13. Algebraic Varieties over the Complex Numbers

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It is not hard to show that there is a unique way to endow all algebraic varieties

over C with a topology such that:

(a) on An = Cn it is just the usual complex topology;

(b) on closed subsets of An it is the induced toplogy;

(c) all morphisms of algebraic varieties are continuous;

(d) it is finer than the Zariski topology.

We call this new topology the complex topology on V . Note that (a), (b), and (c)

determine the topology uniquely for affine algebraic varieties ((c) implies that an isomorphism

of algebraic varieties will be a homeomorphism for the complex topology),

and (d) then determines it for all varieties.

Of course, the complex topology is much finer than the Zariski topology — this

can be seen even on A1. In view of this, the next proposition is little surprising.

Proposition 13.1. Let V be an algebraic variety over C, and let C be a constructible

subset of V (in the Zariski topology); then the closure of C in the Zariski

topology equals its closure in the complex topology.

Proof. Omitted.

For example, if U is an open dense subset of a closed subset Z of V (both for the

Zariski topology), then U is also dense in Z for the complex topology.

The next result helps explain why completeness is the analogue of compactness for

topological spaces.

Proposition 13.2. Let V be an algebraic variety over C; then V is complete (as

an algebraic variety) if and only if it is compact for the complex topology.

Proof. Omitted.

In general, there are many more holomorphic (complex analytic) functions than

there are polynomial functions on a variety over C. For example, by using the exponential

function it is possible to construct many holomorphic functions on C that

are not polynomials in z, but all these functions have nasty singularities at the point

at infinity on the Riemann sphere. In fact, the only meromorphic functions on the

Riemann sphere are the rational functions. This generalizes.

Theorem 13.3. Let V be a complete nonsingular variety over C. Then V is, in

a natural way, a complex manifold, and the field of meromorphic functions on V (as

a complex manifold) is equal to the field of rational functions on V .

Proof. Omitted.

This provides one way of constructing compact complex manifolds that are not

algebraic varieties:fi nd such a manifoldM of dimension n such that the transcendence

degree of the field of meromorphic functions on M is < n. For a torus Cg/Λ of

dimension g ≥ 1, this is typically the case. However, when the transcendence degree

of the field of meromorphic functions is equal to the dimension of manifold, then M

can be given the structure, not necessarily of an algebraic variety, but of something

152 Algebraic Geometry: 13. Algebraic Varieties over the Complex Numbers

more general, namely, that of an algebraic space. Roughly speaking, an algebraic

space is an object that is locally an affine algebraic variety, where locally means for

the ´etale “topology” rather than the Zariski topology.

One way to show that a complex manifold is algebraic is to embed it into projective

space.

Theorem 13.4. Any closed analytic submanifold of Pn is algebraic.

Proof. Omitted.

Corollary 13.5. Any holomorphic map from one projective algebraic variety to

a second projective algebraic variety is algebraic.

Proof. Let ϕ: V → W be the map. Then the graph Γϕ of ϕ is a closed subset of

V × W, and hence is algebraic according to the theorem. Since ϕ is the composite

of the isomorphism V → Γϕ with the projection Γϕ → W, and both are algebraic, ϕ

itself is algebraic.

Since, in general, it is hopeless to write down a set of equations for a variety (it is a

fairly hopeless task even for an abelian variety of dimension 3), the most powerful way

we have for constructing varieties is to first construct a complex manifold and then

prove that it has a natural structure as a algebraic variety. Sometimes one can then

show that it has a canonical model over some number field, and then it is possible

to reduce the equations defining it modulo a prime of the number field, and obtain a

variety in characteristic p.

For example, it is known that Cg/Λ (Λ a lattic in Cg) has the structure of an

algebraic variety if and only if there is a skew-symmetric form ψ on Cg having certain

simple properties relative to Λ. The variety is then an abelian variety, and all abelian

varieties over C are of this form.

References

Mumford 1966, I.10.

Shafarevich 1994, Book 3.

Algebraic Geometry: 14. Further Reading 153

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