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An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts.
Fix a field k with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes R and C with respect to their usual absolute values, as well as fields of Puiseux series with respect to their natural valuations.
Let U be an open subset of kn, and let f1, ..., fk be a collection of analytic functions on U. Denote by Z the common vanishing locus of f1, ..., fk, that is, let Z = { x | f1(x) = ... = fk(x) = 0 }. Z is an analytic variety.
Suppose that the structure sheaf of U is . Then Z has a structure sheaf , where is the ideal generated by f1, ..., fk. In other words, the structure sheaf of Z consists of all functions on U modulo the possible ways they can differ outside of Z.
An analytic space is a locally ringed space such that around every point x of X, there exists an open neighborhood U such that is isomorphic (as locally ringed spaces) to an analytic variety with its structure sheaf. Such an isomorphism is called a local model for X at x.
An analytic mapping or morphism of analytic spaces is a morphism of locally ringed spaces.
This definition is similar to the definition of a scheme. The only difference is that for a scheme, the local models are spectra of rings, whereas for an analytic space, the local models are analytic varieties. Because of this, the basic theories of analytic spaces and of schemes are very similar. Furthermore, analytic varieties have much simpler behavior than arbitrary commutative rings (for example, analytic varieties are defined over fields and are always finite-dimensional), so analytic spaces behave very similarly to finite-type schemes over a field.
Every point in an analytic space has a local dimension. The dimension at x is found by choosing a local model at x and determining the local dimension of the analytic variety at the point corresponding to x.
Every point in an analytic space has a tangent space. If x is a point of X and mx is ideal sheaf of all functions vanishing at x, then the cotangent space at x is mx / mx2. The tangent space is (mx / mx2)*, the dual vector space to the cotangent space. Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces.
The dimension of the tangent space at x is called the embedding dimension at x. By looking at a local model it is easy to see that the dimension is always less than or equal to the embedding dimension.
An analytic space is called smooth at x if it has a local model at x which is an open subset of kn for some n. The analytic space is called smooth if it is smooth at every point, and in this case it is an analytic manifold. The subset of points at which an analytic space is not smooth is a closed analytic subset.
An analytic space is reduced if every local model for the space is defined by a radical sheaf of ideals. An analytic space X which isn't reduced has a reductionXred, a reduced analytic space with the same underlying topological space. There is a canonical morphism r : Xred→X. Every morphism from X to a reduced analytic space factors through r.
An analytic space is normal if every stalk of the structure sheaf is a normal ring (meaning an integrally closed integral domain). In a normal analytic space, the singular locus has codimension at least two. When X is a local complete intersection at x, then X is normal at x.
Non-normal analytic spaces can be smoothed out into normal spaces in a canonical way. This construction is called the normalization. The normalization N(X) of an analytic space X comes with a canonical map ν : N(X) →X. Every dominant morphism from a normal analytic space to X factors through ν.
An analytic space is coherent if its structure sheaf is a coherent sheaf. A coherent sheaf of -modules is called a coherent analytic sheaf. For example, on a coherent space, locally free sheaves and sheaves of ideals are coherent analytic sheaves.
Analytic spaces over algebraically closed fields are coherent. In the complex case, this is known as the Oka coherence theorem. This is not true over non-algebraically closed fields; there are examples of real analytic spaces that are not coherent.
In some situations, the concept of an analytic space is too restrictive. This is often because the ground field has additional structure that is not captured by analytic sets. In these situations, there are generalizations of analytic spaces which allow more flexibility in the local model spaces.
For example, over the real numbers, consider the circle x2 + y2 = 1. The circle is an analytic subset of the analytic space R2. But its projection onto the x-axis is the closed interval [−1, 1], which is not an analytic set. Therefore the image of an analytic set under an analytic map is not necessarily an analytic set. This can be avoided by working with subanalytic sets, which are much less rigid than analytic sets but which are not defined over arbitrary fields. The corresponding generalization of an analytic space is a subanalytic space. (However, under mild point-set topology hypotheses, it turns out that subanalytic spaces are essentially equivalent to subanalytic sets.)
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, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets.
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables, which the Mathematics Subject Classification has as a top-level heading.
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative". The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors, there is an equivalent notion of an ample divisor.
In algebraic geometry, a closed immersion of schemes is a morphism of schemes that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X. The latter condition can be formalized by saying that is surjective.
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 algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets , where each is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether.
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If is a morphism of geometric or algebraic objects, the corresponding cotangent complex can be thought of as a universal "linearization" of it, which serves to control the deformation theory of . It is constructed as an object in a certain derived category of sheaves on using the methods of homotopical algebra.
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold that allows the presence of singularities. Complex analytic varieties are locally ringed spaces that are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.
This is a glossary of algebraic geometry.
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times the restriction of s for any f in O(U) and s in F(U).
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents S-derivations in the sense: for any -modules F, there is an isomorphism
In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf together with a linear functional
