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#### Topics in Algebraic and Analytic Geometry

#### Notes from a course of Phillip Griffiths

**By Phillip A. Griffiths, John Adams**

**Princeton University Press and University of Tokyo Press**

**Copyright © 1974 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-08151-9

All rights reserved.

ISBN: 978-0-691-08151-9

CHAPTER 1

§1 Sheaf theory, ringed spaces

On Cn, the space of n complex variables, there is a sequence of sheaves defined in the usual topology:

Ocont [contains] Odiff [contains] Ohol [contains] Oalg

where for an open set U [subset] Cn:

Γ(U, Ocont) = continuous complex-valued functions defined on U.

Γ(U, Odiff) = C∞ complex-valued functions defined on U.

Γ(U, Ohol) = holomorphic functions defined on U.

Γ(U, Oalg) = rational holomorphic functions defined on U.

(A holomorphic function φ on U is said to be rational just in case each a in U has a neighborhood W in U such that there are two polynomials p,q with q nowhere zero in W , with φ = p/q in W. In fact, because the polynomial ring C [z1, ..., zn] is a unique factorization domain, one can always take W=U.)

If W is an open set properly contained in the open set U then there is always an element of Γ(W, Ocont) which does not extend to U , and similarly for Odiff. But Ohol, and Oalg, behave differently: We will prove a little theorem about this.

I.A THEOREM (Hartog's removable singularities theorem) In case n ≥ 2 And W = U less a point, every holomorphic function on W extends to U.

We may suppose that W = U - {(0, ..., 0)}. If δ is a small positive number such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is contained in U we define for each f holomorphic in W

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for z in the interior of the polydisk. It will suffice to show that f = f1 in the interior of the polydisk less its center. But for a point inside the polydisk with z2 ≠ 0 the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is valid, so f = f1 where both are defined.

This type of behavior is more pronounced in the case of Oalg. Every holomorphic rational function on W will extend to U unless there is a polynomial with zeroes in U but no zeroes in W.

The sheaf Oalg may naturally be restricted to a coarser topology on Cn, the Zariski topology . A set in Cn is a Zariski closed set in case it is the locus of zeroes of a set of polynomials – one can always take the set of polynomials to be an ideal in the polynomial ring. The Zariski closed set associated to an ideal I will be denoted V(I) ("variety of I") . The complement of a Zariski closed set is a Zariski open set. This defines a topology on Cn, coarser than the usual topology.

Two algebraic facts about this topology are

(1) V(I)= φ just in case I=C[z1, ..., zn]

(2) V(I)= V(J) just in case rad (I)= rad (J) .

The first of these facts is called the Hubert Nullstellensatz. For a proof of this, and the deduction of (2) from (1), see Safarevic [35], or Lefschetz [20].

Henceforth when we consider the sheaf Oalg, on Cn it will usually be with respect to the Zariski topology, sometimes denoted CnZar.

We have met with four examples (Cn, Ocont), (Cn, Odiff), (Cn, Ohol) (CnZar, Oalg) of a topological space with a sheaf of rings. For any topological space X there is another example, (X, OC), where OC is the sheaf of complex-valued (not necessarily continuous) functions.

A topological space with a sheaf of rings on it, (X, O), such that O is a subsheaf of the sheaf OC, is called a ringed space.

A morphism of ringed spaces (X, OX) and (Y, OY) is a continuous map φ: X [right arrow] Y such that, for any open set U [subset] Y, and f [member of] Γ(U, OY), foφ[member of] Γ-1 (U), OX). An isomorphism of ringed spaces (X, OX) and (Y, OY) is a homeomorphism of φ [right arrow] Y such that Γ(φ-1 (U), OX) = Γ(U, OY) for all open U in Y.

If U is an open subset on the ringed space (X, OX), then OX|U makes U into a ringed space so that the natural inclusion U [right arrow] X induces a morphism of ringed spaces.

The terminology of ringed spaces provides a convenient, general way to speak of the class of distinguished functions singled out on a topological space by some special structure on that space. For example, we could define a continuous or differentiable manifold to be a Hausdorff ringed space (X, O) such that each point has an open neighborhood which is isomorphic, with its induced ringed space structure, to either (Rn, Ocont) or Rnn, Odiff). Then a morphism between the ringed spaces associated to two differentiable manifolds, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the same thing as a differentiable map in the usual sense.

We define a complex manifold as a Hausdorff ringed space , (X, O) , such that each point has a neighborhood which is isomorphic to (U, Ohol), with U an open subset in Cn. A morphism of ringed spaces between two complex manifolds is called a holomorphic mapping.

The stalk of an arbitrary point a in a ringed space (X, O) is denoted Oa. In the examples of (Cn, Ocont), (Cn, Odiff) (Cn, Ohol), (CnZar, Oalg) the stalk of a point is always a local ring. For example, the stalk of the origin in (Cn, Ohol) is the ring C < z1, ..., zn> of convergent power series in ? variables; the stalk of the origin in (CnZar, Oalg) is the localization of the polynomial ring C[z1, ..., zn] at the ideal (z1, ..., zn). We shall be dealing with such local ringed spaces almost exclusively. A ringed space is local just in case it has the properties

(1) A distinguished function which is non-zero at some point is also nonzero in some neighborhood of the point.

(2) The reciprocal of a non-zero distinguished function defined in a neighborhood of a point is also distinguished over a perhaps smaller neighborhood.

If (X, O) is a local ringed space and f [member of] Γ(X, O) then {x [member of] X:f(x)=0} is closed in X.

The chief motivation for introducing the terminology of ringed spaces is in the study of analytic and algebraic sets. An algebraic set in Cn is just a Zariski closed set. An analytic set in Cn is characterized by being locally the locus of zeroes of some finite number of holomorphic functions-that is, there is a covering {Ui} of Cn such that the intersection of the set with Ui is {x [member of] Ui : fij (x) = 0, j = 1, ..., N} for some fij [member of] Γ(Ui Ohol). If X is an (resp. analytic) set in Cn, one can form a sheaf of ideals on the ringed space (CnZar, Oalg) (resp. (Cn, Ohol)) by IX(U) = {f [member of] Γ(U, O): f = 0 on X [intersection]U}. The quotient sheaf O/IX will be a sheaf of rings supported on the closed subspace X. It will be a subsheaf of the sheaf of complex functions, so this makes X into a ringed space. Thus any algebraic (resp. analytic) subset of Cn has a natural ringed space structure - in fact a local ringed space structure, since it arises by taking a quotient from a sheaf with local rings for stalks.

The same procedure can be followed to define analytic subsets of open sets in Cn, and to show how any such analytic set has a natural ringed space structure. We could also do this in the algebraic case, but in this case we would get nothing new, as we shall see later.

An algebraic variety is by definition a ringed space which is locally isomorphic to the ringed space defined by an algebraic subset of Cn. An analytic space, or analytic variety, is a ringed space which is locally isomorphic to the ringed space defined by an analytic subset of an open set in Cn. A morphism of two algebraic varieties (or holomorphic rational mapping ) is a ringed space morphism; similarly, for the morphisms of analytic spaces, which are called holomorphic mappings.

When working with analytic spaces it is usually helpful to require them to be Hausdorff spaces if possible. There is an analogous condition which we can put on algebraic varieties . First note that, given two algebraic varieties (or analytic spaces) X and Y, one can form their product as an algebraic variety (or analytic space), X × Y : If X and Y were algebraic subsets of Cn (D this construction would be clear, and the general case reduces to this and patching. In the analytic case one reduces to X, Y analytic subsets of open sets in Cn. For a proof in the algebraic case, see Serre [27]. For an analytic space X, the condition of Hausdorffness is equivalent to the diagonal map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being closed. If Y is an algebraic variety one says that Y is separated just in case the diagonal map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is closed (in the Zariski topology on Y × Y, which is not the product topology).

Algebraic subvarieties of algebraic varieties, and analytic subsets of analytic spaces are defined by analogy with Cn, and given ringed space structure. Any algebraic subvariety of a separated algebraic variety is itself separated, just as any subspace of a Hausdorff space is Hausdorff. CnZar is separated. From now on we will assume that all our analytic spaces are Haudorff and all our algebraic varieties are separated.

We will consider some examples of analytic and algebraic subsets of C2. A polynomial in two variables defines an algebraic subset of C2, V(p(z1, z2))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A pair of polynomials in two variables, f(z1, z2) will, providing that neither is constant and they have no common factor, define a finite set of points. Any algebraic subset of C2 is either like the first example or the second, or a union of both. See Safarevic [35] or Lefschetz [20].

Let X be an algebraic subset of C2. We propose to describe the sheaf of ideals IX on (C2Zar, Oalg). For any f [member of] C[z1, z2] we denote the Zariski open set by C2 by D(f). Then Γ(D(i), Oalg) = C[z1, z2]f = C[z1, z2, z3]/(1-z3 f). Now an element g/fn Γ(D)(f)) will vanish on X just in case g vanishes on X, and this implies that IX(D)(f)) = IX (C2)[GAMMA(D)(f)). Since the sets D(f) form a basis for the Zariski topology on C2, we find that the sheaf of ideals IX is generated over Oalg by its global sections.

As this discussion suggests, the algebraic subsets of the Zariski open subsets of Cn are exactly the Zariski closed subsets, that is, the closed subsets inherited from the Zariski topology on Cn. For if Cn - V(I)=U is a Zariski open then Γ(U, Oalg) = C[z1, ..., zn] localized by the multiplicative subset I. If we define an algebraic subset of U to be the locus of zeroes of an ideal in Γ(U, Oalg), then there will be an ideal J [subset] C [z1, ..., zn] such that I = JΓ(U, Oalg), so V(I)= V(J) [intersection] U. This also shows that a subset of Cn which is locally given (in Zariski opens) as the zeroes of rational functions is a globally defined algebraic set.

Another point to notice is that, for any polynomial f [member of] C[z1, ..., z2], D(f) = CnZar - V(I), is isomorphic, as ringed space, to a closed algebraic subset of Cn+1: It will be isomorphic to V(1 - zn+1 f). This shows that any open subset of an algebraic variety is an algebraic variety.

As an example of an analytic subset of C2, consider the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This set will not be algebraic, and as a ringed space will be isomorphic to (C, Ohol).

§ 2 Local structure of analytic and algebraic sets

A discussion of the local properties of analytic subsets of open sets in requires some preparation. We shall mostly limit our discussion to the case of analytic varieties defined by a single equation. What we need is the

I. B THEOREM (Weierstrass preparation and division theorems)

Let f/n be a holomorphic function defined in a neighborhood of the closed polydisk Δn(r). Suppose that the function of one variable defined by y(zn) = f(0, ..., 0, zn) in a neighborhood of Δ1(r) is not identically zero and has a zero of order p at 0. Suppose also that f(z1 ..., zn) is never zero when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then

(1) There is a holomorphic function u which is a unit in Δn(r), such that both it and its inverse are bounded on the polydisk, and there are bounded holomorphic functions

a0 (z1, ..., zn-1), ..., ap-1 (z1, ..., zn-1)

on Δn-1(r) such that ai(0, ..., 0) = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2) Given φ holomorphic on Δn(r) there is a function a(z1, ..., zn) holomorphic on Δn(r) and functions b0 b(z1, ..., zn-1), ..., bp-1 (z1, ..., zn-1) holomorphic on Δn(r) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

on Δn(r). There is a constant M, independent of φ, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1) As (z1, ..., zn-1) varies in Δn-1(r) we get a holomorphically varying family of zn holomorphic functions of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], each defined in a neighborhood of Δ1(R)We'll use the notation (z' zn) for a point of Δn(r) = Δn-1(r) × [??](r). Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

= number of zeroes of f(z', zn) in |z| ≤ r

= p identically, by continuity.

Denote by t1 (z'), ..., tp (z') the aforementioned zeroes. The symmetric function σk(z') = (t1(z'))k + ... + (tp(z'))k admits the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and is thus a holomorphic function of z'. The rth elementary symmetric function of the tj(z)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be represented as a polynomial with rational coefficients in the σk (z') hence is holomorphic in z'. Setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it remains to prove that f/π is holomorphic on Δπ(r). Note that f/π is defined and non-vanishing, because for each z' f(z', zn) and π(z', zn) are holomorphic functions of zn with the same zeroes. Note also that we could have done all this on a slightly larger polydisk, so we can assume that f/π is defined on a neighborhood of Δn(r). Then we have the representation in Δn(r)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which shows that f/π is holomorphic.

Since everything could have been done on a slightly larger polycylinder, u and the ai's must be bounded.

*(Continues...)*

Excerpted fromTopics in Algebraic and Analytic GeometrybyPhillip A. Griffiths, John Adams. Copyright © 1974 Princeton University Press. Excerpted by permission of Princeton University Press and University of Tokyo Press.

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