Editing Complex geometry (section)

{{Use American English|date = March 2019}}
{{Short description|Study of complex manifolds and several complex variables}}

{{General geometry}}
In [[mathematics]], '''complex geometry''' is the study of [[geometry|geometric]] structures and constructions arising out of, or described by, the [[complex number]]s. In particular, complex geometry is concerned with the study of [[space (mathematics)|spaces]] such as [[complex manifold]]s and [[Complex algebraic variety|complex algebraic varieties]], functions of [[several complex variables]], and holomorphic constructions such as [[holomorphic vector bundles]] and [[coherent sheaf|coherent sheaves]]. Application of transcendental methods to [[algebraic geometry]] falls in this category, together with more geometric aspects of [[complex 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 the [[minimal model program]] and the construction of [[moduli space]]s sets the field apart from differential geometry, where the classification of possible [[smooth manifold]]s is a significantly harder problem. Additionally, the extra structure of complex geometry allows, especially in the [[compact space|compact]] setting, for [[global analysis|global analytic]] results to be proven with great success, including [[Shing-Tung Yau]]'s proof of the [[Calabi conjecture]], the [[Hitchin–Kobayashi correspondence]], the [[nonabelian Hodge correspondence]], and existence results for [[Kähler–Einstein metric]]s and [[constant scalar curvature Kähler metric]]s. These results often feed back into complex algebraic geometry, and for example recently the classification of Fano manifolds using [[K-stability of Fano varieties|K-stability]] has benefited tremendously both from techniques in analysis and in pure [[birational geometry]].

Complex geometry has significant applications to theoretical physics, where it is essential in understanding [[conformal field theory]], [[string theory]], and [[mirror symmetry (string theory)|mirror symmetry]]. It is often a source of examples in other areas of mathematics, including in [[representation theory]] where [[generalized flag varieties]] may be studied using complex geometry leading to the [[Borel–Weil–Bott theorem]], or in [[symplectic geometry]], where [[Kähler manifold]]s are symplectic, in [[Riemannian geometry]] where complex manifolds provide examples of exotic metric structures such as [[Calabi–Yau manifold]]s and [[hyperkähler manifold]]s, and in [[gauge theory (mathematics)|gauge theory]], where [[holomorphic vector bundles]] often admit solutions to important [[differential equation]]s arising out of physics such as the [[Yang–Mills equations]]. Complex geometry additionally is impactful in pure algebraic geometry, where analytic results in the complex setting such as [[Hodge theory]] of Kähler manifolds inspire understanding of [[Hodge structure]]s for [[algebraic variety|varieties]] and [[scheme (algebraic geometry)|schemes]] as well as [[p-adic Hodge theory]], [[deformation theory]] for complex manifolds inspires understanding of the deformation theory of schemes, and results about the [[cohomology]] of complex manifolds inspired the formulation of the [[Weil conjectures]] and [[Grothendieck]]'s [[standard 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 of [[Calabi–Yau manifold]]s, which string theorists predict should have the structure of Lagrangian fibrations through the [[SYZ conjecture]], and the development of [[Gromov–Witten theory]] of [[symplectic manifold]]s has led to advances in [[enumerative geometry]] of complex varieties.

The [[Hodge conjecture]], one of the [[millennium prize problems]], is a problem in complex geometry.<ref>Voisin, C., 2016. The Hodge conjecture. In Open problems in mathematics (pp. 521-543). Springer, Cham.</ref>