Algebraic geometry and analytic geometry

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In mathematics, algebraic geometry and analytic geometry are two closely related subjects. Where 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.

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[edit] Introduction

Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, 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 precisely the polynomial functions (an extension of Liouville's theorem). For if we look at such a function f on C (as a subset of the Riemann sphere), it's an entire function. As the point at infinity must be a pole, we have f(z) < C|z|n for some natural number n and z in a neighboorhood of infinity, say |z| > R. Using Cauchy's integral formula over a circle centered at 0 and with radius r > R, we obtain an upper bound for the Taylor coefficients of f at 0: |ak| < C rn-k. By letting r go to infinity we see that ak = 0 for k > n, so f has to be a polynomial. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere.

[edit] 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.

[edit] Riemann's existence theorem

Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions 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 spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. 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.

This is not proved in GAGA. It is proved in Springer's Introduction to Riemann Surfaces.

[edit] The Lefschetz principle

In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, 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. The precise principle and its proof are due to Tarski and are based in mathematical logic.

This principle allowed to carry over results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed base fields of characteristic 0.

[edit] 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 space 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 set 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.

[edit] Serre's 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 theory. 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.

[edit] Formal statement of GAGA

1. Let (X,\mathcal O_X) a scheme of finite type over \mathbb{C} . Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map \lambda_X: X^{an}\hookrightarrow X . The topology on Xan is called "complex topology" (and is not the subspace topology in general).

2. Suppose \varphi: X\rightarrow Y is a morphism of schemes of locally finite type over \mathbb{C} . Then there exists a continuous map \varphi^{an}: X^{an}\rightarrow Y^{an} such \lambda_Y\circ\varphi^{an} = \varphi\circ \lambda_X.

3. There is a sheaf \mathcal O_X^{an} on Xan such that (X^{an}, \mathcal O_X^{an}) is a ringed space and \lambda_X: X^{an}\rightarrow X becomes a map of ringed spaces. The space (X^{an}, \mathcal O_X^{an}) is called the "analytifiction" of (X,\mathcal O_X) and is an analytic space. For every \varphi: X\rightarrow Y the map \varphi^{an} defined above is a mapping of analytic spaces. Furthermore, the map \varphi \mapsto \varphi^{an} maps open immersions into open immersions. If X = \mathbb{C}[X_1,\ldots,X_n] then X^{an} = \mathbb{C}^n and \mathcal O_X^{an}(U) for every polydisc U is a suitable quotient of the space of holomorphic functions on U.

4. For every sheaf \mathcal F on X (called algebraic sheaf) there is a sheaf \mathcal F^{an} on Xan (called analytic sheaf) and a map of sheaves of \mathcal O_X -modules \lambda_X^*: \mathcal F\rightarrow (\lambda_X)_* \mathcal F^{an} . The sheaf \mathcal F^{an} is defined as \lambda_X^{-1} \mathcal F \otimes_{\lambda_X^{-1} \mathcal O_X} \mathcal O_X^{an} . The correspondence \mathcal F \mapsto \mathcal F^{an} defines an exact functor from the category of sheaves over (X, \mathcal O_X) to the category of sheaves of (X^{an}, \mathcal O_X^{an}) .

The following two statements are the the hearth of Serre's GAGA theorem (as extended by Grothendieck, Neeman et. al.)

5. If f: X\rightarrow Y is an arbitrary morphism of schemes of finite type over \mathbb{C} and \mathcal F is coherent then the natural map (f_* \mathcal F)^{an}\rightarrow f_*^{an} \mathcal F^{an} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves (R^i f_* \mathcal F)^{an} \cong R^i f_*^{an} \mathcal F^{an} in this case.

6. Now assume that Xan is hausdorff and compact. If \mathcal F, \mathcal G are two coherent algebraic sheaves on (X, \mathcal O_X) and if f: \mathcal F^{an} \rightarrow \mathcal G^{an} is a map of sheaves of \mathcal O_X^{an} modules then there exists a unique map of sheaves of \mathcal O_X modules \varphi: \mathcal F\rightarrow \mathcal G with f = \varphi^{an} . If \mathcal R is a coherent analytic sheaf of \mathcal O_X^{an} modules over Xan then there exists a coherent algebraic sheaf \mathcal F of \mathcal O_X -modules and an isomorphism \mathcal F^{an} \cong \mathcal R .

[edit] Moishezon manifolds

A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions 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 spaces. Moishezon in 1967 showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric.

[edit] References

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