Inmathematics,complex geometry is the study ofgeometric structures and constructions arising out of, or described by, thecomplex numbers. In particular, complex geometry is concerned with the study ofspaces such ascomplex manifolds andcomplex algebraic varieties, functions ofseveral complex variables, and holomorphic constructions such asholomorphic vector bundles andcoherent sheaves. Application of transcendental methods toalgebraic geometry falls in this category, together with more geometric aspects ofcomplex analysis.

Complex geometry sits at the intersection of algebraic geometry,differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through theminimal model program and the construction ofmoduli spaces sets the field apart from differential geometry, where the classification of possiblesmooth manifolds is a significantly harder problem. Additionally, the extra structure of complex geometry allows, especially in thecompact setting, forglobal analytic results to be proven with great success, includingShing-Tung Yau's proof of theCalabi conjecture, theHitchin–Kobayashi correspondence, thenonabelian Hodge correspondence, and existence results forKähler–Einstein metrics andconstant scalar curvature Kähler metrics. These results often feed back into complex algebraic geometry, and for example recently the classification of Fano manifolds usingK-stability has benefited tremendously both from techniques in analysis and in purebirational geometry.

Complex geometry has significant applications to theoretical physics, where it is essential in understandingconformal field theory,string theory, andmirror symmetry. It is often a source of examples in other areas of mathematics, including inrepresentation theory wheregeneralized flag varieties may be studied using complex geometry leading to theBorel–Weil–Bott theorem, or insymplectic geometry, whereKähler manifolds are symplectic, inRiemannian geometry where complex manifolds provide examples of exotic metric structures such asCalabi–Yau manifolds andhyperkähler manifolds, and ingauge theory, whereholomorphic vector bundles often admit solutions to importantdifferential equations arising out of physics such as theYang–Mills equations. Complex geometry additionally is impactful in pure algebraic geometry, where analytic results in the complex setting such asHodge theory of Kähler manifolds inspire understanding ofHodge structures forvarieties andschemes as well asp-adic Hodge theory,deformation theory for complex manifolds inspires understanding of the deformation theory of schemes, and results about thecohomology of complex manifolds inspired the formulation of theWeil conjectures andGrothendieck'sstandard conjectures. On the other hand, results and techniques from many of these fields often feed back into complex geometry, and for example developments in the mathematics of string theory and mirror symmetry have revealed much about the nature ofCalabi–Yau manifolds, which string theorists predict should have the structure of Lagrangian fibrations through theSYZ conjecture, and the development ofGromov–Witten theory ofsymplectic manifolds has led to advances inenumerative geometry of complex varieties.

TheHodge conjecture, one of themillennium prize problems, is a problem in complex geometry.^[1]

Broadly, complex geometry is concerned withspaces andgeometric objects which are modelled, in some sense, on thecomplex plane. Features of the complex plane andcomplex analysis of a single variable, such as an intrinsic notion oforientability (that is, being able to consistently rotate 90 degrees counterclockwise at every point in the complex plane), and the rigidity ofholomorphic functions (that is, the existence of a single complex derivative implies complex differentiability to all orders) are seen to manifest in all forms of the study of complex geometry. As an example, every complex manifold is canonically orientable, and a form ofLiouville's theorem holds oncompact complex manifolds orprojective complex algebraic varieties.

Complex geometry is different in flavour to what might be calledreal geometry, the study of spaces based around the geometric and analytical properties of thereal number line. For example, whereassmooth manifolds admitpartitions of unity, collections of smooth functions which can be identically equal to one on someopen set, and identically zero elsewhere, complex manifolds admit no such collections of holomorphic functions. Indeed, this is the manifestation of theidentity theorem, a typical result in complex analysis of a single variable. In some sense, the novelty of complex geometry may be traced back to this fundamental observation.

It is true that every complex manifold is in particular a real smooth manifold. This is because the complex plane${\displaystyle \mathbb{C}}$ is, after forgetting its complex structure, isomorphic to the real plane${\displaystyle {\mathbb{R}}^2}$. However, complex geometry is not typically seen as a particular sub-field ofdifferential geometry, the study of smooth manifolds. In particular,Serre'sGAGA theorem says that everyprojectiveanalytic variety is actually analgebraic variety, and the study of holomorphic data on an analytic variety is equivalent to the study of algebraic data.

This equivalence indicates that complex geometry is in some sense closer toalgebraic geometry than todifferential geometry. Another example of this which links back to the nature of the complex plane is that, in complex analysis of a single variable, singularities ofmeromorphic functions are readily describable. In contrast, the possible singular behaviour of a continuous real-valued function is much more difficult to characterise. As a result of this, one can readily studysingular spaces in complex geometry, such as singular complexanalytic varieties or singular complex algebraic varieties, whereas in differential geometry the study of singular spaces is often avoided.

In practice, complex geometry sits in the intersection of differential geometry, algebraic geometry, andanalysis inseveral complex variables, and a complex geometer uses tools from all three fields to study complex spaces. Typical directions of interest in complex geometry involveclassification of complex spaces, the study of holomorphic objects attached to them (such asholomorphic vector bundles andcoherent sheaves), and the intimate relationships between complex geometric objects and other areas of mathematics and physics.

Complex geometry is concerned with the study ofcomplex manifolds, andcomplex algebraic andcomplex analytic varieties. In this section, these types of spaces are defined and the relationships between them presented.

Acomplex manifold is atopological space${\displaystyle X}$ such that:

• ${\displaystyle X}$ isHausdorff andsecond countable.
• ${\displaystyle X}$ is locallyhomeomorphic to an open subset of${\displaystyle {\mathbb{C}}^n}$ for some${\displaystyle n}$. That is, for every point${\displaystyle p\in X}$, there is anopen neighbourhood${\displaystyle U}$ of${\displaystyle p}$ and a homeomorphism${\displaystyle \phi :U\to V}$ to an open subset${\displaystyle V\subseteq {\mathbb{C}}^n}$. Such open sets are calledcharts.
• If${\displaystyle (U_1,\phi )}$ and${\displaystyle (U_2,\psi )}$ are any two overlapping charts which map onto open sets${\displaystyle V_1,V_2}$ of${\displaystyle {\mathbb{C}}^n}$ respectively, then thetransition function${\displaystyle \psi \circ {\phi }^{-1}:\phi (U_1\cap U_2)\to \psi (U_1\cap U_2)}$ is abiholomorphism.

Notice that since every biholomorphism is adiffeomorphism, and${\displaystyle {\mathbb{C}}^n}$ is isomorphism as areal vector space to${\displaystyle {\mathbb{R}}^{2n}}$, every complex manifold of dimension${\displaystyle n}$ is in particular a smooth manifold of dimension${\displaystyle 2n}$, which is always an even number.

In contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. Anaffine complex analytic variety is a subset${\displaystyle X\subseteq {\mathbb{C}}^n}$ such that about each point${\displaystyle p\in X}$, there is an open neighbourhood${\displaystyle U}$ of${\displaystyle p}$ and a collection of finitely many holomorphic functions${\displaystyle f_1,\dots ,f_k:U\to \mathbb{C}}$ such that${\displaystyle X\cap U=\{z\in U\mid f_1(z)=\cdots =f_k(z)=0\}=Z(f_1,\dots ,f_k)}$. By convention we also require the set${\displaystyle X}$ to beirreducible. A point${\displaystyle p\in X}$ issingular if theJacobian matrix of the vector of holomorphic functions${\displaystyle (f_1,\dots ,f_k)}$ does not have full rank at${\displaystyle p}$, andnon-singular otherwise. Aprojective complex analytic variety is a subset${\displaystyle X\subseteq {\mathbb{C}\mathbb{P}}^n}$ ofcomplex projective space that is, in the same way, locally given by the zeroes of a finite collection of holomorphic functions on open subsets of${\displaystyle {\mathbb{C}\mathbb{P}}^n}$.

One may similarly define anaffine complex algebraic variety to be a subset${\displaystyle X\subseteq {\mathbb{C}}^n}$ which is locally given as the zero set of finitely many polynomials in${\displaystyle n}$ complex variables. To define aprojective complex algebraic variety, one requires the subset${\displaystyle X\subseteq {\mathbb{C}\mathbb{P}}^n}$ to locally be given by the zero set of finitely manyhomogeneous polynomials.

In order to define a general complex algebraic or complex analytic variety, one requires the notion of alocally ringed space. Acomplex algebraic/analytic variety is a locally ringed space${\displaystyle (X,{\mathcal{O}}_X)}$ which is locally isomorphic as a locally ringed space to an affine complex algebraic/analytic variety. In the analytic case, one typically allows${\displaystyle X}$ to have a topology that is locally equivalent to the subspace topology due to the identification with open subsets of${\displaystyle {\mathbb{C}}^n}$, whereas in the algebraic case${\displaystyle X}$ is often equipped with aZariski topology. Again we also by convention require this locally ringed space to be irreducible.

Since the definition of a singular point is local, the definition given for an affine analytic/algebraic variety applies to the points of any complex analytic or algebraic variety. The set of points of a variety${\displaystyle X}$ which are singular is called thesingular locus, denoted${\displaystyle X^{sing}}$, and the complement is thenon-singular orsmooth locus, denoted${\displaystyle X^{nonsing}}$. We say a complex variety issmooth ornon-singular if it's singular locus is empty. That is, if it is equal to its non-singular locus.

By theimplicit function theorem for holomorphic functions, every complex manifold is in particular a non-singular complex analytic variety, but is not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a projective complex algebraic variety. When a complex variety is non-singular, it is a complex manifold. More generally, the non-singular locus ofany complex variety is a complex manifold.

Complex manifolds may be studied from the perspective of differential geometry, whereby they are equipped with extra geometric structures such as aRiemannian metric orsymplectic form. In order for this extra structure to be relevant to complex geometry, one should ask for it to be compatible with the complex structure in a suitable sense. AKähler manifold is a complex manifold with a Riemannian metric and symplectic structure compatible with the complex structure. Every complex submanifold of a Kähler manifold is Kähler, and so in particular every non-singular affine or projective complex variety is Kähler, after restricting the standard Hermitian metric on${\displaystyle {\mathbb{C}}^n}$ or theFubini-Study metric on${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ respectively.

Other important examples of Kähler manifolds includeRiemann surfaces,K3 surfaces, andCalabi–Yau manifolds.

Serre's GAGA theorem asserts that projective complex analytic varieties are actually algebraic. Whilst this is not strictly true for affine varieties, there is a class of complex manifolds that act very much like affine complex algebraic varieties, calledStein manifolds. A manifold${\displaystyle X}$ is Stein if it is holomorphically convex and holomorphically separable (see the article on Stein manifolds for the technical definitions). It can be shown however that this is equivalent to${\displaystyle X}$ being a complex submanifold of${\displaystyle {\mathbb{C}}^n}$ for some${\displaystyle n}$. Another way in which Stein manifolds are similar to affine complex algebraic varieties is thatCartan's theorems A and B hold for Stein manifolds.

Examples of Stein manifolds include non-compact Riemann surfaces and non-singular affine complex algebraic varieties.

A special class of complex manifolds ishyper-Kähler manifolds, which areRiemannian manifolds admitting three distinct compatibleintegrable almost complex structures${\displaystyle I,J,K}$ which satisfy thequaternionic relations${\displaystyle I^2=J^2=K^2=IJK=-\mathrm{Id}}$. Thus, hyper-Kähler manifolds are Kähler manifolds in three different ways, and subsequently have a rich geometric structure.

Examples of hyper-Kähler manifolds includeALE spaces,K3 surfaces,Higgs bundle moduli spaces,quiver varieties, and many othermoduli spaces arising out ofgauge theory andrepresentation theory.

As mentioned, a particular class of Kähler manifolds is given by Calabi–Yau manifolds. These are given by Kähler manifolds with trivial canonical bundle${\displaystyle K_X={\mathrm{\Lambda }}^n{T}_{1,0}^{\ast }X}$. Typically the definition of a Calabi–Yau manifold also requires${\displaystyle X}$ to be compact. In this caseYau's proof of theCalabi conjecture implies that${\displaystyle X}$ admits a Kähler metric with vanishingRicci curvature, and this may be taken as an equivalent definition of Calabi–Yau.

Calabi–Yau manifolds have found use instring theory andmirror symmetry, where they are used to model the extra 6 dimensions of spacetime in 10-dimensional models of string theory. Examples of Calabi–Yau manifolds are given byelliptic curves, K3 surfaces, and complexAbelian varieties.

A complexFano variety is a complex algebraic variety withample anti-canonical line bundle (that is,${\displaystyle K_X^{\ast }}$ is ample). Fano varieties are of considerable interest in complex algebraic geometry, and in particularbirational geometry, where they often arise in theminimal model program. Fundamental examples of Fano varieties are given by projective space${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ where${\displaystyle K=\mathcal{O}(-n-1)}$, and smooth hypersurfaces of${\displaystyle {\mathbb{C}\mathbb{P}}^n}$ of degree less than${\displaystyle n+1}$.

Toric varieties are complex algebraic varieties of dimension${\displaystyle n}$ containing an opendense subset biholomorphic to${\displaystyle ({\mathbb{C}}^{\ast }{)}^n}$, equipped with an action of${\displaystyle ({\mathbb{C}}^{\ast }{)}^n}$ which extends the action on the open dense subset. A toric variety may be described combinatorially by itstoric fan, and at least when it is non-singular, by amoment polytope. This is a polygon in${\displaystyle {\mathbb{R}}^n}$ with the property that any vertex may be put into the standard form of the vertex of the positiveorthant by the action of${\displaystyle \mathrm{GL}(n,\mathbb{Z})}$. The toric variety can be obtained as a suitable space which fibres over the polytope.

Many constructions that are performed on toric varieties admit alternate descriptions in terms of the combinatorics and geometry of the moment polytope or its associated toric fan. This makes toric varieties a particularly attractive test case for many constructions in complex geometry. Examples of toric varieties include complex projective spaces, and bundles over them.

Due to the rigidity of holomorphic functions and complex manifolds, the techniques typically used to study complex manifolds and complex varieties differ from those used in regular differential geometry, and are closer to techniques used in algebraic geometry. For example, in differential geometry, many problems are approached by taking local constructions and patching them together globally using partitions of unity. Partitions of unity do not exist in complex geometry, and so the problem of when local data may be glued into global data is more subtle. Precisely when local data may be patched together is measured bysheaf cohomology, andsheaves and theircohomology groups are major tools.

For example, famous problems in the analysis of several complex variables preceding the introduction of modern definitions are theCousin problems, asking precisely when local meromorphic data may be glued to obtain a global meromorphic function. These old problems can be simply solved after the introduction of sheaves and cohomology groups.

Special examples of sheaves used in complex geometry include holomorphicline bundles (and thedivisors associated to them),holomorphic vector bundles, andcoherent sheaves. Since sheaf cohomology measures obstructions in complex geometry, one technique that is used is to prove vanishing theorems. Examples of vanishing theorems in complex geometry include theKodaira vanishing theorem for the cohomology of line bundles on compact Kähler manifolds, andCartan's theorems A and B for the cohomology of coherent sheaves on affine complex varieties.

Complex geometry also makes use of techniques arising out of differential geometry and analysis. For example, theHirzebruch-Riemann-Roch theorem, a special case of theAtiyah-Singer index theorem, computes theholomorphic Euler characteristic of a holomorphic vector bundle in terms of characteristic classes of the underlying smooth complex vector bundle.

One major theme in complex geometry isclassification. Due to the rigid nature of complex manifolds and varieties, the problem of classifying these spaces is often tractable. Classification in complex and algebraic geometry often occurs through the study ofmoduli spaces, which themselves are complex manifolds or varieties whose points classify other geometric objects arising in complex geometry.

The termmoduli was coined byBernhard Riemann during his original work on Riemann surfaces. The classification theory is most well known for compact Riemann surfaces. By theclassification of closed oriented surfaces, compact Riemann surfaces come in a countable number of discrete types, measured by theirgenus${\displaystyle g}$, which is a non-negative integer counting the number of holes in the given compact Riemann surface.

The classification essentially follows from theuniformization theorem, and is as follows:^[2][3][4]

• g = 0:${\displaystyle {\mathbb{C}\mathbb{P}}^1}$
• g = 1: There is a one-dimensional complex manifold classifying possible compact Riemann surfaces of genus 1, so-calledelliptic curves, themodular curve. By theuniformization theorem any elliptic curve may be written as a quotient${\displaystyle \mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})}$ where${\displaystyle \tau }$ is a complex number with strictly positive imaginary part. The moduli space is given by the quotient of the group${\displaystyle \mathrm{PSL}(2,\mathbb{Z})}$ acting on theupper half plane byMöbius transformations.
• g > 1: For each genus greater than one, there is a moduli space${\displaystyle {\mathcal{M}}_g}$ of genus g compact Riemann surfaces, of dimension${\displaystyle {\dim }_{\mathbb{C}}{\mathcal{M}}_g=3g-3}$. Similar to the case of elliptic curves, this space may be obtained by a suitable quotient ofSiegel upper half-space by the action of the group${\displaystyle \mathrm{Sp}(2g,\mathbb{Z})}$.

Complex geometry is concerned not only with complex spaces, but other holomorphic objects attached to them. The classification of holomorphic line bundles on a complex variety${\displaystyle X}$ is given by thePicard variety${\displaystyle \mathrm{Pic}(X)}$ of${\displaystyle X}$.

The picard variety can be easily described in the case where${\displaystyle X}$ is a compact Riemann surface of genus g. Namely, in this case the Picard variety is a disjoint union of complexAbelian varieties, each of which is isomorphic to theJacobian variety of the curve, classifyingdivisors of degree zero up to linear equivalence. In differential-geometric terms, these Abelian varieties are complex tori, complex manifolds diffeomorphic to${\displaystyle (S^1{)}^{2g}}$, possibly with one of many different complex structures.

By theTorelli theorem, a compact Riemann surface is determined by its Jacobian variety, and this demonstrates one reason why the study of structures on complex spaces can be useful, in that it can allow one to solve classify the spaces themselves.

• Bivector (complex)
• Calabi–Yau manifold
• Cartan's theorems A and B
• Complex analytic space
• Complex Lie group
• Complex polytope
• Complex projective space
• Cousin problems
• Deformation Theory#Deformations of complex manifolds
• Enriques–Kodaira classification
• GAGA
• Hartogs' extension theorem
• Hermitian symmetric space
• Hodge decomposition
• Hopf manifold
• Imaginary line (mathematics)
• Kobayashi metric
• Kobayashi–Hitchin correspondence
• Kähler manifold
• ${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-lemma
• Lelong number
• List of complex and algebraic surfaces
• Mirror symmetry
• Multiplier ideal
• Projective variety
• Pseudoconvexity
• Several complex variables
• Stein manifold

• Huybrechts, Daniel (2005).Complex Geometry: An Introduction. Springer.ISBN 3-540-21290-6.
• Griffiths, Phillip;Harris, Joseph (1994),Principles of algebraic geometry, Wiley Classics Library, New York:John Wiley & Sons,ISBN 978-0-471-05059-9,MR 1288523
• Hörmander, Lars (1990) [1966],An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo:North-Holland,ISBN 0-444-88446-7,MR 1045639,Zbl 0685.32001
• S. Kobayashi, K. Nomizu.Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.
• E. H. Neville (1922)Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions,Cambridge University Press.

• Complex manifolds
• Several complex variables
• Algebraic geometry
• Complex geometry

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