Part 3

Algebraic Methods

CHAPTER 14

Coherent sheaves on Projective Space

In this chapter, we develop some algebraic tools for studying sheaves on pro-jective spaces. In the last section, we compare the algebraic and analytic points ofview. The ideas in this chapter were introduced by Serre in two landmark papers,usually abreviated FAC [S1] and GAGA [S2].

14.1. Cohomology of line bundles on Pn

Let k be a field, and let (Pnk ,OPn) denote projective space over k viewed as

an algebraic variety over k 1.3 (this includes the case of k = C). Then Pnk has a

covering by n + 1 open affine sets Ui = xi 6= 0, where xi are the homogenouscoordinates. Ui can be identified with affine n-space with coordinates

x0

xi

, . . .xi

xi

. . .xn

xi

In particular, O(Ui) is the polynomial ring in these variables. Our goal is to computecohomology of OPn(i) using the Cech complex with respect to this covering. But weneed to show that this a Leray covering. Since the intersection of a finite numberof the Ui’s is affine, this will follow from:

Theorem 14.1.1 (Serre). If X is an affine variety, then

Hi(X,OX) = 0

for all i > 0.

This will be proven in a more general form in section 16.2.

Theorem 14.1.2 (Serre). Let Si be the space of homogeneous degree i polyno-mials in the variables x0, . . . xn, then

a) H0(Pn,OP(d)) ∼= Sd (in particular it is 0 if d < 0).b) Hi(Pn,OP(d)) = 0 if 0 6= i 6= n.c) Hn(Pn,OP(d)) ∼= S−d−n−1.

Proof. A complete proof can be found in [Har, III.5], from a somewhat moregeneral point of view. We use Cech cohomology with respect to the covering Ui.Our first task is to identify this with a complex of polynomials. Let S = k[x0, . . . xn].All localizations are subsets of the field K = k(x0, . . . xn), so they can be compared.The localization

S

[1

xi

]=

⊕

d

S

[1

xi

]

d

has a natural Z-grading, where the dth component is

S

[1

xi

]

d

=∑

e

Sd+e

xei

.

141

142 14. COHERENT SHEAVES ON PROJECTIVE SPACE

Note the that degree 0 piece of S[1/xi] is exactly k[x0/xi, . . . xn/xi] = O(Ui).Similar statements apply to localizations S[1/xi1xi2 . . .] along monomials.

We start with the description of O(d) given in section 6.3, where we identify asection of O(d)(U) with collection of functions fi ∈ O(U ∩ Ui) satisfying

fi = (xj/xi)dfj

Formally cross multiplying yields

(25) xdi fi = xd

jfj

It will be more convenient, to identify O(d)(Ui) = S[1/xi]d, by sending fi to xdi fi.

Equation 25 then simply says these elements agree in K. By carrying out simi-lar identifications for elements of O(d)(Ui ∩ Uj) etcetera, we can realize the Cechcomplex for O(d) as the degree d piece of the complex

(26)⊕

i

S

[1

xi

]→

⊕

i<j

S

[1

xixj

]→ . . .

where the differentials are defined as alternating sums of the inclusions along thelines of section 6.3.

Notice that Hi(Pn,O(d)) is automatically zero when i > n because the Cechcomplex has length n, this takes care of part of (b). An element of H0(Pn,O(d))can be represented by a (n + 1)-tuple (p0/xe

0, p1/xe1, . . .) of rational expressions of

degree d, where the pi’s are polynomials, satisfyingp0

xe0

=p1

xe1

=p2

xe2

. . .

This forces the polynomials pi to be divisible by xei . Thus the H0(Pn,O(d)) can be

identified with Sd as claimed in (a).⊕d Hn(Pn,O(d)) is S[ 1

x0...xn] modulo the the space of coboundaries B, i e. the

image of the previous term in (26). S[ 1x0...xn

] is spanned by monomials xi00 . . . xin

n

with arbitrary integer exponents. The image B is the space spanned by those mono-mials where at least one of the exponents is nonnegative. Therefore the quotientcan be identified with the complementary submodule spanned by monomials withnegative exponents. In particular

Hn(Pn,O(d)) ∼=⊕

i0+...in=d;i1,...in<0

kxi00 . . . xin

n .

This is isomorphic to S−d−n−1 via

xi00 . . . xin

n 7→ x−i0−10 . . . x−in−1

n

This proves (c).It remains to finish the proof (b). We will be content to work this out for

n = 2. The general case can be found in [Har, pp 227-228]. Our treatment willbe elementary but a little ad hoc. A 1-cocycle is given by a triple pij/(xixj)

e ofrational expressions of degree d satisfying

p02

(x0x2)e=

p01

(x0x1)e+

p12

(x1x2)e

or

(27) xe1p02 = xe

2p01 + xe0p12

14.2. COHERENCE 143

To show that ⊕H1(P2,O(d)) vanishes, it suffices to find polynomials qi sat-isfying

pij

(xixj)e=

qi

xei

−qj

xej

or

pij = xejqi − xe

i qj

(27) implies that p02 lies in the ideal (xe0, x

e2). Similarly p01 ∈ (xe

0, xe1) and

p12 ∈ (xe1, x

e2). Therefore, we can find polynomials such that

p01 = xe1q0 − xe

0q1

p12 = xe2q

′1 − xe

1q′2

Substituting back into (27) shows that q1 − q′1 = xe1s for some polynomial s. Now

setting q2 = q′2 + xe2s will do the job. ¤

Exercises

1. Compute Hi(P1,OP1(d)) directly using Mayer-Vietoris with respect to U0, U1without refering to above arguments.

2. Generalize the above argument to prove that H1(Pn,OPn(d)) = 0.

14.2. Coherence

Let (X,R) be a ringed space. We need to isolate a class of R-modules withgood finiteness properties.

Definition 14.2.1. Given a ringed space (X,R), an R-module E is coherentif

1. E is locally finitely generated, i. e., each point has a neighbourhood U suchthat R|nU surjects onto E|U for some n < ∞.

2. If R|nU → E|U is a surjection, the kernel is finitely generated.

Standard properties of coherent sheaves on ringed spaces can be found in [EGA](the definition given in [Har] is only valid for noetherian schemes).

Proposition 14.2.2. Given a short exact sequence of R-modules

0 → E1 → E2 → E3 → 0

If any two of the Ei’s are coherent, then so is the third. The tensor product, andsheaf Hom of two coherent modules is coherent.

Proof. [EGA, chap 0, 5.3]. ¤

Corollary 14.2.3. The collection of coherent R-modules and morphisms be-tween them forms an Abelian category.

The structure sheaf R need not be coherent in general. However, it is for anoetherian scheme.

Example 14.2.4. If M is a finitely generated module over a noetherian ringR, then M is coherent on SpecR.

144 14. COHERENT SHEAVES ON PROJECTIVE SPACE

Proposition 14.2.5. All coherent sheaves on SpecR arise as above. Moreprecisely, the functor M → M gives an equivalence between the category of finitelygenerated R-modules and the category of coherent sheaves on SpecR. The inversefunctor is the global section functor M 7→ Γ(SpecR,M).

Proof. [Har, p. 113]. ¤

In the analytic case, we have:

Theorem 14.2.6 (Oka). If (X,OX) is a complex manifold, then OX is coher-ent.

Proof. [GR, p. 59] ¤

Corollary 14.2.7. Any locally free OX-module is coherent. The ideal sheafof submanifold Y ⊂ X is coherent.

For algebraic varieties, we want to be able pass back and forth between thelanguages of Serre varieties and Grothendieck’s schemes. Let X be an algebraic va-riety (in Serre’s sense) over an algebraically closed field k. Consider the embeddingof ι : X → Xsch into a scheme given in theorem 2.4.5.

Lemma 14.2.8. The functor M 7→ ι∗M induces an equivalence of categoriesbetween the categories of coherent sheaves over Xsch and X.

From this point on, we are going drop the distinction between X and Xsch. Acoherent submodule I ⊂ OX is called a coherent ideal sheaf. Define

V (I) = x ∈ X | (OX/I)x 6= 0

Lemma 14.2.9. The set V (I) is closed, and (V (I), (OX/I)|V (I)) is a scheme.

We call (V (I), (OX/I)|V (I)) the closed subscheme defined by I.We state two basic results about the cohomology of coherent sheaves in the

analytic setting. A complex manifold is called Stein1 if it can be embedded intosome CN . For example, any nonsingular complex affine variety is Stein.

Theorem 14.2.10 (Cartan’s theorem B). Let E be a coherent sheaf on the Steinmanifold then Hi(X, E) = 0 for all i > 0.

Proof. [Hrm, 7.4.3] ¤

Theorem 14.2.11 (Cartan-Serre). If E is a coherent sheaf on a compact com-plex manifold X then the cohomology groups Hi(X, E) are finite dimensional.

Proof. See [GR] for the general case. When E is a vector bundle, a proof canbe given using Hodge theory [GH, W]. ¤

The analogous statements in the algebraic category will be proved later.

1 This definition is nonstandard but equivalent to the usual ones, see [Hrm, 5.1.5, 5.3.9].

14.3. COHERENT SHEAVES ON Pn 145

14.3. Coherent Sheaves on Pn

Our goal, in this section, is to describe all coherent sheaves on Pn = Pnk . Let

S = k[x0, . . . xn] = ⊕Si be as in section 14.1. Let π : An+1 − 0 → Pnk be the

canonical the projection.Let M be a Z-graded S-module, which means that in addition to being an

S-module, it possessess a decomposition M = ⊕i∈ZMi into subspaces such thatSiMj ⊂ Mi+j . If M is also finitely generated as an S-module, then each of thecomponent Mi is finite dimensional, and Mi = 0 for i << 0. Starting with afinitely generated graded module M , we can produce a coherent sheaf M on An+1,restrict it to An+1 − 0, and push this down to Pn to get an OPn-module M =

π∗(M |An+1−0). This will not be coherent, but it will a sum of coherent sheaves

M = ⊕dM(d) that will be defined shortly.The basic open sets for the Zariski topology are of the form π(D(f)) with f

a homogenous polynomial. If f ∈ Sa, then M(D(f)) = M [1/f ]. has a Z-grading,where elements of degree d ∈ Z, M [1/f ]d, are generated by the fractions m/f i withm ∈ Md+ai. This can be applied to S to yield a grading on the ring S[1/f ], andthe degree 0 component is a subring. For example,

S

[1

xi

]

0

= k

[x0

xi

, . . .xn

xi

]

is the coordinate ring of Ui This is not a coincidence:

Lemma 14.3.1. OPn(π(D(f))) ∼= S[1/f ]0.

Proof. We prove one direction. An element of S[1/f ]0 is a linear combinationof fractions g/f i with g homogenous of the same degree as the denominator. Inparticular, the function is

[a0, . . . an] 7→g(a0, . . . an)

f(a0, . . . an)i

is well defined and regular on π(D(f)). ¤

For M as above, M [1/f ] is a graded S[1/f ]-module. In particular, it followsthat each M [1/f ]d is an S[1/f ]0-module.

Lemma 14.3.2. There is an OPn-module M(d) such that

M(d)(π(D(f))) = M [1/f ]d

and

M =⊕

d∈Z

M(d)

Proof. We define M(d)(U) to be the submodule of M(U) consisting of sec-tions m such that its restriction lies in M [1/f ]d for every π(D(f)) ⊂ U . ¤

A homomorphism of graded S-modules f : M → N is a module homomorphism

which preserves the grading i.e. f(Mi) ⊆ Ni. The construction M(d) is clearly

functorial for such maps. We define M = M(0). Given a graded module M , we

146 14. COHERENT SHEAVES ON PROJECTIVE SPACE

define M(d) to be M with the shifted grading (M(d))i = Mi+d Given two gradedmodules M,N , let us say that they are stably isomorphic or M ∼ N if and only if

⊕

i≥i0

Mi∼=

⊕

i≥i0

Ni

as graded S-modules for some i0.Now let us go in reverse. Suppose that M is a coherent OP-module. Define

Γ∗(M) = Γ(An+1 − 0, π∗M).

This is a module over the ring Γ∗(OPn) which is given by:

Lemma 14.3.3. Γ∗(OPn) = S

Proof. Γ∗(OPn) is the ring of regular functions on An+1−0. Such a functioncan be represented by ratio of polynomials f/g such that g(a) 6= 0 for all a ∈An+1 − 0. This forces g to be a nonzero constant. ¤

We need a grading on this module. For this, we have to consider the action ofthe group k∗. Given t ∈ k∗, let µt denote the automorphism of An+1 −0 inducedby scalar multiplication by t. Since π µt = π, given σ ∈ Γ∗(M), we can identify

µ∗t σ ∈ Γ(µ∗

t π∗M) = Γ∗(M)

When M = OPn , the action is given explicitly by µ∗t (f(x0, . . . xn)) = f(tx0, . . . txn).

The grading on S can be recovered from this, since Si is precisely the space ofpolynomials on which each t ∈ k∗ acts by ti. In general, the ith component Γi(M)can be defined similarly as the (simultaneous) eigenspace of the µ∗

t with eigenvalueti. In particular, Γ0(M) consists of the space of k∗-invariant sections of π∗M, andthis can be identified with Γ(M). An similar interpretation for Γi(M) is given inthe exericses.

We summarize the main properties.

Theorem 14.3.4.1. Given a finitely generated graded S-module M , the OPn-modules M(d) are

coherent.2. S(d) ∼= OPn(d).

3. M(d) ∼= M(d).

4. The operations M → M(d) are exact functors.

5. Γ∗(M) ∼ M

6. Γ∗(M) ∼= M.

Proof. A proof can be found in [Har, II,5] using the description of Γ∗ givenin the exercises. ¤

Corollary 14.3.5. Every coherent OPn-module is isomorphic to M for somefinitely generated graded S-module M .

It is not always true that Γ∗(M) ∼= M as the following example shows.

Example 14.3.6. Let M be a graded module which is finite dimensional over

k, so Mi = 0 for i >> 0. Then M = 0.

Example 14.3.7. Let I ⊂ S be a homogenous ideal. Then I defines a closedsubscheme of Pn which is a subvariety if I is prime.

14.4. COHOMOLOGY OF COHERENT SHEAVES 147

The scheme above depends only the graded ring S/I and is usually denote byProj S/I. See [Har] for further details.

Theorem 14.3.8. Any coherent sheaf E on Pn fits into an exact sequence

0 → Er → En−1 → . . . E0 → E → 0

where r ≤ n and each Ei is a sum of a finite number of line bundles OPn(j).

Proof. This is a consequence of the Hilbert syzygy theorem [E, 1.13] whichsays that any finitely generated graded S-module has a finite free graded resolutionof length at most n. ¤

For many standard examples, it is possible to construct explicit resolutions ofthis type.

Example 14.3.9. The tangent sheaf fits into an exact sequence

0 → OPn

0

B

B

@

x0

x1

. . .

1

C

C

A

−→ OPn(1)n+1 → TPn → 0

This will be justified later.

Exercises

1. Let M(i) = M [1/xi]0 viewed as an S[1/xi]0-module. Check that M(i) is

isomorphic to the restriction of M to Ui. Prove coherence of M .2. Check that example 14.3.6 works as stated.3. Given a pair of graded S-modules M and N , define a grading on their

tensor product M ⊗ N = M ⊗S N by

(M ⊗ N)i =∑

a+b=i

Ma ⊗ Nb

Prove that(a) M ⊗ S(d) = M(d).(b) (M ⊗ N)(i) ∼= M(i) ⊗S(i)

N(i)

(c) M ⊗ N ∼= M ⊗OPn N4. Given a coherent OPn-module M, define M(d) = M⊗O(d). Check that

Γd(M) = Γ(M(d)) (hint: µ∗t (f ⊗ g) = µ∗

t (f) ⊗ µ∗t (g)). Thus Γ∗(M) =

⊕dΓ(M(d)).

14.4. Cohomology of coherent sheaves

Given a coherent sheaf, E on Pn, let E(d) = E ⊗ O(d). This is again coherent,

and consistent with the notation M(i) by the previous exercises.

Theorem 14.4.1. If E is a coherent sheaf on Pn, then Hi(Pn, E) is finite di-mensional for each i, and is zero for i > n. Furthermore Hi(Pn, E(d)) = 0 ford >> 0 and i > 0.

148 14. COHERENT SHEAVES ON PROJECTIVE SPACE

Proof. We prove this by induction on r, where r is the length of the shortestresolution given by theorem 14.3.8. If r = 0, we are done by theorem 14.1.2.Suppose, we know the theorem for all r′ < r. Choose a resolution

0 → Er → En−1 → . . . E0 → E → 0

and let E ′ = ker[E0 → E ]. We have exact sequences

0 → Er → En−1 → . . . E1 → E ′ → 0

and

0 → E ′ → E0 → E → 0

The first sequence implies that H∗(Pn, E ′) is finite dimensional by the inductionhypothesis. The second sequences yields a long exact sequence

. . . Hi(Pn, E0) → Hi(Pn, E) → Hi+1(Pn, E ′) . . .

which finishes the proof. The vanishing of Hi(Pn, E(d)) for i > n and for d >> 0and can be proved by induction in similar manner. Details are left as exercise. ¤

The following gives an analogue to theorem 14.2.11.

Corollary 14.4.2. If E is a coherent sheaf on a projective variety X, then thecohomology groups Hi(X, E) are finite dimensional.

The Euler characteristic of a sheaf F on a space X is

χ(F) =∑

(−1)i dimHi(X,F)

provided that the sum is finite. The advantage of working with the Euler charac-teristic is the following:

Lemma 14.4.3. If

0 → F1 → F2 → F3 → 0

is an exact sequence,

χ(F2) = χ(F1) + χ(F3)

Theorem 14.4.4. If E is a coherent sheaf on Pn, then i → χ(E(i)) is a polyno-mial in i. If X is a subvariety of Pn

k and E = OX , then this polynomial has degreedimX.

Proof. From theorem 14.1.2,

χ(OPn(i)) =

dim Si if i ≥ 0

(−1)n dimS−d−n−1 otherwise

=

(n + i

n

),

which is a polynomial of degree n. This implies that χ(E(i)) is degree n polynomialwhen E is sum of line bundles OPn(j). In general, theorem 14.3.8 implies that forany coherent E

χ(E(i)) =

r∑

j=0

(−1)jχ(Ej(i))

where Ej are sums of line bundles. This shows χ(E(i)) is degree n polynomial ingeneral.

14.5. GAGA 149

For the second, we use induction on dimX and the relation

χ(OH∩X(i)) = χ(OX(i)) − χ(OX(i − 1))

which follows from the sequence

0 → OX(−1) → OX → OH∩X → 0,

where H is a general hyperplane. ¤

χ(E(i)) is called the Hilbert polynomial of E . It has a elementary algebraicinterpretation

Corollary 14.4.5 (Hilbert). Let M be a finitely generated graded S-module,then for i >> 0, dimMi is given by a polynomial, and this coincides with the Hilbertpolynomial χ(M(i)).

Exercises

1. Let f be a homogeneous polynomial of degree d in S = k[x0, x1, x2]. Thencorresponding to the exact sequence of graded modules

0 → S(−d) ∼= Sf → S → S/(f) → 0

there is an exact sequence of sheaves

0 → OP2(−d) → OP2 → OX → 0.

Prove that

dimH1(X,OX) =(d − 1)(d − 2)

2

2. Let f be a homogeneous polynomial of degree d in 4 variables. Repeat theabove for the first and second cohomology groups.

3. Prove the remaining parts of theorem 14.4.1.

14.5. GAGA

Let k = C throughout this section.Let (Pn,OPn) denote complex projective space as an algebraic variety, and

(Pnan,OPn

an) projective space as a complex manifold. In other words, Pn (resp. Pn

an)is endowed with the Zariski (strong) topology and OPn (resp. OPn

an) is the sheaf of

algebraic (resp. holomorphic) functions. We have a morphism of ringed spaces

ι : (Pnan,OPan

) → (Pn,OP)

Given an OP-module E , define the OPan-module Ean = ι∗E . Ean is coherent if E

is. This operation gives a functor between the categories of coherent OP and OPan

modules. We have the following fact, which we will need below:

Lemma 14.5.1. This functor is exact. If E 6= 0, then Ean 6= 0.

Proof. The point is that OPnan,x is a faithfully flat extension of OPn,x [S2].

This implies both statements since on the stalk level, we have Ean,x = Ex ⊗OPnanx.

¤

150 14. COHERENT SHEAVES ON PROJECTIVE SPACE

Let us compare analytic and algebraic line bundles on P1. Since Hi(P1an,OP1

an) =

0 for i > 0 by the Hodge theorem for example, the exponential sequence yields

H1(P1,O∗P1

an) ∼= H2(P1

an, Z) = Z

Thus the set of isomorphism classes of analytic line bundles coincide with the setOP1(d)an of isomorphism classes of algebraic line bundles. Let us compute theircohomology.

Cartan’s theorem B (theorem 14.2.10) shows that U0, U1 is a Leray coveringfor OP1(d)an (of course, there are easier ways to see this). Set z = x1/x0 andidentify this with a trivializing section OP1(d)an(U0) (i.e. the image of 1 under andisomorphism OP1

an|U0

∼= OP1(d)an|U0.) Then zd extends to a trivializing section of

OP1(d)an(U1). Thus the Cech complex can be identified with a complex

∞∑

n=0

anzn

⊕

∞∑

n=0

bnz−n

0

@

1−zd

1

A

−→

∞∑

n=−∞

cnzn

where the above series converge in U0, U1 and U0 ∩ U1 respectively. From this itfollows easily that H0(OP1(d)an) can be identified with the space of polynomialsof degree d in z, and H1(OP1(d)an) with the space spanned by z−d+1, . . . z−1.Therefore

Hi(OP1(d)an) ∼= Hi(OP1(d))

We have succeed in proving the following “GAGA” theorem in a special case.

Theorem 14.5.2 (Serre). There is an isomorphism

Hi(Pn, E) ∼= Hi(Pnan, Ean)

Proof. We outline the proof, a complete argument can be found in [S2]. Thekey steps are as follows:

Step 1. The isomorphism holds for OPn :From theorem 14.1.2, we see that Hi(Pn,O) = 0 if i > 0 and C if i = 0.

From 6.2, it follows that Hi(Pn, C) is zero if i is odd and generated by theclass of linear subspace otherwise. It follows that the Hodge structure onHi(Pn

an, C) is of type (i/2, i/2) when i is even. Therefore Hi(Pnan,OPn

an)

which is the (0, i) part of Hi(Pnan, C) is zero if i > 0. The same reasoning

shows that H0(Pan,OPnan

) = H0(Pan, C) = C.Step 2. The isomorphism holds for OPn(d):

Let H ⊂ Pn be a hyperplane. By induction on n, we can assume thatthe result holds for H. Then we have an exact sequence

0 → OPn(d − 1) → OPn(d) → OH(d) → 0

which yields a diagram

. . . Hi(OP(d − 1)) → Hi(OP(d)) → Hi(OH(d)) . . .↓ ↓ ↓

Hi(OPan(d − 1)) → Hi(OPan

(d)) → Hi(OHan(d))

By the induction assumption and the five lemma, we see that the isomor-phism holds for OPn(d) if and only if it holds for OPn(d − 1). Combiningthis with step 1 finishes the proof.

14.5. GAGA 151

Step 3. The isomorphism holds for any coherent E :We do this by descending induction on i. When i > n, Hi(Pn, E) = 0

by theorem 14.4.1. For the analogous result on the analytic side, we need toappeal to theorem 14.2.10, which implies that the standard covering Ujof Pn by affine spaces is a Leray covering with respect to E . This impliesthe vanishing of Hi(Pn

an, Ean) for i > n, since the Cech complex for Ujhas length n + 1.

We have an exact sequence

0 → R → F → E → 0

where R is coherent and F is a direct sum of line bundles by theorem 14.3.8.The previous step, induction and the five lemma will finish the argument.

¤

The GAGA theorem can fail for nonprojective varieties. For example, H0(OCnan

)is the space of holomorphic functions on Cn which is much bigger than the spaceH0(OCn) of polynomials.

Let X ⊆ Pn be a nonsingular subvariety. Then the sheaf of algebraic p-formsΩp

X can be viewed as a coherent OP-module. Then (ΩpX)an is the sheaf of holomor-

phic p-forms. In particular, (OX)an is the sheaf of holomorphic functions of theassociated complex manifold Xan. Thus:

Corollary 14.5.3. dimHq(X,ΩpX) coincides with the Hodge number hpq of

the manifold Xan.

There is another part to the GAGA theorems which is even more surprising.

Theorem 14.5.4 (Serre). The functor E 7→ Ean induces an equivalence be-tween the categories of coherent OPn and OPn

anmodules. In particular, any analytic

coherent sheaf arises from an algebraic sheaf.

Proof of theorem 14.5.4. We outline the steps. We will refer to a coher-ent OPn-module (respectively OPn

an-module) as an algebraic (respectively analytic)

sheaf.

Step 1. Given a coherent analytic sheaf E on Pn, there exists a constant d0 suchthat E(d) is generated by global sections for d ≥ d0.

The proof goes by induction on n. Thus we can assume that thisstatement (and step 2 which will follow from this) is valid for hyperplane.Given x ∈ Pn, choose a hyperplane H passing through x. Consider thesequence

0 → E(d − 1) → E(d) → E|H(d) → 0

By induction, H1(E|H(d)) = 0 for d ≥ d1. Therefore, for d ≥ d1 we get asequence of surjections

H1(E(d − 1)) → H1(E(d)) → H1(E(d + 1)) . . .

Since these spaces are finite dimensional (theorem 14.2.11), these mapsmust stabilize to isomorphisms for all d ≥ d2 ≥ d1. Thus

H0(E(d)) → H0(E|H(d))

is surjective for d ≥ d2. By induction, we can assume that E|H(d) has isglobally generated for d ≥ d0 ≥ d2. Thus E(d) will have a section whichwill not vanish at x.

152 14. COHERENT SHEAVES ON PROJECTIVE SPACE

Step 2. Given a coherent analytic sheaf E on Pn, there exists a constant d0 suchthat Hi(Pn, E(d)) = 0 for d ≥ d0.

Arguing as in the proof of theorem 14.5.2, we see that Hi(Pn, E(d)) = 0for i > n and any d. From step 1, we have an exact sequence

0 → R → F → E → 0

where R is coherent and F = H0(E(a)) ⊗C O(−a). In particular, F is is adirect sum of line bundles, so by theorems 14.5.2 and 14.1.2,

Hi(E(d)) ∼= Hi+1(R(d))

Thus this step follows by descending on i.Step 3. If E ,F are algebraic, then Hom(E ,F) ∼= Hom(Ean,Fan).

The left and right hand sides are the spaces of global sections ofHom(F , E) and

Hom(F , E)an = Hom(Fan, Ean)

respectively. These spaces are isomorphic by theorem 14.5.2.Step 4. Any analytic sheaf E arises from an algebraic sheaf.

By applying step 1 twice, we see that E fits into an exact sequence

F1f→ F2 → E → 0

where Fi are direct sums of line bundles. In particular, the Fi are algebraic.It follows from the previous step that f is a morphism of the underlyingalgebraic sheaves. Consequently E is the cokernel of an algebraic map, andhence also algebraic.

¤

Corollary 14.5.5 (Chow’s theorem). Every complex submanifold of Pn is anonsingular projective algebraic subvariety.

Proof. The holomorphic ideal sheaf J of a submanifold X of Pn is a coher-ent OPn

an-module. Thus J = Ian, for some coherent submodule I ⊂ OPn . The

subvariety defined by I is easily seen to coincide with X. ¤

For example, this implies the assertion we made some time ago that all compactRiemman surfaces are algebraic curves.

Corollary 14.5.6. A holomorphic map between nonsingular projective alge-braic is a morphism of varieties.

Proof. Apply Chow’s theorem to the graph of the map. ¤

Exercises

1. If X is a smooth projective variety, show that H1(X,O∗X) ∼= H1(Xan,O∗

Xan).

CHAPTER 15

Computation of some Hodge numbers

The GAGA theorem 14.5.2 allows us compute Hodge numbers by working inthe algebraic setting. We look at a few examples.

15.1. Hodge numbers of Pn

Let S = k[x0, . . . xn] and P = Pnk for some field k. We first need to determine

the cotangent sheaf.

Proposition 15.1.1. There is an exact sequence

0 → Ω1P → OP(−1)n+1 → OP → 0

Proof. Let us start with Ω1P

and construct the graded S-module

Ω = Γ∗(Ω1P) = Γ(An+1 − 0, π∗Ω1

P),

where π : An+1 −0 → P is the projection. This can be realized as the submoduleof the cotangent module ΩS = ⊕Sdxi

∼= Sn+1 of S consisting of those forms whichannhilate the tangent spaces of the fibers of π. These tangent spaces are generatedby the Euler vector field

∑xi

∂∂xi

. Thus a 1-form∑

fidxi lies in Ω if and only if it∑fixi = 0.Next, we have to check the gradings. ΩS has a natural grading, such that dxi

lie in degree 0. Ω also has a grading such that the degree 0 elements of Ω[1/xi]correspond to sections in Ω1

P(Ui). These are generated by

d

(xj

xi

)=

xidxj − xjdxi

x2i

Thus the gradings on ΩS and Ω are off by a shift of one.To conclude, we have an exact sequence of graded modules

(28) 0 → Ω → ΩS(−1) → m → 0

where m = (x0, . . . xn) and the first map sends dxi to xi. Since m ∼ S, (28) impliesthe result. ¤

Dualizing yields 14.3.9.

Proposition 15.1.2. Given an exact sequence of locally free sheaves

0 → A → B → C → 0

If A has rank one, then

0 → A ⊗ ∧p−1C → ∧pB → ∧pC → 0

is exact for any p. If C has rank one, then

0 → ∧pA → ∧pB → ∧p−1A ⊗ C → 0

153

154 15. COMPUTATION OF SOME HODGE NUMBERS

is exact for any p.

Proof. See [Har, II, 5.16]. ¤

Corollary 15.1.3. There is an exact sequence

0 → ΩpP→ OP(−p)(

n+1p ) → Ωp−1

P→ 0

Proof. This follows from the above proposition and proposition 15.1.1, to-gether with the isomorphism

∧p[OP(−1)n+1] ∼= OP(−p)(n+1

p )

¤

This corollary can be understood from another point of view. Using thenotation introduced in the proof of proposition 15.1.1, we can extend the mapΩS(−1) → m to an exact sequence

(29) 0 → [∧n+1ΩS ](−n − 1)δ→ . . . [∧2ΩS ](−2)

δ→ ΩS(−1)

δ→ m → 0

where

δ(dxi1 ∧ . . . dxip) =

∑(−1)pxij

dxi1 ∧ . . . dxij∧ . . . dxip

is contraction with the Euler vector field. The sequence is called the Koszul complexand it is one of the basic workhorses of homological algebra [E, chap 17]. Theassociated sequence of sheaves is

0 → [∧n+1On+1P

](−n − 1) → . . . [∧2On+1P

](−2) → [On+1P

](−1) → OP → 0

If we break this up into short exact sequences, then we obtain exactly the sequencesin corollary 15.1.3.

Proposition 15.1.4.

Hq(P,ΩpP) =

k if p = q ≤ n0 otherwise

When k = C, this gives a new proof of the formula for Betti numbers of Pn

given in section 6.2.

Exercises

1. Prove proposition 15.1.4.

15.2. Hodge numbers of a hypersurface

We now let X ⊂ Pn = P be a nonsingular hypersurface defined by a degree dpolynomial.

Proposition 15.2.1. The restriction map

Hq(Pn,ΩpP) → Hq(Xn,Ωp

X)

is an isomorphism when p + q < n − 1.

When k = C (which we assume for the remainder of this section), this propo-sition can be deduced from corollary 13.4.3, the canonical Hodge decomposition(10.2.4) and GAGA. As corollary, we can calculate many of the Hodge numbers ofX.

15.2. HODGE NUMBERS OF A HYPERSURFACE 155

Corollary 15.2.2. The Hodge numbers hpq(X) = δpq, where δpq is the Kro-necker symbol, when n − 1 6= p + q < 2n − 2.

Proof. For p + q < n− 1, this follows from the above proposition and propo-sition 15.1.4. For p + q > n − 1, this follows from GAGA and 9.2.3. ¤

This leaves the middle Hodge numbers, which are determined by the Eulercharacteristics:

Corollary 15.2.3. hp,n−1−p(X) = (−1)n−1−pχ(ΩpX) + (−1)n

Let us work out a few cases. From the sequence

0 → OPn(−d) → OPn → OX → 0

we obtain, the Hilbert polynomial of X

(30) χ(OX(i)) =

(i + n

n

)−

(i + n − d

n

).

Setting i = 0 yields the first Euler characteristic. To proceed we need

Lemma 15.2.4. 0 → OX(−d) → Ω1P|X → Ω1

X → 0 is exact (recall that Ω1P|X is

shorthand for the inverse image.)

Proof. We have a natural epimorphism Ω1P|X → Ω1

X corresponding to re-striction of 1-forms. We just have to determine the kernel. Let f be a definingpolynomial of X, and let Ω = Γ∗(Ω

1P) and Ω = Γ∗(Ω

1X). We embed Ω as a submod-

ule of ΩS(−1) as in the proof of proposition 15.1.1. In particular, the symbols dxi

have degree 1. Then ker[Ω/fΩ → Ω] is a free S/(f) module generated by

df =∑

i

∂f

∂xi

dxi

Thus it is isomorphic to S/(f)(−d). ¤

As a consequence of all of this,

χ(Ω1X) = χ(Ω1

P|X) − χ(OX(−d))

= (n + 1)χ(OX(−1)) − χ(OX) − χ(OX(−d))

= (n + 1)[

(n − 1

n

)−

(n − 1 − d

n

)] − 1 − 2

(n − d

n

)−

(n − 2d

n

).

by (30).We can calculate all the Hodge numbers this way in principle. However, we will

give the formulas for the whole lot in terms of a generating function. Let hpq(d)denote the pqth Hodge number of smooth hypersurface of degree d in Pp+q+1.Define the formal power series

H(d) =∑

pq

(hpq(d) − δpq)xpyq

in x and y. Then:

Theorem 15.2.5 (Hirzebruch).

H(d) =1

(1 + x)(1 + y)

[(1 + x)d − (1 + y)d

(1 + y)dx − (1 + x)dy− 1

]

156 15. COMPUTATION OF SOME HODGE NUMBERS

Hirzebruch [Hrz] deduces a slightly different identity from his general Riemman-Roch theorem. The above form of the identity appears in [SGA7, exp X1]. Similarformulas are available for complete intersections.

Exercises

1. Calculate the hodge numbers of a degree d surface in P3.2. Show that the generating for h0,q obtained by setting x = 0 in H(d) is

correct.

15.3. Machine computations

The formulas of the previous section are easily implemented on computer andare rather efficient to use when X is a hypersurface (or a complete intersection).There is, in principle, a method for computing Hodge numbers of a general X ⊂ Pn

given explicit equations for it. In rough outline, one can proceed as follows:

• View the sheaves ΩpX as coherent sheaves on Pn. These can be given an

explicit presentation. For example by combining

Ω1X = Ω1

Pn/(IΩ1Pn + dI),

with the formulas from section 15.1, where I is the ideal sheaf of X.• Resolve this as in theorem 14.3.8.• Calculate cohomology using the resolution.

Nowdays, it is possible to do these calculations on a machine using packagessuch as Macaulay2 [M2]. Below is part of a Macaulay 2 session for computing H12

and H22 of the Fermat quartic threefold x4 + y4 + z4 + u4 + v4 = 0 over Q. Thecommands should be more or less self explanitory. The answer H12 = Q30 andH22 = Q can be checked against the formulas from the previous section.

i1 : S = QQ[x,y,z,u,v];

i2 : I = ideal (x^4+y^4+z^4+u^4+v^4);

i3 : X = Proj(S/I);

i4 : Om1 = cotangentSheaf(1,X);

i5 : HH^2 Om1

30

o5 = QQ

i6 : Om2 = cotangentSheaf(2,X);

i7 : HH^2 Om2

1

o7 = QQ

Exercises

1. Do more examples in Macaulay 2. (Look up the commands HH, cotan-gentSheaf... first.)

CHAPTER 16

Deformation invariance of Hodge numbers

In this chapter, our goal is to prove that Hodge numbers of a family of smoothcomplex projective varieties stays constant. While this can be proved purely ana-lytically, we develop much of background using Grothendieck’s language of schemeswhich gives a particularly elegant approach to families.

16.1. Families of varieties via schemes

We gave a definition of schemes 2.4 earlier without giving much geometricmotivation. Varieties often occur in families as we have already seen. As a simpleexample, let f(x, y, t) = y2 −xt be a polynomial over a field k. We can view this asdefining a family of parabolas in the xy-plane A2 parameterized by t ∈ A1. Whent = 0, we get a degenerate parabola y2 = 0 which is the “doubled” x-axis. It isimpossible to capture this fully within the category of varieties, but it makes perfectsense with schemes. Here, we have a map of rings

k[t] → k[x, y, t]/(f(x, y, t))

which induces a morphism of schemes

π : Spec k[x, y, t]/(f) → Spec k[t] = A1

We can view this as the family of schemes

Spec k[x, y, t]/f(x, y, a) | a ∈ k

given by the fibers of π.More generally, let R = O(Y ) be the coordinate ring be an affine variety Y

over an algebraically closed field k. Affine space over R is

AnR = SpecR[x1, . . . xn] ∼= An

k × Y.

Let I ⊂ R[x1, . . . xn] be an ideal. It is generated by polynomials

fj(x, y) =∑

fj,i1,...in(y)xi1

1 . . . xinn .

We get a morphism

SpecR[x1, . . . xn]/I → SpecR = Y

We can view this as a family of subschemes of Ank

SpecR[x1, . . . xn]/I ⊗ R/ma = Spec k[x1, . . . xn]/(fj(x, a))

parameterized by points a ∈ Y . If (fj(x, a)) is a radical ideal in k[x1, . . . xn], thiswould be a subvariety of An

k , but in general it be a closed subscheme.Similarly, a homogenous ideal I ⊂ R[x1, . . . xn] = S gives a family Proj S/I of

subschemes of Pnk . If I has homogenous generators fj(x, y). We can loosely define

Proj S/I as the family of subschemes defined by by ideals (fj(x, a)) as a ∈ Y varies.

157

158 16. DEFORMATION INVARIANCE OF HODGE NUMBERS

A bit more formally, the projective space over R PnR = Pn

k × Y can be constructedby gluing n + 1 copies of affine space

Ui,R = SpecR[x0/xi, . . . xi/xi . . . xn/xi]

together. Given a homogenous polynomial in f(x, y) ∈ S = R[x1, . . . xn], we get apolynomial in f(x1/xi, . . . xn/xi, y) in the above ring. Thus a homogenous ideal Idefines ideals in Ii ⊂ R[x1/xi, . . .] by applying this substitution on the generators.The scheme ProjS/I is defined by gluing the affine schemes defined Ij together.See [Har] for a more precise account of Proj.

16.2. Cohomology of Affine Schemes

Let R be a commutative ring.

Definition 16.2.1. An R-module I is injective if and only if for any for injec-tive map N → M of R-modules, the induced map HomR(M, I) → HomR(N, I) issurjective.

Example 16.2.2. If R = Z then I is injective provided that it is divisible i.e.if a = nb has a solution for every a ∈ I and n ∈ Z − 0.

A standard result of algebra is the following. A proof can be found in severalplaces, for example [E, p. 627].

Theorem 16.2.3 (Baer). Every module is (isomorphic to) a submodule of aninjective module.

Recall construction of the sheaf M of section 2.6.

Proposition 16.2.4. If I is injective and R is noetherian, then I is flasque.

Proof. Suppose that U = D(f) is a basic open set. Given an element of I(U),it can be written as a fraction x/fn. Consider the map R → I which sends 1 7→ x.Since I is injective, this extends to a map R[1/fn] → I. In other words, x/fn liesin I.

For a general open set U , express U = ∪D(fi). We have a diagram

I(U)i

//∏

I((D(fi))

I(X)

r

OO

s

99s

ss

ss

ss

ss

s

where i is injective. s is surjective by the previous paragraph, therefore r is surjec-tive. ¤

Theorem 16.2.5 (Grothendieck). Let X = specR, then for any R-module,

Hi(X, M) = 0

Proof. Let M be an R-module, then it embeds into an injective module I.Let N = I/M . By lemma 2.6.5, M 7→ M is exact, therefore

0 → M → I → N → 0

is exact. As Hi(X, I) = 0 for i > 0, we obtain

I → N → H1(X, M) → 0,

16.3. SEMICONTINUITY OF COHERENT COHOMOLOGY 159

and

Hi+1(X, M) ∼= Hi(X, N).

I → N is surjective, therefore H1(X, M) = 0. Note that this argument can beapplied to any module, in particular to N . This implies that

H2(X, M) ∼= H1(X, N) = 0.

We can kill the all higher cohomology groups in the same fashion. ¤

16.3. Semicontinuity of coherent cohomology

Let R = O(Y ) be the coordinate ring of an affine variety, and let S = R[x0, . . . xn] =⊕Si with the usual grading. Given a finited generated graded R-module M . We canconstruct a coherent sheaf M on Pn

R = Pnk × Y as in section 14.3. Theorem 14.3.4

generalizes to this setting [Har, II, ex. 5.9]. For each point a ∈ Y , we get a agraded k[x0, . . . xn]-module Ma = M ⊗ R/ma. Thus we get a family of coherent

sheaves Ma parameterized by Y . This can be constructed sheaf theoretically. Letia : Pn

k → Pnk × Y be the inclusion x 7→ (x, a), then Ma = i∗aM .

Example 16.3.1. Let M be a finitely generated graded k[x0, . . . xn]-module,

then R⊗M is a finitely generated graded S-module. R ⊗ M is a “constant” family

sheaves; we have R ⊗ Ma = M . This can be constructed geometrically. Let π :

Pnk × Y → Y be the projection. Then R ⊗ M = π∗(M). A special case is OPn

k×Y (i)

which is the constant family π∗OPnk.

Example 16.3.2. Let I ⊂ S be a homogeneous ideal, this gives rise to a familyof projective schemes of Pn

k with ideal sheaves Ia.

We now ask the basic question, suppose that M as above, how do the dimensionsof the cohomology groups

hi(Ma) = dim Hi(Ma)

vary with a? We look at some examples.

Example 16.3.3. If M is constant family, then clearly a 7→ hi(Ma) is a con-stant function of a.

In general, this is not constant. Here is a more typical example:

Example 16.3.4. Set R = k[s, t], and choose three points

p1 = [0, 0, 1], p2 = [s, 0, 1], p3 = [0, t, 1]

in P2k with s, t variable. Let I be the ideal sheaf of the union of these points in

P2k × A2

k. This can also be described as I, where I is product

(x, y, z − 1)(x − s, y, z − 1)(x, y − t, z − 1) ⊂ R[x, y, z]

Consider I(1) = I(1). The global sections of this sheaf correspond to the space oflinear forms vanishing at p1, p2, p3. Such a form can exist only when the points arecolinear, and it is unique upto scalars unless the points coincide. Thus

h0(I(1)(s,t)) =

2 if s = t = 01 if s = 0 or t = 0 but not both0 if s 6= 0 and t 6= 0

160 16. DEFORMATION INVARIANCE OF HODGE NUMBERS

In the above example, the sets where cohomology jumps are Zariski closed.This can be reformulated as saying that the function h0(I(1)(s,t)) is upper semi-continuous for the Zariski topology. Let’s try to show this in general for the topcohomology (which is technically easier). Suppose that M is a finitely graded mod-ule S = R[x0, . . . xn]-module, with R = O(Y ). Choose a graded presentation

⊕

j

S(ij) →⊕

m

S(ℓm) → M → 0

The first map is given by a matrix of polynomials. In order for this presentation tobe useful to us, we want

⊕

j

S(ij)y →⊕

m

S(ℓm)y → My → 0

to stay exact for any y ∈ Y . This to leads to concept of flatness.

Definition 16.3.5. An R-module T is flat if the functor N 7→ T ⊗R N is exact(note that it always right exact).

Lemma 16.3.6. If T is flat, and

0 → A → B → T → 0

is exact, then0 → A ⊗ C → B ⊗ C → T ⊗ C → 0

is exact for any module C.

Proof. This follows immediately from properties of the Tor functor. See forexample [E, pp162-172]. ¤

Now suppose that M is a flat R-module. Break the above presentation intoexact sequences

0 → K →⊕

m

S(ℓm) → M → 0

⊕

j

S(ij) → K → 0

The above lemma, shows that

0 → Ky →⊕

m

S(ℓm)y → My → 0

is exact, and we also have a surjection⊕

j

S(ij)y → Ky → 0

Applying Hn to the corresponding sequences of sheaves, and splicing the resultingsequences yields

⊕

j

Hn(Pnk ,O(ij))

A(y)−→

⊕

m

Hn(Pnk ,O(ℓm)) → Hn(Pn, My) → 0

Note that the first two spaces are constant. Thus we can view the map betweenthem, A(y), as matrix depending algebraically on y (i.e. it comes from the evalua-tion of a matrix over R). The (r+1)× (r+1) minors of A(y) define a closed subsetYr ⊂ Y . Set N =

∑hn(Oℓm). Then we obtain

hn(My) = N − rank(A(y)) ≥ N − r

16.4. SMOOTH FAMILIES 161

if and only if y ∈ Yr. Thus we have shown that hn(My) is upper semicontinuous.In general,

Theorem 16.3.7 (Grothendieck). If M is a flat finitely generated graded R-module, then

y 7→ hi(My)

is upper semicontinuous, and the function

y 7→ χ(My) =∑

(−1)ihi(My)

is locally constant.

Proof. See [Har, 12.8] and [EGA, III]. ¤

Grauert has established analogous results in the analytic setting, see [GPR].We say that the subscheme Z ⊂ Pn

k × Y is flat over Y , if OZ is given by a flatR-module The geometric meaning of flatness is a little elusive. However the abovetheorem implies

Corollary 16.3.8. If Y is flat over Z, then the Hilbert polynomial χ(OY (d))is locally constant.

It follows from this that the fibers of Y → Z have the same dimension. Thus,for example, the blow up of point on surface is not flat. If fact, constancy of theHilbert polynomial characterizes flatness [Har][9.9].

16.4. Smooth families

We introduced flatness as techinical assumption without really giving any ex-amples. We now introduce a stronger notion, where the geometric meaning is clear.To simply matters, we restrict out attention to nonsingular varieties.

Definition 16.4.1. A morphism f : X → Y of nonsingular varieties is smoothif for every point x, the induced map Tx → Ty is surjective.

Example 16.4.2. Let f : Amk → An

k be given by f(x) = (f1(x), . . . fn(x)). wherefi are polynomials. Then f is smooth if and only if the Jacobian (∂fi/∂xj) has rankn. When this is satisfied, it is clear that the fibers f−1(y) are nonsingular varietiesof dimension m − n.n

Theorem 16.4.3. If f : X → Y is a smooth morphism of nonsingular varieties,then f is flat and the fibers f−1(y) are all nonsingular varieties.

Thus we can view a smooth morphism as a family of nonsingular varieties overa nonsingular parameter space. We can define an equivalence relation on nonsin-gular projective varieties weaker than isomorphism. We say that two nonsingularprojective varieties X1,X2 are deformations of each other if there is a smooth mapof nonsingular varieties f : Z → Y such that the fibers are all projective varietiesand Xi = f−1(yi) for points yi ∈ Y . This generates an equivalence relation thatwe will call deformation equivalence. For example, any two nonsingular hypersur-faces Xi ⊂ Pn of degree d are deformation equivalent, since they members of thethe family constructed below. Let Vd be the space of homogeneous polynomials ofdegree d in n + 1 variables. ∆ ⊂ P(Vd) the closed set of singular hypersurfaces.Then

U = (x, [f ]) ∈ Pn × (P(Vd) − ∆) | f(x) = 0

162 16. DEFORMATION INVARIANCE OF HODGE NUMBERS

Then U → P(Vd) − ∆ is a smooth map containing all nonsingular hypersurfaces asfibers.

Since any elliptic curve can be realized as a smooth cubic in P2, it follows thatany two elliptic curves are deformation equivalent. Considerably deeper is the factthat any two smooth projective curves of the same genus g ≥ 2 are deformationequivalent. This can be proved with the help of the Hilbert scheme. A weakformulation of its defining property is as follows:

Theorem 16.4.4 (Grothendieck). Fix a polynomial p ∈ Q[t]. There is a pro-jective scheme H = Hilbp

Pn with a flat family U → H of closed subschemes of Pn

with Hilbert polynomial p, such that every closed subscheme with Hilbert polynomialp occurs as a fiber of U exactly once.

For example, the space P(Vd) above is the Hilbert scheme HilbpPn with

p(i) = χ(OPn(i)) − χ(OPn(i − d)) =

(n + i

n

)−

(n + i − d

n

).

Choose N ≥ 3, then for a smooth projective curve X of genus g, ω⊗NX is very ample.

The set of such curves in P(g−1)(2N−1) is parameterized by an open subset of theHilbert scheme with p(t) = (2g − 2)Nt + (1 − g). This set can be shown to beirreducible. See [HM] for further details.

16.5. Deformation invariance of Hodge numbers

In this section, we revert to working over C.

Theorem 16.5.1 (Kodaira-Spencer). If two complex nonsingular projective va-rieties are deformation equivalent, then their Hodge numbers are the same.

Proof. Let f : X → Y be a smooth projective morphism of nonsingularvarieties. Then by theorem 16.3.7, there there are constants gpq such that

(31) hpq(Xt) ≥ gqp

for all t ∈ X with equality on an a nonempty open set U . Choose a points t ∈ Uand s /∈ U . Then since Xt and Xs are diffeomorphic 12.1.1, they have the sameBetti numbers. Therefore by the Hodge decomposition (theorem 9.2.4),

∑

pq

hpq(Xs) =∑

pq

gpq

and this implies that (31) is an equality. ¤

This kind of result is not true of all the invariants considered so far. Any ellipticcurve can be embedded as a cubic in P2, thus any two elliptic curves are deformationequivalent. Likewise for the products of elliptic curves with themselves. But wesaw in section 11.3 that the Picard number of E × E was not constant. Thereforeit not a deformation invariant. Other examples of this phenomenon are providedby:

Theorem 16.5.2 (Noether-Lefschetz). Let d ≥ 4 then there exists a surfaceX ⊂ P3 of degree d with Picard number 1.

Remark 16.5.3. This is true “for almost all” surfaces.

16.5. DEFORMATION INVARIANCE OF HODGE NUMBERS 163

Proof. We sketch the proof. Since H1(X,OX) = 0 for any surface in P3,c1 : Pic(X) → H2(X, Z) is injective. On the other hand, H2(X,OX) 6= 0 if Xis has degree d ≥ 4, therefore c1 is not onto. Choose a Lefschetz pencil Xtt∈P1

of surfaces, let X → P1 be the incidence variety and let U ⊂ P1 parameterize Xt

smooth 13.3. The set of all curves lying on some Xt is a parameterized by a count-able union of Hilbert schemes. Each irreducible component will either parameterizecurves lying on a fixed Xt or curves varying over all Xt. Therefore for all but count-ably many t, any curve lying on a an Xt will propogate to all the members of thepencil. Choose such a nonexceptional t0 ∈ U . The group c1(Pic(Xt0)) is the groupof curves on Xt0 , and this is stable under the action of π1(U, t0) since any suchcurve can be moved along a loop avoiding the exceptional t’s. By theorem 13.3.9,H2(Xt0 , Q) = im(H2(P3)) ⊕ V , where V is the generated by vanishing cycles, andπ1(U) acts irreducibly on this. Since C = c1(Pic(Xt0)) ⊗ Q contains im(H2(P3)),it follows that either c1(Pic(Xt0)) equals im(H2(P3)) or it equals H2(Xt0). Thelast case is impossible, so c1(Pic(Xt0)) = im(H2(P3)) = Q. ¤

Exercises

1. Construct a smooth quartic X ∈ P3 containing a line L. It can be shownthat L2 = −2, assume this and prove that the Picard number of X is atleast 2.