Algebraic geometry and analytic geometry
Encyclopedia
In mathematics
, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry
studies algebraic varieties
, analytic geometry deals with complex manifold
s and the more general analytic space
s defined locally by the vanishing of analytic function
s 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.
s, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.
For example, it is easy to prove that the analytic functions from the Riemann sphere
to itself are either
the rational functions or the identically infinity function (an extension of Liouville's theorem
). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem
is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere
.
theory shows that a compact
Riemann surface has enough meromorphic function
s on it, making it an algebraic curve
. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological space
s are classified by permutation representations of the fundamental group
of the complement of the ramification point
s. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field
.
, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field
K of characteristic
0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski
and are based in mathematical logic
.
This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.
that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology
is closed in the Zariski topology
." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.
. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre
, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves
with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves.
Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.
such that the field of meromorphic function
s on M has transcendence degree equal to the complex dimension
of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic space
s. In 1967, Boris Moishezon
showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
studies algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
, analytic geometry deals with complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s and the more general analytic space
Analytic space
An analytic space in the theory of functions of several complex variables is a generalization of the concept of an analytic manifold: heuristically, an analytic space is a set which is locally isomorphic to an analytic variety, while an analytic manifold is locally isomorphic to a complex euclidean...
s defined locally by the vanishing of analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Background
Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functionHolomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way.
For example, it is easy to prove that the analytic functions from the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
to itself are either
the rational functions or the identically infinity function (an extension of Liouville's theorem
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
.
Important results
There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order.Riemann's existence theorem
Riemann surfaceRiemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
theory shows that a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Riemann surface has enough meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on it, making it an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s are classified by permutation representations of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the complement of the ramification point
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
s. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
.
The Lefschetz principle
In the twentieth century, the Lefschetz principle, named for Solomon LefschetzSolomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
K of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
and are based in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
.
This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.
Chow's theorem
Chow's theorem, proved by W. L. Chow. is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology
Strong topology
In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:* the final topology on the disjoint union...
is closed in the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.
GAGA
Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theoryHodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves.
Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.
Formal statement of GAGA
- Let
be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map λX: Xan → X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). - Suppose φ: X → Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map φan: Xan → Yan such λY ° φan = φ ° λX.
- There is a sheaf
on Xan such that 
is a ringed space and λX: Xan → X becomes a map of ringed spaces. The space 
is called the "analytification" of 
and is an analytic space. For every φ: X → Y the map φan defined above is a mapping of analytic spaces. Furthermore, the map φ ↦ φan maps open immersions into open immersions. If X = C[x1,...,xn] then Xan = Cn and 
for every polydisc U is a suitable quotient of the space of holomorphic functions on U. - For every sheaf
on X (called algebraic sheaf) there is a sheaf 
on Xan (called analytic sheaf) and a map of sheaves of 
-modules 
. The sheaf 
is defined as 
. The correspondence 
defines an exact functor from the category of sheaves over 
to the category of sheaves of 
.
The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) - If f: X → Y is an arbitrary morphism of schemes of finite type over C and
is coherent then the natural map 
is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves 
in this case. - Now assume that Xan is hausdorff and compact. If
are two coherent algebraic sheaves on 
and if 
is a map of sheaves of 
modules then there exists a unique map of sheaves of 
modules 
with f = φan. If 
is a coherent analytic sheaf of 
modules over Xan then there exists a coherent algebraic sheaf 
of 
-modules and an isomorphism 
.
Moishezon manifolds
A Moishezon manifold M is a compact connected complex manifoldComplex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
such that the field of meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s on M has transcendence degree equal to the complex dimension
Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic space
Algebraic space
In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory...
s. In 1967, Boris Moishezon
Boris Moishezon
Boris Gershevich Moishezon was a Ukrainian-born Soviet mathematician. He left the Soviet Union in 1972 for Tel Aviv, and in 1977 moved to Columbia University, where he was a professor of mathematics until his death sixteen years later...
showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.