermat quintic threefold
In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation . This t
Hitchin functional
The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. and are the original articles of the Hitchin functio
Andreotti–Grauert theorem
In mathematics, the Andreotti–Grauert theorem, introduced by Andreotti and Grauert, gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensio
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.
Complex affine space
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notio
Branched covering
In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
Calabi conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Cala
Bergman metric
In differential geometry, the Bergman 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 of whic
Differential of the first kind
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic variet
Positive form
In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).
Trivial cylinder
In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds. In Floer homology and its variants, chain complexes or differential graded
Ddbar lemma
In complex geometry, the lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The -lemma is a result of Hodge theory and the Kähler
Cox–Zucker machine
The Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines whether a given set of sections provides a basis (up to torsion) for the Mordell–Weil group
Nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after and Carlos Simpson) is a correspondence between Higgs bundles and r
Kähler identities
No description available.
Circular points at infinity
In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the com
Positive current
In mathematics, more particularly in complex geometry,algebraic geometry and complex analysis, a positive currentis a positive (n-p,n-p)-form over an n-dimensional complex manifold,taking values in di
Constant scalar curvature Kähler metric
In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is
Hodge–de Rham spectral sequence
In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) is an alternative term sometimes used to describe the Frölicher spectral sequence (named afte
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described i
Quintic threefold
In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space. Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-
Hopf manifold
In complex geometry, a Hopf manifold is obtainedas a quotient of the complex vector space(with zero deleted) by a free action of the group ofintegers, with the generator of acting by holomorphic contr
Holomorphic vector bundle
In mathematics, a 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 : E → X is holomorphic. Fund
Period domain
In mathematics, a period domain is a parameter space for a polarized Hodge structure. They can often be represented as the quotient of a Lie group by a compact subgroup.
Andreotti–Vesentini theorem
In mathematics, the Andreotti–Vesentini separation theorem, introduced by Aldo Andreotti and Edoardo Vesentini states that certain cohomology groups of coherent sheaves are separated.
Lelong number
In mathematics, the Lelong number is an invariant of a point of a complex analytic variety that in some sense measures the local density at that point. It was introduced by Lelong. More generally a cl
Complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the stud
Hermitian manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a
Complexification
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending t
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural
Remmert–Stein theorem
In complex analysis, a field in mathematics, the Remmert–Stein theorem, introduced by Reinhold Remmert and Karl Stein, gives conditions for the closure of an analytic set to be analytic. The theorem s
Bismut connection
In mathematics, the Bismut connection is the unique connection on a complex Hermitian manifold that satisfies the following conditions, 1. * It preserves the metric 2. * It preserves the complex str
Gibbons–Hawking space
In mathematical physics, a Gibbons–Hawking space, named after Gary Gibbons and Stephen Hawking, is essentially a hyperkähler manifold with an extra U(1) symmetry. (In general, Gibbons–Hawking metrics
Holomorphic Lefschetz fixed-point formula
In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bu
Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical phys
Frölicher spectral sequence
In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory
Holomorphic curve
In mathematics, 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 theory addresses the question
Poincaré–Lelong equation
In mathematics, the Poincaré–Lelong equation, studied by Lelong, is the partial differential equation on a Kähler manifold, where ρ is a positive (1,1)-form.
Hermitian connection
In mathematics, a Hermitian connection is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric on , meaning that for all smooth vector fields
Polar homology
In complex geometry, a polar homology is a group which captures holomorphic invariants of a complex manifold in a similar way to usual homology of a manifold in differential topology. Polar homology w
Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the
Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questio
Stable map
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given sympl
Automorphic function
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifol
Andreotti–Frankel theorem
In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel, states that if is a smooth, complex affine variety of complex dimension or, more generally, if is any
Twistor space
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation . It was described in the 1960s by Roger Penrose and
Hasse–Witt matrix
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the pr
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential i
Dolbeault cohomology
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a
CR manifold
In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more ge
Skoda–El Mir theorem
The Skoda–El Mir theorem is a theorem of complex geometry, stated as follows: Theorem (Skoda, El Mir, Sibony). Let X be a complex manifold, and E a closed complete pluripolar set in X. Consider a clos
Fujiki class C
In algebraic geometry, a complex manifold is called Fujiki class C if it is to a compact Kähler manifold. This notion was defined by .
Relative canonical model
In the mathematical field of algebraic geometry, the relative canonical model of a singular variety of a mathematical object where is a particular canonical variety that maps to , which simplifies the
Complex torus
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 product of some number N circles). Here N
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the ori
Kobayashi metric
In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyp