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 study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and 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 spaces sets the field apart from differential geometry, where the classification of possible smooth manifolds is a significantly harder problem. Additionally, the extra structure of complex geometry allows, especially in the compact setting, for 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 metrics and constant scalar curvature Kähler metrics. These results often feed back into complex algebraic geometry, and for example recently the classification of Fano manifolds using 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. 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 manifolds are symplectic, in Riemannian geometry where complex manifolds provide examples of exotic metric structures such as Calabi–Yau manifolds and hyperkähler manifolds, and in gauge theory, where holomorphic vector bundles often admit solutions to important differential equations 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 structures for varieties and 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 manifolds, which string theorists predict should have the structure of Lagrangian fibrations through the SYZ conjecture, and the development of Gromov–Witten theory of symplectic manifolds 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.

Idea

thumb|A typical example of a complex space is the [[complex projective line. It may be viewed either as the sphere, a smooth manifold arising from differential geometry, or the Riemann sphere, an extension of the complex plane by adding a point at infinity.]]

Broadly, complex geometry is concerned with spaces and geometric objects which are modelled, in some sense, on the complex plane. Features of the complex plane and complex analysis of a single variable, such as an intrinsic notion of orientability (that is, being able to consistently rotate 90 degrees counterclockwise at every point in the complex plane), and the rigidity of holomorphic 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 of Liouville's theorem holds on compact complex manifolds or projective complex algebraic varieties.

Complex geometry is different in flavour to what might be called real geometry, the study of spaces based around the geometric and analytical properties of the real number line. For example, whereas smooth manifolds admit partitions of unity, collections of smooth functions which can be identically equal to one on some open set, and identically zero elsewhere, complex manifolds admit no such collections of holomorphic functions. Indeed, this is the manifestation of the identity 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 <math>\mathbb{C}</math> is, after forgetting its complex structure, isomorphic to the real plane <math>\mathbb{R}^2</math>. However, complex geometry is not typically seen as a particular sub-field of differential geometry, the study of smooth manifolds. In particular, Serre's GAGA theorem says that every projective analytic variety is actually an algebraic 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 to algebraic geometry than to differential 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 of meromorphic 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 study singular spaces in complex geometry, such as singular complex analytic 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, and analysis in several complex variables, and a complex geometer uses tools from all three fields to study complex spaces. Typical directions of interest in complex geometry involve classification of complex spaces, the study of holomorphic objects attached to them (such as holomorphic vector bundles and coherent sheaves), and the intimate relationships between complex geometric objects and other areas of mathematics and physics.

Definitions

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

A complex manifold of dimension <math>n</math> is a topological space <math>X</math>, endowed with an open cover <math>(U_\alpha)_{\alpha\in A}</math>, and a family of homeomorphisms <math>(\phi_\alpha\colon U_\alpha\to V_\alpha)_{\alpha\in A}</math> from <math>U_\alpha</math> to an open subset <math>V_\alpha</math> of <math>\mathbb{C}^n</math>such that:

<math>X</math> is Hausdorff and second countable.
If <math>(U_1,\varphi)</math> and <math>(U_2,\psi)</math> are any two overlapping charts which map onto open sets <math>V_1, V_2</math> of <math>\mathbb{C}^n</math> respectively, then the transition function <math>\psi \circ \varphi^{-1}:\varphi(U_1\cap U_2) \to \psi(U_1\cap U_2)</math> is a biholomorphism. 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 <math>\mathbb{C}^n</math> or the Fubini-Study metric on <math>\mathbb{CP}^n</math> respectively.

Other important examples of Kähler manifolds include Riemann surfaces, K3 surfaces, and Calabi–Yau manifolds.

Stein 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, called Stein manifolds. A manifold <math>X</math> 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 <math>X</math> admitting a proper holomorphic embedding into <math>\mathbb{C}^n</math> for some <math>n</math>. Another way in which Stein manifolds are similar to affine complex algebraic varieties is that Cartan's theorems A and B hold for Stein manifolds

g = 0: <math>\mathbb{CP}^1</math>
g = 1: There is a one-dimensional complex manifold classifying possible compact Riemann surfaces of genus 1, so-called elliptic curves, the modular curve. By the uniformization theorem any elliptic curve may be written as a quotient <math>\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})</math> where <math>\tau</math> is a complex number with strictly positive imaginary part. The moduli space is given by the quotient of the group <math>\operatorname{PSL}(2,\mathbb{Z})</math> acting on the upper half plane by Möbius transformations.
g > 1: For each genus greater than one, there is a moduli space <math>\mathcal{M}_g</math> of genus g compact Riemann surfaces, of dimension <math>\dim_{\mathbb{C \mathcal{M}_g = 3g-3</math>. Similar to the case of elliptic curves, this space may be obtained by a suitable quotient of Siegel upper half-space by the action of the group <math>\operatorname{Sp}(2g, \mathbb{Z})</math>.

Complex surfaces

The tripartition of Riemann surfaces according to the genus has a higher-rank analog : Any <math>n</math>-dimensional compact manifold <math>X</math> has a Kodaira dimension which can take <math>n+1</math> values

:<math>\kappa(X)\in\{-\infty, 0,1,\dots,n\}.</math>

Compact manifolds of maximal Kodaira dimension (<math>\kappa(X)=\dim(X)</math>) are called manifolds of general type.

In complex dimension 2, this basic classification can be refined: surfaces that are not of general type are subdivided into 9 classes, which leads to the Enriques-Kodaira classification of compact surfaces into ten classes.

Holomorphic line bundles

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 <math>X</math> is given by the Picard variety <math>\operatorname{Pic}(X)</math> of <math>X</math>.

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

By the Torelli 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.

Complex geometry

Idea

Definitions

Stein manifolds

Complex surfaces

Holomorphic line bundles

Minimal model program

See also

References

Sources