Complex geometry

Source: Wikipedia, the free encyclopedia.

In

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,

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

of complex varieties.

The

millennium prize problems, is a problem in complex geometry.[1]

Idea

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

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 is, after forgetting its complex structure, isomorphic to the real plane . However, complex geometry is not typically seen as a particular sub-field of

, 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

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

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 analytic varieties
. In this section, these types of spaces are defined and the relationships between them presented.

A complex manifold is a topological space such that:

  • is
    second countable
    .
  • is locally
    homeomorphic
    to an open subset of for some . That is, for every point , there is an
    open neighbourhood
    of and a homeomorphism to an open subset . Such open sets are called charts.
  • If and are any two overlapping charts which map onto open sets of respectively, then the transition function is a biholomorphism.

Notice that since every biholomorphism is a diffeomorphism, and is isomorphism as a

real vector space
to , every complex manifold of dimension is in particular a smooth manifold of dimension , which is always an even number.

In contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. An affine complex analytic variety is a subset such that about each point , there is an open neighbourhood of and a collection of finitely many holomorphic functions such that . By convention we also require the set to be

irreducible
. A point is singular if the
Jacobian matrix
of the vector of holomorphic functions does not have full rank at , and non-singular otherwise. A projective complex analytic variety is a subset of complex projective space that is, in the same way, locally given by the zeroes of a finite collection of holomorphic functions on open subsets of .

One may similarly define an affine complex algebraic variety to be a subset which is locally given as the zero set of finitely many polynomials in complex variables. To define a projective complex algebraic variety, one requires the subset to locally be given by the zero set of finitely many

homogeneous polynomials
.

In order to define a general complex algebraic or complex analytic variety, one requires the notion of a

locally ringed space
. A complex algebraic/analytic variety is a locally ringed space which is locally isomorphic as a locally ringed space to an affine complex algebraic/analytic variety. In the analytic case, one typically allows to have a topology that is locally equivalent to the subspace topology due to the identification with open subsets of , whereas in the algebraic case is often equipped with a Zariski 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 which are singular is called the singular locus, denoted , and the complement is the non-singular or smooth locus, denoted . We say a complex variety is smooth or non-singular if it's singular locus is empty. That is, if it is equal to its non-singular locus.

By the implicit 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 of any complex variety is a complex manifold.

Types of complex spaces

Kähler manifolds

Complex manifolds may be studied from the perspective of differential geometry, whereby they are equipped with extra geometric structures such as a

symplectic 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. A Kähler 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 or the
Fubini-Study metric
on 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 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 being a complex submanifold of for some . 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.

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

Hyper-Kähler manifolds

A special class of complex manifolds is

hyper-Kähler manifolds, which are Riemannian manifolds admitting three distinct compatible integrable almost complex structures
which satisfy the quaternionic relations . Thus, hyper-Kähler manifolds are Kähler manifolds in three different ways, and subsequently have a rich geometric structure.

Examples of hyper-Kähler manifolds include ALE spaces, K3 surfaces, Higgs bundle moduli spaces, quiver varieties, and many other moduli spaces arising out of gauge theory and representation theory.

Calabi–Yau manifolds

A real two-dimensional slice of a quintic Calabi–Yau threefold

As mentioned, a particular class of Kähler manifolds is given by Calabi–Yau manifolds. These are given by Kähler manifolds with trivial canonical bundle . Typically the definition of a Calabi–Yau manifold also requires to be compact. In this case Yau's proof of the Calabi conjecture implies that admits a Kähler metric with vanishing Ricci curvature, and this may be taken as an equivalent definition of Calabi–Yau.

Calabi–Yau manifolds have found use in

Abelian varieties
.

Complex Fano varieties

A complex Fano variety is a complex algebraic variety with ample anti-canonical line bundle (that is, is ample). Fano varieties are of considerable interest in complex algebraic geometry, and in particular birational geometry, where they often arise in the minimal model program. Fundamental examples of Fano varieties are given by projective space where , and smooth hypersurfaces of of degree less than .

Toric varieties

Toric varieties
are complex algebraic varieties of dimension containing an open
dense subset
biholomorphic to , equipped with an action of which extends the action on the open dense subset. A toric variety may be described combinatorially by its toric fan, and at least when it is non-singular, by a
moment
polytope
. This is a polygon in with the property that any vertex may be put into the standard form of the vertex of the positive orthant by the action of . 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.

Techniques in complex geometry

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 by

cohomology groups
are major tools.

For example, famous problems in the analysis of several complex variables preceding the introduction of modern definitions are the Cousin 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 holomorphic

coherent 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 the Kodaira vanishing theorem for the cohomology of line bundles on compact Kähler manifolds, and Cartan'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, the

holomorphic Euler characteristic
of a holomorphic vector bundle in terms of characteristic classes of the underlying smooth complex vector bundle.

Classification in complex geometry

One major theme in complex geometry is classification. 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 of moduli spaces, which themselves are complex manifolds or varieties whose points classify other geometric objects arising in complex geometry.

Riemann surfaces

The term moduli was coined by

genus
, which is a non-negative integer counting the number of holes in the given compact Riemann surface.

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

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 is given by the

Picard variety
of .

The picard variety can be easily described in the case where 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 , 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.

See also

References

  1. ^ Voisin, C., 2016. The Hodge conjecture. In Open problems in mathematics (pp. 521-543). Springer, Cham.
  2. ^ Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.
  3. ^ Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.
  4. ^ Donaldson, S. (2011). Riemann surfaces. Oxford University Press.
  • .
  • 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
    .