Principles of Algebraic Geometry

By Phillip Griffiths and Joseph Harris

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Aug 21, 2014
• ISBN: 9781118626320

Book preview

0-471-05059-8

PREFACE

Algebraic geometry is among the oldest and most highly developed subjects in mathematics. It is intimately connected with projective geometry, complex analysis, topology, number theory, and many other areas of current mathematical activity. Moreover, in recent years algebraic geometry has undergone vast changes in style and language. For these reasons there has arisen about the subject a reputation of inaccessibility. This book gives a presentation of some of the main general results of the theory accompanied by—and indeed with special emphasis on—the applications to the study of interesting examples and the development of computational tools.

A number of principles guided the preparation of the book. One was to develop only that general machinery necessary to study the concrete geometric questions and special classes of algebraic varieties around which the presentation was centered.

A second was that there should be an alternation between the general theory and study of examples, as illustrated by the table of contents. The subject of algebraic geometry is especially notable for the balance provided on the one hand by the intricacy of its examples and on the other by the symmetry of its general patterns; we have tried to reflect this relationship in our choice of topics and order of presentation.

A third general principle was that this volume should be self-contained. In particular any hard result that would be utilized should be fully proved. A difficulty a student often faces in a subject as diverse as algebraic geometry is the profusion of cross-references, and this is one reason for attempting to be self-contained. Similarly, we have attempted to avoid allusions to, or statements without proofs of, related results. This book is in no way meant to be a survey of algebraic geometry, but rather is designed to develop a working facility with specific geometric questions. Our approach to the subject is initially analytic: Chapters 0 and 1 treat the basic techniques and results of complex manifold theory, with some emphasis on results applicable to projective varieties. Beginning in Chapter 2 with the theory of Riemann surfaces and algebraic curves, and continuing in Chapters 4 and 6 on algebraic surfaces and the quadric line complex, our treatment becomes increasingly geometric along classical lines. Chapters 3 and 5 continue the analytic approach, progressing to more special topics in complex manifolds.

Several important topics have been entirely omitted. The most glaring are the arithmetic theory of algebraic varieties, moduli questions, and singularities. In these cases the necessary techniques are not fully developed here. Other topics, such as uniformization and automorphic forms or monodromy and mixed Hodge structures have been omitted, although the necessary techniques are for the most part available.

We would like to thank Giuseppe Canuto, S. S. Chern, Maurizio Cornalba, Ran Donagi, Robin Hartshorne, Bill Hoffman, David Morrison, David Mumford, Arthur Ogus, Ted Shifrin, and Loring Tu for many fruitful discussions; Ruth Suzuki for her wonderful typing; and the staff of John Wiley, especially Beatrice Shube, for enormous patience and skill in converting a very rough manuscript into book form.

PHILLIP GRIFFITHS

JOSEPH HARRIS

May 1978

Cambridge, Massachusetts

0

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* 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, with

For U , write C∞(U) for the set of C∞ functions defined on Ufor the set of Cof U.

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 , 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

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 , we denote the first term ∂f fare differential operators invariant under a complex linear change of coordinates. A C∞ function f on an open set U is called holomorphic 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 z, it is holomorphic in z1 as well. Moreover, in the open subset |z1| > r′ of U – V, F(z1, z2) = f(z1, z2) by Cauchy’s formula; thus F|U – V = f.

Q.E.D.

Hartogs’ theorem applies to many pairs of sets V ⊂ U ; it is commonly applied in the form

A holomorphic function on the complement of a point in an open set (n > 1) extends to a holomorphic function in all of U.

Weierstrass Theorems and Corollaries

In one variable, every analytic function has a unique local representation

from which we see in particular that the zero locus of f is discrete. Similarly, the Weierstrass theorems give local representations of holomorphic functions in several variables, from which we get a picture of the local geometry of their zero sets.

Suppose we are given a function f(z1,…, zn–1, w, with f(0,…, 0) = 0. Assume that f does not vanish identically on the w-axis, i.e., the power series expansion for f around the origin contains a term a · wd with a ≠ 0 and d ≥ 1; clearly this will be the case for most choices of coordinate system.

For suitable r, δ, and ε > 0, then, |f(0, w)| ≥ δ > 0 for |w| = r, and consequently |f(z, w)| ≥ δ/2 for |w| = r, ||z|| ≤ ε. Now if w = b1,…, bd are the roots of f(z, w) = 0 for |w| < r, by the residue theorem

so the power sums ∑bi(z)q are analytic functions of z for ||z|| < ε. Let σ1(z),…, σd(z) be the elementary symmetric polynomials in b1,…, bd; σ1,…, σd can be expressed as polynomials in the power sums ∑bi(z)q. Thus the function

is holomorphic in ||z|| < ε, |w| < r, and vanishes on exactly the same set as f. The quotient

is defined and holomorphic in ||z|| < ε, |w| < r, at least outside the zero set of f and g. Moreover, for fixed z, h(z, w) has only removable singularities in the disc |w| < r, so h can be extended to a function in all of ||z|| < ε, |w| < r and analytic in w for each fixed z, as well as in the complement of the zero locus. Writing

we see that h is holomorphic in z as well.

DEFINITION. A Weierstrass polynomial in w is a polynomial of the form

We have proved the existence part of the

Weierstrass Preparation Theorem. If f is holomorphic around the origin in and is not identically zero on the w-axis, then in some neighborhood of the origin f can be written uniquely as

where g is a Weierstrass polynomial of degree d in w and h(0) ≠ 0.

The uniqueness is clear, since the coefficients of any Weierstrass polynomial g vanishing exactly where f does are given as polynomials in the integrals

We see from the Weierstrass theorem that the zero locus of a function f, is for most choices of coordinate system z1,…, zn–1, w the zero locus of a Weierstrass polynomial

Now, the roots bi(z) of the polynomial g(z, ·) are, away from those values of z for which g(z, ·) has a multiple root, locally single-valued holomorphic functions of z. Since the discriminant of g(z, ·) is an analytic function of z,

The zero locus of an analytic function f(z1,…, zn–, w), not vanishing identically on the w-axis, projects locally onto the hyperplane (w = 0) as a finite-sheeted cover branched over the zero locus of an analytic function.

As a corollary of the preparation theorem, we have the

Riemann Extension Theorem. Suppose f(z, w) is holomorphic in a disc and g(z, w) is holomorphic in – {f = 0} and bounded. Then g extends to a holomorphic function on Δ.

Proof (in a neighborhood of 0). Assume that the line z = 0 is not contained in {f = 0}. As before, we can find r, ε, and δ > 0 such that |f(0, w)| ≥ δ > 0 for |w| = r and ε such that |f(z, w)| > δ/2 for ||z|| < ε, |w| = r; f then has zeros only in the interior of the discs z = z0, |w| ≤ r. By the one-variable Riemann extension theorem, we can extend g in |z| < ε, |w| < r, holomorphic away from {f = 0} and holomorphic in w everywhere. As before, we write

is holomorphic in z as well.

Q.E.D.

We recall some facts and definitions from elementary algebra:

Let R be an integral domain, i.e., a ring such that for u∈ R, u u = 0. An element u ∈ R is a unit ∈ R such that u 1; u is irreducible , w ∈ R, u · w is a unit or w is a unit. R is a unique factorization domain (UFD) if every u ∈ R can be written as a product of irreducible elements u1,…, ul, the ui’s unique up to multiplication by units. The main facts we shall use are

1. R R[t] is a UFD (Gauss’ lemma).

2. If R is a UFD and u∈ R[t] are relatively prime, then there exist relatively prime elements α, β ∈ R[t], γ ≠ 0 ∈ R, such that

γ is called the resultant of u .

n,z denote the ring of holomorphic functions defined in some neighborhood of z n nn is an integral domain by the identity theorem, and moreover is a local ring whose maximal ideal m is {f: f(0) = 0}. f n is a unit if and only if f(0) ≠ 0. The first result is

Proposition. n is a UFD.

Proof. n–1 is a UFD and let f n. We may assume f is regular with respect to w = zn; i.e., f(0,…, 0, w0. Write

where u n and g n–1[wn–1[w] is a UFD by Gauss’ lemma, and so we can write g as a product of irreducible elements g1, …, gm n–1[w]

where the factors gi are uniquely determined up to multiplication by units. Now suppose we write f as a product of irreducible elements f1,…, fk n Each fi must be regular with respect to w, and we can write

with ui n–1. We have

with g both Weierstrass polynomials; by the Weierstrass preparation theorem

n–1[ware the same, up to units, as the gi. Thus the expression (*) represents a unique factorization of f n.

Q.E.D.

Proposition. If f and g are relatively prime in n,0, then for ||z|| < ε, f and g are relatively prime in n,z.

Proof. We may assume that f and g are regular with respect to zn and are both Weierstrass polynomials; for each fixed zn–1 sufficiently small we have f(z′, zn0 in zn. Now we can write

with α, β n–1[w], γ n.

If for some small z, f(z0) = g(z0) = 0 and f and g have a common factor h(z′, znn,z0 with h(z0) = 0, then

But then h(z01,…, z0n–1, zn) vanishes identically in zn, contradicting our assumption that f(z0,…, z0n–1, zn0.

Q.E.D.

We now prove the

Weierstrass Division Theorem. Let n–1[w] be a Weierstrass polynomial of degree k in w. Then for any n, we can write

with r(z, w) a polynomial of degree < k in w.

Proof. For ε, δ > 0 sufficiently small, define for ||z|| < ε, |w| < δ,

h is clearly holomorphic, and hence so is r = f – gh. We have

But (u – w) divides [g(z, u) – g(z, w)] as polynomials in w; thus

is a polynomial in w of degree < k. Since the factor w appears only in p in the expression for r(z, w), we see that r(z, w) is a polynomial of degree < k in w. Explicitly, if

then

where

Q.E.D.

Corollary (Weak Nullsteilensatz). If n is irreducible and n vanishes on the set f(z, w) = 0, then f divides h in n.

Proof. First, we may assume f is a Weierstrass polynomial of degree k in w. Since f is irreducible, f and ∂f/∂w n–1[w] (degwf > degw∂f/∂w); thus we can write

If, for a given z0, f(z0, w) has a multiple root u, we have

thus: f(z, w) has k distinct roots in w for γ(z) ≠ 0.

Now by the division theorem, we can write

But for any z0 outside the locus (γ = 0), f(z0, w) and hence h(z0, w) have at least k distinct roots in w. Since degree r < k, this implies r(z0, w[w]; it follows that r ≡ 0 and h = f·g.

Q.E.D.

Analytic Varieties

. We say a subset V of an open set U is an analytic variety in U if, for any p ∈ U, there exists a neighborhood U′ of p in U such that V ∩ U′ is the common zero locus of a finite collection of holomorphic functions f1,…, fk on U′. In particular, V is called an analytic hypersurface if V is locally the zero locus of a single nonzero holomorphic function f.

An analytic variety V ⊂ U is said to be irreducible if V cannot be written as the union of two analytic varieties V1, V2 ⊂ U with V1, V2 ≠ V; it is said to be irreducible at p ∈ V if V ∩ U′ is irreducible for small neighborhoods U′ of p in U. Note first that if f n n, then the analytic hypersurface V = {f(z) = 0} given by f in a neighborhood of 0 is irreducible at 0: if V = V1, V2, with V1, V2 analytic varieties ≠ V, then there exist f1, fn with f1 (respectively f2) vanishing identically on V1 (respectively V2) but not on V2 (respectively V1). By the Nullstellensatz, f must divide the product f1·f2; since f is irreducible, it follows that f must divide either f1 or f2, i.e., either V1 ⊃ V, or V2 ⊃ V, a contradiction. In addition to the basic picture of an analytic hypersurface (p. 9) we see that

1. Suppose V ⊂ U is an analytic hypersurface, given by V = {f(z) = 0} in a neighborhood of 0 ∈ Vn is a UFD, we can write

with fi n; if we set Vi = {fi(z) = 0} then we have

with Vi irreducible at 0. Thus if p is any point on any analytic hypersurface V ⊂ U , V can be expressed uniquely in some neighborhood U′ of p as the union of a finite number of analytic hypersurfaces irreducible at p.

2. Let W ⊂ U be an analytic variety given in a neighborhood Δ of 0 ∈ W as the zero locus of two functions f,g n. If W contains no analytic hypersurface through 0, then f and g n; if W does not contain the line {z′ = 0}, then by taking linear combinations we may assume that neither {f(z) = 0) or {g(z) = 0} contains {z′ = 0}, and hence that f and g are Weierstrass polynomials in zn. Let

be the resultant of f and g. We claim that the image of W is just the locus of γ. To see this, write

with the degree of r strictly less than the degree of g. Then

Now, if for some z n–1, γ vanishes at z but f and g have no common zeros along the line π–1(z), it follows that r vanishes at all the zeros of g in π–1(z); since deg(r) < deg(g), this implies that r, and hence β + hf, vanish identically on π–1(z). Thus r and β + hf both are zero on the inverse image of any component of the zero locus of γ other than π(W); but r and β + hf are relatively prime and so have no common components. We see then that π(W) is an analytic hypersurface in a neighborhood of the origin in n–1, and, reiterating our basic description of analytic hypersurfaces, that projection of W onto a suitably chosen (n − 2)-plane nn expresses W locally as a finite-sheeted branched cover of a neighborhood of the origin in n–2.

3. Last, let V ⊂ U n be an analytic variety irreducible at 0 ∈ V n, π(V n–1. Write

near 0. Then the functions fi n n, since otherwise V would be contained in the common locus of two relatively prime functions, and by assertion 2, π(V n–1. If we let g(z) be the greatest common divisor of the fi’s, then we can write

Since V is irreducible at 0 and since the locus {fi(z)/g(z) = 0, all i} cannot contain {g(z) = 0}, we must have

i.e., V is an analytic hypersurface near 0.

The results 1, 2, and 3 above, together with our basic picture of an analytic hypersurface, give us a picture of the local behavior of those analytic varieties cut out locally by one or two holomorphic functions. In fact, the same picture is in almost all respects valid for general analytic varieties, but to prove this requires some relatively sophisticated techniques from the theory of several complex variables. Since the primary focus of the material in this book is on the codimension 1 case, we will for the time being simply state here without proof the analogous results for general analytic varieties:

1. If V ⊂ U is any analytic variety and p ∈ V, then in some neighborhood of p, V can be uniquely written as the union of analytic varieties Vi irreducible at p with Vi Vj.

2. Any analytic variety can be expressed locally by a projection map as a finite-sheeted cover of a polydisc Δ branched over an analytic hypersurface of Δ.

3. If V does not contain the line z1 = ··· = zn–1 = 0, then the image of a neighborhood of 0 in V under the projection map π:(z1,…, zn) → (z1,…, znn–1.

The difficulties in proving these results are more technical than conceptual. For example, to prove assertion 3, note that if V by functions f1,…, fk, then π(Vn–1 by the resultants of all pairs of relatively prime linear combinations of the fi. The problem then is to show that the zero locus of an arbitrary collection of holomorphic functions in a polydisc is in fact given by a finite number of holomorphic functions in a slightly smaller polydisc. Granted assertions 3 and 1, 2 is not hard to prove by a sequence of projections.

All of these facts will follow from the proper mapping theorem, which we shall state in the next section and prove in Chapter 3.

Finally, several more foundational results in several complex variables will be proved by the method of residues in Chapter 5.

2. COMPLEX MANIFOLDS

Complex Manifolds

DEFINITION. A complex manifold M is a differentiable manifold admitting an open cover {Uα} and coordinate maps φα : Uα such that φα ° φβ–1 is holomorphic on φβ(Uα ∩ Uβfor all α, β.

A function on an open set U ⊂ M is holomorphic if, for all α, f·φα–1 is holomorphic on φα(U ∩ Uα. Likewise, a collection z = (z1,…, zn) of functions on U ⊂ M is said to be a holomorphic coordinate system if φα ° z–1 and z ° φα–1 are holomorphic on z(U ∩ Uα) and φα(U ∩ Uα), respectively, for each α; a map f : M → N of complex manifolds is holomorphic if it is given in terms of local holomorphic coordinates on N by holomorphic functions.

Examples

1. A one-dimensional complex manifold is called a Riemann surface.

n+1. A line l n+1 is determined by any Z ≠ 0 ∈ l, so we can write

On the subset Ui = {[Z] : Zi of lines not contained in the hyperplane (Zi = 0), there is a bijective map φi given by

On (zj ≠ 0) = φi(Uj ∩ Ui,

has the structure of a complex manifold, called complex projective space. The coordinates Z = [Z0,…, Zn] are called homogeneous coordinates ; the coordinates given by the maps φi are called Euclidean coordinatesn∪ {∞}.

kn; the image of such a map is called a linear subspace n+1 is again called a hyperplanenkn+1 is called a kin these terms: for example, the span of a collection {pin+1 spanned by the lines π–1(pi); k points are said to be linearly independent nis a (k – l)-plane.

nn+1 modulo scalar multiplication; it is thus itself a projective space, called the dual projective *.

obtained by adding on the hyperplane H is (z1,…, zn)→[1, z1,…, zn]; H has equation (Z.

. It is compact if and only if k = 2nis called a complex torus.

In general, if π : M → N is a topological covering space and N is a complex manifold, then π gives M the structure of a complex manifold as well; if M is a complex manifold and the deck transformations of M are holomorphic, then N inherits the structure of a complex manifold from M.

Another example of this construction is the Hopf surface. The Hopf surface is the simplest example of a compact complex manifold that cannot be imbedded in projective space of any dimension.

Let M be a complex manifold, p ∈ M any point, and z = (z1,…, zn) a holomorphic coordinate system around p. There are three different notions of a tangent space to M at p, which we now describe:

1. TR,p(M) is the usual real tangent space to M at p, where we consider M as a real manifold of dimension 2n. TR,p(M-linear derivations on the ring of real-valued C∞ functions in a neighborhood of p; if we write zi = xi + iyi,

is called the complexified tangent space to M at p-linear derivations in the ring of complex-valued C∞ functions on M around p. We can write

where, as before,

is called the holomorphic tangent space to M at p. It can be realized as the subspace of T ,p(M) consisting of derivations that vanish on antiholomorphic functions (i.e., f is holomorphic), and so is independent of the holomorphic coordinate system (z1,…, znis called the antiholomorphic tangent space to M at p; clearly

Observe that for M, N complex manifolds any C∞ map f : M → N induces a linear map

and hence a map

. In fact, a map f : M → N is holomorphic if and only if

for all p ∈ M.

Note also that since T ,p(M) is given naturally as the real vector space T ,p(M, the operation of conjugation sending ∂/∂zi to ∂/∂ i is well-defined and

It follows that the projection

-linear isomorphism. This last feature allows us to do geometry purely in the holomorphic tangent space. For example, let z(t) (0 ≤ t ≤ 1 ) be a smooth arc in the complex z-plane. Then z(t) = x(ty(t), and the tangent to the arc may be taken either as

or

and these two correspond under the projection T ) → T).

Now let M, N be complex manifolds, z = (z1,…, zn) be holomorphic coordinates centered at p ∈ M, w = (w1,…, wm) holomorphic coordinates centered at q ∈ N and f : M → N a holomorphic map with f(p) = q. Corresponding to the various tangent spaces to M and N at p and q, we have different notions of the Jacobian of f, as follows:

1. If we write zi = xi yi and wα = ua a, then in terms of the bases {∂/∂xi, ∂/∂yi} and {∂/∂uα, ∂/∂ α} for T ,p(M) and T ,q(N), the linear map f* is given by the 2m × 2n matrix

In terms of the bases {∂/∂zi, ∂/∂ i} and {∂/∂α, ∂/∂ α} for T ,p(M) and T ,q(N), f* is given by

where

and that if m = n, then

i.e., holomorphic maps are orientation preserving. We take the natural orientation to be given by the 2n-form

it is clear that if φα : Uα , φβ : Uβ are holomorphic coordinate maps on the complex manifold M, the pullbacks via φα and φβ agree on Uα ∩ Uβ. Thus any complex manifold has a natural orientation which is preserved under holomorphic maps.

Submanifolds and Subvarieties

Now that we have established the relations among the various Jacobians of a holomorphic map, it is not hard to prove the

Inverse Function Theorem. Let U, V be open sets in with 0 ∈ U and f : U → V a holomorphic map with nonsingular at 0. Then f is one-to-one in a neighborhood of 0, and f–1 is holomorphic at f(0).

Proof. at 0, by the ordinary inverse function theorem f has a C∞ inverse f–1 near 0. Now we have

so

Since (∂fk/∂zi) is nonsingular, this implies ∂fjk = 0 for all j, k, so f–1 is holomorphic.

Q.E.D.

Similarly, we have the

Implicit Function Theorem. Given n with

there exist functions n–k such that in a neighborhood of 0 in ,

Proof. Again, by the C∞ implicit function theorem we can find C∞ functions w1,…, wk with the required property; to see that they are holomorphic we write, for z = (zk+1,…, zn), k + 1 ≤ α ≤ n,

Q.E.D.

One special feature of the holomorphic case is the following:

Proposition. If f : U → V is a one-to-one holomorphic map of open sets in then , i.e., f–1 is holomorphic.

Proof. We prove this by induction on n; the case n = 1 is clear. Let z = (z1,…, zn) and w = (w1,…, wn) be coordinates on U and V(f) has rank k at 0 ∈ U; we may assume that the matrix ((∂fi/∂zj)(0))0≤i,j≤k is nonsingular. Set

is a holomorphic coordinate system for U near 0. But now f one-to-one onto the locus (w1 = · · · = wk is singular at 0; by the induction hypothesis, either k = 0 or k = n. We see then that the Jacobian matrix of f vanishes identically wherever its determinant is zero, i.e., that f (f)| = 0 to a single point in V. Since f (f(f)| ≠ 0.

Q.E.D.

is one-to-one but does not have a C∞ inverse.

Now we can make the

DEFINITION. A complex submanifold S of a complex manifold M is a subset S ⊂ M given locally either as the zeros of a collection f1,…, fk (f) = k, or as the image of an open set U n–k under a map f : U → M (f) = n – k.

The implicit function theorem assures us that the two alternate conditions of the definition are in fact equivalent, and that the submanifold S has naturally the structure of a complex manifold of dimension n – k.

DEFINITION. An analytic subvariety V of a complex manifold M is a subset given locally as the zeros of a finite collection of holomorphic functions. A point p ∈ V is called a smooth point of V if V is a submanifold of M near p, that is, if V is given in some neighborhood of p by holomorphic functions f1,…, fk (f) = k;, the locus of smooth points of V is denoted V*. A point p ∈ V – V* is called a singular point of V; the singular locus V – V* of V is denoted Vs. V is called smooth or nonsingular if V = V*, i.e., if V is a submanifold of M.

In particular, if p is a point of an analytic hypersurface V ⊂ M given in terms of local coordinates z by the function f, we define the multiplicity multp(V) to be the order of vanishing of f at p, that is, the greatest integer m such that all partial derivatives

We should mention here a piece of terminology that is pervasive in algebraic geometry: the word generic. When we are dealing with a family of objects parametrized locally by a complex manifold or an analytic subvariety of a complex manifold, the statement that a (or the) generic member of the family has a certain property means exactly that the set of objects in the family that do not have that property is contained in a subvariety of strictly smaller dimension.

In general, it will be clear how the objects in our family are to be parametrized. One exception will be a reference to "the generic k: until the section on Grassmannians, we have—at least officially—no way of parametrizing linear subspaces of projective space. The fastidious reader may substitute the linear span of the generic (k ."

A basic fact about analytic subvarieties is the

Proposition. Vs is contained in an analytic subvariety of M not equal to V.

Proof. For p ∈ V let k be the largest integer such that there exist k functions f1,…, fk in a neighborhood U of p vanishing on V(f) has a k × k minor not everywhere singular on Von V. Let U′ ⊂ U and V′ the locus f1 = · · · = fk = 0. Then V′ = V ∩ U′ is a complex submanifold of U′, and for any holomorphic function f vanishing on V the differential df ≡ 0 on V′, i.e., f is constant on V′. It follows that for q ∈ V′ near p, V = V′ is a manifold in a neighborhood of q and so Vs ⊂ (|∂fi/∂zj)1≤i,j≤k| = 0).

Q.E.D.

It is in fact the case that Vs is an analytic subvariety of M—if we choose local defining functions f1,…, fl for V carefully, Vs will be the common zero locus of the determinants of the k × k (f). For our purposes, however, we simply need to know that the singular locus of an analytic variety is comparatively small, and so we will not prove this stronger assertion.

We state one more result on analytic varieties:

Proposition. An analytic variety V is irreducible if and only if V* is connected.

Proof. One direction is clear: if V = V1 ∪ V2 with V1, VV analytic varieties, then ( V1 ∩ V2) ⊂ Vs, so V* is disconnected.

The converse is harder to prove in general; since we will use it only for analytic hypersurfaces, we will prove it in this case. Suppose V* is disconnected, and let {Vi} denote the connected components of Vis an analytic variety. Let p be any point, f a defining function for V near p, and z = (z1, … zn) local coordinates around p; we may assume that f is a Weierstrass polynomial of degree k in zn.

Write

then for Δ some polydisc around p n–1, the projection map π : (z1,…, zn(z1,…, zn–) expresses Vi ∩ (Δ – (g = 0)) as a covering space of Δ′ – (g = 0). Let {wv(z′)} denote the zn-coordinates of the points in π–1(z′) for z′ = (z1,…, zn–1) ∈ Δ′ – (g = 0) and let σ1(z′),…, σk(z′) denote the elementary symmetric functions of the wv. The functions σi are well-defined and bounded on Δ′ – (g = 0), and so extend to Δ′; the function

is thus holomorphic in a neighborhood of p .

Q.E.D.

We take the dimension of an irreducible analytic variety V to be the dimension of the complex manifold V*; we say that a general analytic variety is of dimension k if all of its irreducible components are.

A note: if V ⊂ M is an analytic subvariety of a complex manifold M, then we may define the tangent cone Tp(Vto V at any point p ∈ V as follows: if V = (f = 0) is an analytic hypersurface, and in terms of holomorphic coordinates z1,…, zn on M centered around p we write

with fk(z1,…, zn) a homogeneous polynomial of degree k in z1,…, zn, then the tangent cone to V at p {∂/∂zi} defined by

In general, then, the tangent cone to an analytic variety V ⊂ M at p ∈ V is taken to be the intersection of the tangent cones at p to all local analytic hypersurfaces in M containing V. In case V is smooth at p, of course, this is just the tangent space to V at p.

More geometrically, the tangent cone Tp(Vmay be realized as the union of the tangent lines at p to all analytic arcs γ : Δ → V ⊂ M.

The multiplicity of a subvariety V of dimension k in M at a point p, denoted multp(V), is taken to be the number of sheets in the projection, in a small coordinate polydisc on M around p, of V onto a generic k-dimensional polydisc; note that p is a smooth point of V if and only if multp(V) = 1. In general, if W ⊂ M is an irreducible subvariety, we define the multiplicity multW(V) of V along W to be simply the multiplicity of V at a generic point of W.

De Rham and Dolbeault Cohomology

Let M be a differentiable manifold. Let Ap(M) denote the space of differential forms of degree p on M, and Zp(M) the subspace of closed p-forms. Since d² = 0, d(Ap–1(M)) ⊂ Zp(M);, the quotient groups

of closed forms modulo exact forms are called the de Rham cohomology groups of M.

In the same way, we can let Ap(M) and Zp(M) denote the spaces of complex-valued p-forms and closed complex-valued p-forms on M, respectively, and let

be the corresponding quotient; clearly

Now let M be a complex manifold. By linear algebra, the decomposition

of the cotangent space to M at each point z ∈ M gives a decomposition

Correspondingly, we can write

where

A form φ ∈ Ap,q(M) is said to be of type (p, q). By way of notation, we denote by π(p, q) the projection maps

so that for φ ∈ A*(M),

we usually write φ(p,q) for π(p,q)φ.

If φ ∈ Ap,q(M), then for each z ∈ M,

i.e.,

We define the operators

by

accordingly, we have

In terms of local coordinates z = (z1,…, zm), a form φ ∈ An(M) is of type (p, q) if we can write

where for each multiindex I = {i1,…, ip},

are then given by

In particular, we say that a form φ of type (q, 0) is holomorphic φ = 0; clearly this is the case if and only if

with φI(z) holomorphic.

. For f : M → N a holomorphic map of complex manifolds,

and

-closed forms of type (p, q

on Ap,q(M), and we have

accordingly, we define the Dolbeault cohomology groups to be

Note in particular that if f : M → N is a holomorphic map of complex manifolds, f induces a map

n is exact assures us that the de Rham groups are locally trivial. Analogously, a fundamental fact about the Dolbeault groups is the

-Poincaré Lemma. For Δ = Δ(r) a poly cylinder in ,

Proof. First note that if

-closed form, then the forms

are again closed, and that if

then

vanish.

We first show that if φ -closed (0, q)-form on Δ = Δ(r), then for any s < r, we can find ψ ∈ A⁰,q–1(Δ(sψ = φ in Δ(s). To see this, write

we claim that if φ ≡ 0 modulo (d 1,…, d k)—that is, if φ1 ≡ 0 for I {1,…, k}—then we can find η ∈ A⁰,q–1(Δ(s)) such that

this will clearly be sufficient. So assume φ ≡ 0 modulo (d 1,…, d k) and set

, with φ2 ≡ 0 modulo (d 1,…, d k–1). If l > kφ2 contains no terms with a factor d k ∧ d lφ φφ2 = 0, it follows that

for l > k and I such that k ∈ I.

Now set

where

By the proposition on p. 5, we have

and for l > k,

Thus

in Δ(s) as was desired.

-Poincaré lemma let {ri} be a monotone increasing sequence tending to r. By the first step, we can find ψk ∈ A⁰,qψk = φ in Δ(rk, ρk a C∞ bump function ≡ 1 on Δ(rk) and having compact support in Δ(rk—the problem is to show that we can choose {ψk} so that they converge suitably on compact sets. We do this by induction on q. Suppose we have ψk as above. Take α ∈ A⁰,qα = φ in Δ(rk+1); then

and, if q ≥ 2, then by the induction hypothesis we can find β ∈ A⁰,q–2(Δ) with

Set

ψkα = φ in Δ(rk+1) and

Thus the sequence {ψk} chosen in this way converges uniformly on compact sets.

It remains to consider the case q = 1. Again, say ψk ∈ Cψk = φ in Δ(rk), α ∈ Cα = φ in Δ(rk+1); then ψk – α is a holomorphic function in Δ(rk. Truncate this series expansion to obtain a polynomial β with

and set

ψk α = φ in Δ(rk+1), ψk+1 – ψk is holomorphic in Δ(rk), and

so ψ = limψk ψ = φ.

Q.E.D.

Note that the proof works for r = ∞.

We leave it as an exercise for the reader to prove, using a similar argument with annuli and Laurent expansions, that

where Δ* is the punctured disc Δ – {0}.

Calculus on Complex Manifolds

Let M be a complex manifold of dimension n. A hermitian metric on M is given by a positive definite hermitian inner product

on the holomorphic tangent space at z for each z ∈ M, depending smoothly on z—that is, such that for local coordinates z on M the functions

are C∞. Writing ( , )z for

the hermitian metric is given by

A coframe for the hermitian metric is an n-tuple of forms (φ1,…, φn) of type (1, 0) such that

by ( , )z . From this description it is clear that coframes always exist locally: we can construct one by applying the Gram-Schmidt process to the basis (dz1,…, dznat each z.

-linear isomorphism

we see that for a hermitian metric ds² on M,

is a Riemannian metric on M, called the induced Riemannian metric of the hermitian metric. When we speak of distance, area, or volume on a complex manifold with hermitian metric, we always refer to the induced Riemannian metric.

We also see that since the quadratic form

is alternating, it represents a real differential form of degree 2; ω Im ds² is called the associated (1, 1)-form of the metric.

Explicitly, if (φ1,…, φn) is a coframe for ds², we write

where αi, βi are real differential forms; then

The induced Riemannian metric is given by

and the associated (1, 1)-form of the metric is given by

. Indeed, any real differential form ω of type (1, 1) on M gives a hermitian form H. The form H will be positive definite—i.e., will induce a hermitian metric on M—if and only if for every z ∈ M ,

Such a differential form ω is called a positive (1, 1)-form; in terms of local holomorphic coordinates z = (z1,…, zn) on M, a form ω is positive if

with H(z) = (hij(z)) a positive definite hermitian matrix for each z.

If S ⊂ M is a complex submanifold, then for z ∈ S we have a natural inclusion

consequently a hermitian metric on M induces the same on S by restriction. More generally, if f : N → M is any holomorphic map such that

is injective for all z ∈ N, a metric on M induces a metric on N by setting

Note that in this case we can always find, for U ⊂ N small, a coframe (φ1,…, φn) on f(U) ⊂ M with φk+1,…, φn ∈ Ker f, then f*φ1,…, f*φk form a coframe on U for the induced metric on N. The associated (1, 1)-form ωN on N is thus given by

i.e., the associated (1, 1)-form of the induced metric on N is the pullback of the associated (1, 1)-form of the metric on M.

Examples

given by

is called the Euclidean or standard ²n.

by

is again called the Euclidean metric .

3. Let Z0,…, Zn n+1 and denote by π nthe standard projection map. Let U be an open set and Z : U n+1 – {0} a lifting of U, i.e., a holomorphic map with π ° Z = id; consider the differential form

If Z′ : U n+1 {0} is another lifting, then

with f a nonzero holomorphic function, so that

Therefore ω is independent of the lifting chosen; since liftings always exist locally, ω . Clearly ω is of type (1,1). To see that ω is positive, first note that the unitary group U(n and leaves the form ω invariant, so that ω is positive everywhere if it is positive at one point. Now let {wi = Zi/Z0} be coordinates on the open set U0 = (Zand use the lifting Z = (1, w1,…, wn) on U0; we have

At the point [1, 0,…, 0],

Thus ω , called the Fubini-Study metric.

The Wirtinger Theorem. The interplay between the real and imaginary parts of a hermitian metric now gives us the Wirtinger theorem, which expresses another fundamental difference between Riemannian and hermitian differential geometry. Let M be a complex manifold, z = (z1,…, zn) local coordinates on M, and

a hermitian metric on M with associated (1, 1)-form ω. Write φi = αi βi; then the associated Riemannian metric on M is

and the volume element associated to Re(ds²) is given by

On the other hand, we have

so that the nth exterior power

Now let S ⊂ M be a complex submanifold of dimension d. As we have observed, the (1, 1)-form associated to the metric induced on S by ds² is just ω|S, and applying the above to the induced metric on S, we have the

Wirtinger Theorem

The fact that the volume of a complex submanifold S of the complex manifold M is expressed as the integral over S of a globally defined differential form on M is quite different from the real case. For a C∞ arc

², for example, the element of arc length is given by

².

To close this section, we discuss integration over analytic subvarieties of a complex manifold M. To begin with, we define the integral of a differential form φ on M over a possibly singular subvariety V to be the integral of φ over the smooth locus V* of V. The first thing to prove is the

Proposition. V* has finite volume in bounded regions.

Proof. Since the question is local and the volume increases by increasing the metric, it is sufficient to prove it for V with the Euclidean metric. Suppose V is of dimension k so that, in a neighborhood of 0, V meets each of the coordinate (n – k)-planes (zi1, = zi2 = · · · = zik is

and so for c /2)k(–1)k(k – ¹)/²·k!

Thus it will suffice to prove that

for I = {1,…, k}, Δ a small polydisc around the origin. But the projection map

expresses V* as a d-sheeted branched cover of Δ′ = π(Δ) and consequently

Q.E.D.

Note again the contrast to the C∞ case, where the set of manifold points of the zero locus of a smooth function — e.g., f(y) = (e–y–2 – 1) sin(1/y)—need not have locally finite area.

As a corollary of the proof, we see that for any region U ⊂ M compact and φ ∈ A),

An obvious but fundamental observation is that if V* has dimension k, Ap,q(V*) = 0 for p or q > k; consequently for any form φ,

We can now prove

Stokes’ Theorem for Analytic Varieties. For M a complex manifold, V ⊂ M an analytic subvariety of dimension k, and φ a differential form of degree 2k – 1 with compact support in M,

Proof. The question is local, i.e., it will be sufficient to show that for every p ∈ V, there exists a neighborhood U of p

For any p ∈ V, we can find a coordinate system z = (z1,…, zn) and a polycylinder Δ around p such that the projection map π: (z1,…, zn) → (z1,…, zk) expresses V ∩ Δ as a branched cover of Δ′ = π(Δ), branched over an analytic hypersurface D ⊂ Δ′. Let Tε be the ε-neighborhood of D in Δ′ and

,

Thus to prove the result, we simply have to prove that the volume of ∂π–1(Tε) → 0 as ε → 0. But ∂π–1(Tε) is a finite cover