Complex geometry

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 ${\displaystyle \mathbb {C} }$ 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 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.

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 is a topological space ${\displaystyle X}$ such that:

• ${\displaystyle X}$ is Hausdorff and second countable.
• ${\displaystyle X}$ is locally homeomorphic to an open subset of ${\displaystyle \mathbb {C} ^{n}}$ for some ${\displaystyle n}$. That is, for every point ${\displaystyle p\in X}$, there is an open neighbourhood ${\displaystyle U}$ of ${\displaystyle p}$ and a homeomorphism ${\displaystyle \varphi :U\to V}$ to an open subset ${\displaystyle V\subseteq \mathbb {C} ^{n}}$. Such open sets are called charts.
• If ${\displaystyle (U_{1},\varphi )}$ 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 the transition function ${\displaystyle \psi \circ \varphi ^{-1}:\varphi (U_{1}\cap U_{2})\to \psi (U_{1}\cap U_{2})}$ is a biholomorphism.

Notice that since every biholomorphism is a diffeomorphism, and ${\displaystyle \mathbb {C} ^{n}}$ is isomorphism as a real 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. An affine 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 be irreducible. A point ${\displaystyle p\in X}$ is singular if the Jacobian matrix of the vector of holomorphic functions ${\displaystyle (f_{1},\dots ,f_{k})}$ does not have full rank at ${\displaystyle p}$, and non-singular otherwise. A projective complex analytic variety is a subset ${\displaystyle X\subseteq \mathbb {CP} ^{n}}$ 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 ${\displaystyle \mathbb {CP} ^{n}}$.

One may similarly define an affine 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 a projective complex algebraic variety, one requires the subset ${\displaystyle X\subseteq \mathbb {CP} ^{n}}$ 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 ${\displaystyle (X,{\mathcal {O}}_{X})}$ 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 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 ${\displaystyle X}$ which are singular is called the singular locus, denoted ${\displaystyle X^{sing}}$, and the complement is the non-singular or smooth locus, denoted ${\displaystyle X^{nonsing}}$. 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.

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 sheaf cohomology, and sheaves and their 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 line bundles (and the divisors associated to them), holomorphic vector bundles, and 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 Hirzebruch-Riemann-Roch theorem, a special case of the Atiyah-Singer index theorem, computes the holomorphic 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 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.

• 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
• ${\displaystyle \partial {\bar {\partial }}}$-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.