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.
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.
In mathematics, the Jacobian varietyJ(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety.
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.
In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle Ω on V.
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are derived from the notion of divisibility in the integers and algebraic number fields.
In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
In mathematics, a Weierstrass point on a nonsingular algebraic curve defined over the complex numbers is a point such that there are more functions on , with their poles restricted to only, than would be predicted by the Riemann–Roch theorem.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions.
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by Rosenhain (1851)
The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.
In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory.
