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FOUNDATIONAL MATERIAL

In this chapter we sketch the foundational material from several complex variables, complex manifold theory, topology, and differential geometry that will be used in our study of algebraic geometry. While our treatment is for the most part self-contained, it is tacitly assumed that the reader has some familiarity with the basic objects discussed. The primary purpose of this chapter is to establish our viewpoint and to present those results needed in the form in which they will be used later on. There are, broadly speaking, four main points:

1. The Weierstrass theorems and corollaries, discussed in Sections 1 and 2. These give us our basic picture of the local character of analytic varieties. The theorems themselves will not be quoted directly later, but the picture—for example, the local representation of an analytic variety as a branched covering of a polydisc—is fundamental. The foundations of local analytic geometry are further discussed in Chapter 5. 2. Sheaf theory, discussed in Section 3, is an important tool for relating the analytic, topological, and geometric aspects of an algebraic variety. A good example is the exponential sheaf sequence, whose individual terms , , and * reflect the topological, analytic, and geometric structures of the underlying variety, respectively. 3. Intersection theory, discussed in Section 4, is a cornerstone of classical algebraic geometry. It allows us to treat the incidence properties of algebraic varieties, a priori a geometric question, in topological terms. 4. Hodge theory, discussed in Sections 6 and 7. By far the most sophisticated technique introduced in this chapter, Hodge theory has, in the present context, two principal applications: first, it gives us the Hodge decomposition of the cohomology of a Kähler manifold; then, together with the formalism introduced in Section 5, it gives the vanishing theorems of the next chapter.

1. RUDIMENTS OF SEVERAL COMPLEX VARIABLES

Cauchy’s Formula and Applications

NOTATION. We will write z = (z1,…, zn) for a point in , with

For U an open set in , write C∞(U) for the set of C∞ functions defined on U; for the set of C∞ functions defined in some neighborhood of the closure of U.

The cotangent space to a point in is spanned by {dxi, dyi}; it will often be more convenient, however, to work with the complex basis

and the dual basis in thetangent space

With this notation, the formula for the total differential is

In one variable, we say a C∞ function f on an open set U ⊂ is holomorphic if f satisfies the Cauchy-Riemann equations . Writing , this amounts to

We say f is analytic if, for all z0 ∈ U, f has a local series expansion in z – z0, i.e.,

in some disc Δ(z0, ε)= {z : |z – z0| < ε}, where the sum converges absolutely and uniformly. The first result is that f is analytic if and only if it is holomorphic; to show this, we use the

Cauchy Integral Formula. For Δ a disc in ,

where the line integrals are taken in the counterclockwise direction (the fact that the last integral is defined will come out in the proof).

Proof. The proof is based on Stokes’ formula for a differential form with singularities, a method which will be formalized in Chapter 3. Consider the differential form

we have for z ≠ w

and so

Let Δε = Δ(z, ε) be the disc of radius ε around z. The form η is C∞ in Δ – Δε, and applying Stokes’ theorem we obtain

Setting w – z = reiθ,

which tends to f(z) as ε→0; moreover,

so

Thus is absolutely integrable over Δ, and

as ε→0; the result follows.

Q.E.D.

Now we can prove the

Proposition. For U an open set in and f ∈ C∞(U), f is holomorphic if and only if f is analytic.

Proof. Suppose first that ∂f/∂ = 0. Then for z0 ∈ U, ε sufficiently small, and z in the disc Δ = Δ(z0, ε) of radius ε around z0,

so, setting

we have

for z ∈ Δ, where the sum converges absolutely and uniformly in any smaller disc.

Suppose conversely that f(z) has a power series expansion

for z ∈ Δ = Δ(z0, ε). Since (∂/∂)(z – z0)n = 0, the partial sums of the expansion satisfy Cauchy’s formula without the area integral, and by the uniform convergence of the sum in a neighborhood of z0 the same is true of f, i.e.,

We can then differentiate under the integral sign to obtain

since for z ≠ w

Q.E.D.

We prove a final result in one variable, that given a C∞ function g on a disc Δ the equation

can always be solved on a slightly smaller disc; this is the

-Poincaré Lemma in One Variable. Given , the function

is defined and C∞ in Δ and satisfies

Proof. For z0 ∈ Δ choose ε such that the disc Δ(z0, 2ε) ⊂ Δ and write

where g1(z) vanishes outside Δ(z0, 2ε) and g2(z) vanishes inside Δ(z0, ε). The integral

is well-defined and C∞ for z ∈ Δ(z0, ε); there we have

Since g1(z) has compact support, we can write

where u = w – z. Changing to polar coordinates u = reiθ this integral becomes

which is clearly defined and C∞ in z. Then

but g1 vanishes on ∂Δ, and so by the Cauchy formula

Q.E.D.

Several Variables

In the formula

for the total differential of a function f on , we denote the first term ∂f and the second term f; ∂ and are differential operators invariant under a complex linear change of coordinates. A C∞ function f on an open set U ⊂ is called holomorphic if f = 0; this is equivalent to f(z1,…, zn) being holomorphic in each variable zi separately.

As in the one-variable case, a function f is holomorphic if and only if it has local power series expansions in the variables zi. This is clear in one direction: by the same argument as before, a convergent power series defines a holomorphic function. We check the converse in the case n = 2; the computation for general n is only notationally more difficult. For f holomorphic in the open set U ⊂ 2, z0 ∈ U, we can fix Δ the disc of radius r around z0 ∈ U and apply the one-variable Cauchy formula twice to obtain, for (z1, z2) ∈ Δ,

Using the series expansion

we find that f has a local series expansion

Q.E.D.

Many results in several variables carry directly over from the one-variable theory, such as the identity theorem: If f and g are holomorphic on a connected open set U and f = g on a nonempty open subset of U, then f = g, and the maximum principle: the absolute value of a holomorphic function f on an open set U has no maximum in U. There are, however, some striking differences between the one- and many-variable cases. For example, let U be the polydisc Δ(r) = {(z1, z2):|z1|, |z2| < r}, and let V ⊂ U be the smaller polydisc Δ(r′) for any r′ < r. Then we have

Hartogs’ Theorem. Any holomorphic function f in a neighborhood of U – V extends to a holomorphic function on U.

Proof. In each vertical slice z1 = constant, the region U – V looks either like the annulus r′ < |z2| < r or like the disc |z2| < r. We try to extend f in each slice by Cauchy’s formula, setting

F is defined throughout U; it is clearly holomorphic in z2, and since , it is holomorphic in z1 as well. Moreover, in the open subset |z1| >...