Ample line bundles and intersection theory.

Tuesday April 14th, Soesterberg Johan de Jong

Introduction. In this talk I will give an overview of the results from intersection theory we need for the proof of Faltings’ theorem. Meanwhile I will try to give a coherent account of (this part of) intersection theory and I will try to show what a beautiful theory it is. I would like to stress here, that it should be possible for anyone with some basic knowledge of (algebraic) geometry to prove all the results mentioned today after reading the first 40 pages of Hartshorne’s Lecture Notes [1].

1. Coherent sheaves, etc. X will always denote a projective variety over the algebraically closed field k. An affine open subvariety of X is an open U ⊂ X such that U is isomorphic as a variety to a closed subvariety Z ⊂ A_{k}^{N} for some N. Example. If X ⊂ P^{n} with homogeneous coordinates T_{0}, …, T_{n} then for any non-zero homogeneous polynomial P (T_{0}, …, T_{n}) the set

{(t_{0} : ⋯: t_{n}) ∈ X(k) | P (t_{0}, …, t_{n}) ≠ 0}

is an affine open subvariety of X.

The affine open subsets form a basis for the topology on X and hence a sheaf on X is determined by the sets of sections over the affine open U ⊂ X. The structure sheaf O_{X} of X is determined by the rule

O_{X}(U) = Γ(U, O_{X}) = Γ(U, O_{U}) = k[T_{1}, …, T_{N}]/I

if U ⊂ X is isomorphic to the affine variety determined by the (prime) ideal I ⊂ k[T_{1}, …, T_{N}].

The other sheaves occurring in the talk today will always be sheaves of O_{X}-modules, that is, they will be sheaves of abelian groups F on X endowed with a multiplication O_{X} ×F → F such that for each open U ⊂ X, F(U) becomes a module over O_{X}(U). A sheaf of O_{X}-modules F is said to be generated by global sections if there exists a family of sections (s_{i})_{i∈I}, s_{i} ∈ Γ(X, F) such that the map of sheaves

⊕_{i∈I}O_{X} ⟶ F ⊕_{i∈I} f_{i} ↦ ∑_{i} f_{i}s_{i}

is surjective (i.e., any local section of F should be locally a finite linear combination of the s_{i}). A coherent sheaf of O_{X}-modules is a sheaf F such that

− for each open affine U ⊂ X, F(U) is a finitely generated O_{X}(U)-module, − if U ⊂ V ⊂ X are open affine then F(U) = F(V) ⊗_{O_{X}(V)} O_{X}(U)

The fundamental theorem on sheaves/coherent sheaves is Theorem For any sheaf F on X, H^{i}(X, F) = 0 for all i > dim X. If F is coherent, then H^{i}(X, F) is a finite dimensional k-vector space for all i.

This theorem allows us to define the Euler-Poincaré characteristic of F:

χ(F) = ∑_{i=0}^{∞}(−1)^{i} dim_{k} H^{i}(X, F).

Using the long exact cohomology sequence, we see that for an exact sequence of coherent sheaves 0 → F_{1} → F_{2} → F_{3} → 0 we have χ(F_{2}) = χ(F_{1}) + χ(F_{3}).

Examples of coherent sheaves.

a) Ideal sheaves. An ideal sheaf is a subsheaf I ⊂ O such that for each U ⊂ X open, I(U) is an ideal of O_{X}(U). It is coherent, since for each open affine U ⊂ X the ring O_{X}(U) is a Noetherian ring.

An ideal sheaf I determines a closed subset Z of X

Z = {x ∈ X | for all x ∈ U ⊂ X and f ∈ I(U): f(x) = 0}

and it determines a sheaf of rings O_{Z} = O_{X}/I. The pair (Z, O_{Z}) is (what we will call) a closed subscheme of X. It is usually denoted by Z.

A closed subvariety is a closed subscheme (Z, O_{Z}) such that the sheaf of algebras O_{Z} is without zero divisors. Example: X = A_{k}^{1}, Γ(X, O_{X}) = k[t] ⊃ I = (t^{n}). In this case we get (Z_{n}, O_{Z_{n}}) with Z_{n} = {0} and Γ(X, O_{Z_{n}}) = k[t]/(t^{n}). Hence Z_{n} ≠ Z_{n′} unless n = n′, and only Z_{1} is a closed subvariety. If F is a coherent sheaf on X then we can construct an ideal sheaf I as follows:

I(U) = {f ∈ O_{X}(U) | multiplication by f: F|_{U} → F|_{U} is the zero map}

The closed subscheme Z associated to this is called the support of F: Z := supp(F). Example: The support of O_{Z} is (Z, O_{Z}).

b) Line bundles. A line bundle or invertible sheaf is a coherent sheaf L on X such that each point x ∈ X has an affine neighborhood U ⊂ X such that L(U) ≅ O_{X}(U) as O_{X}(U)-modules, i.e., L|_{U} ≅ O_{X}|_{U}.

If we have two line bundles L_{1} and L_{2} then we can form the tensor product L_{1} ⊗_{O_{X}} L_{2}. It is again a line bundle. Using the tensor product for multiplication, the set of line bundles up to isomorphism becomes an abelian group with identity O_{X} and inverse L^{−1} = Hom_{O_{X}}(L, O_{X}). This group is denoted by Pic(X), it is the Picard group of X. For morphisms f: X → Y there is a pullback f^{∗}: Pic(Y) → Pic(X) defined as follows: if the line bundle L on Y is trivial on the open sets U_{i} which cover Y and if its transition functions are f_{i,j} ∈ Γ(U_{i} ∩ U_{j}, O_{Y}^{∗}) then f^{∗}L is the line bundle on X which is trivial on the open sets f^{−1}U_{i} and has transition functions f_{i,j} ∘ f ∈ Γ(f^{−1}(U_{i} ∩ U_{j}), O_{X}^{∗}). The pullback is a homomorphism of abelian groups.

2. Ample and very ample line bundles. Recall that a line bundle L on X is said

to be very ample if it is generated by global sections and for a basis s_{0}, …, s_{n} of Γ(X, L) the map ϕ_{s_{0},...,s_{n}}: X → P_{k}^{n} is a closed immersion. The line bundle L is said to be ample if there exists N ∈ N such that L^{⊗N} := L ⊗ ⋯ ⊗ L (N factors L) is very ample. A cohomological criterion for being ample is the following.

Theorem (Serre-Grothendieck) For a line bundle L on X the following are equivalent:

1. L is ample

2. for every coherent sheaf F on X we have H^{i}(X, F ⊗ L^{⊗n}) = 0 for all i > 0, n ≫ 0

3. for every coherent sheaf F on X, the sheaf F ⊗ L^{⊗n} is generated by global sections for n ≫ 0.

Corollary 1 If f: X → Y is a finite surjective morphism of projective varieties and L is a line bundle on Y then L is ample on Y if and only if f^{∗}L is ample on X. The proof uses the theorem above and comparison of cohomology of sheaves on X and on Y. Corollary 2 For any line bundle L on X there exist very ample line bundles L_{1} and L_{2} such that L ≅ L_{1} ⊗ L_{2}^{−1}. Proof Suppose M is very ample on X. For some n ∈ N the line bundle L ⊗ M^{⊗n} is generated by global sections. It is easy to see that L_{1} = L⊗M^{⊗n} ⊗M and L_{2} = M^{⊗(n+1)} are very ample (and L ≅ L_{1} ⊗ L_{2}^{−1}). □

3. Intersection numbers. Let F be a coherent sheaf on X and let L_{1}, …, L_{t} be line bundles on X. Proposition The function (n_{1}, …, n_{t}) ↦ χ(L_{1}^{⊗n_{1}} ⊗ ⋯ ⊗ L_{t}^{⊗n_{t}} ⊗ F) is a numerical polynomial of total degree equal to dim(supp(F)). Proof We prove the proposition by induction on dim(suppF). If dim(suppF) = 0 then we have L_{1}^{⊗n_{1}} ⊗ ⋯ ⊗ L_{t}^{⊗n_{t}} ⊗ F ≅ F since each L_{i} is trivial in a neighborhood of the support of F, and H^{i}(X, F) = 0 unless i = 0 since F is supported in dimension zero. Hence our function is constant and equal to χ(F) = dim_{k} H^{0}(X, F). For the induction step, by Corollary 2 above we may assume that each L_{i} is very ample. Hence we can choose a section s ∈ Γ(X, L_{i}) whose zero set is “transversal” to supp(F), i.e., such that dim((s = 0) ∩ supp(F)) < dim(suppF). Thus we see that the kernel and cokernel of multiplication by s:

0 → K → F →^{·s} F ⊗ L_{i} → Q → 0

are sheaves with lower dimensional support. Hence for all i we have that

χ(L_{1}^{⊗n_{1}} ⊗ ⋯ ⊗ L_{i}^{⊗(n_{i}+1)} ⊗ ⋯ ⊗ L_{t}^{⊗n_{t}} ⊗ F) − χ(L_{1}^{⊗n_{1}} ⊗ ⋯ ⊗ L_{t}^{⊗n_{t}} ⊗ F)

is a polynomial in n_{1}, …, n_{t} of total degree dim(suppF) − 1. ¿From this the result follows. □ Definition If (Z, O_{Z}) is a closed subscheme of dimension d and L_{1}, …, L_{d} are line bundles then we define (L_{1} ⋯ L_{d} ·Z), the intersection number of Z with L_{1}, …, L_{d}, to be the coefficient of n_{1} ⋯ n_{d} in the polynomial χ(L_{1}^{⊗n_{1}} ⊗ ⋯ ⊗ L_{t}^{⊗n_{t}} ⊗ O_{Z}). Remarks a) The intersection number is an integer (any numerical polynomial can be written as a Z-linear combination of the functions ({n_{1},...,n_{d} \atop k_{1},...,k_{d}})).

b) It is additive:

(L_{1} ⊗ L′_{1} ⋯ L_{d} · Z) = (L_{1} ⋯ L_{d} · Z) + (L′_{1} ⋯ L_{d} · Z)

and symmetric: the intersection number is independent of the order of L_{1}, …, L_{d}.

c) Suppose that the irreducible components Z_{i} of Z with dim Z_{i} = dim Z have generic points z_{1}, …, z_{k}. Then the rings O_{Z,z_{i}} (= localisation of O_{Z} at z_{i}) are zero dimensional and of finite length. The multiplicity of Z_{i} in Z is defined as m_{Z_{i}}Z = length of O_{Z,z_{i}}. One can show that

(L_{1} ⋯ L_{d} · Z) = ∑_{i=1}^{k} m_{Z_{i}}Z · (L_{1} ⋯ L_{d} · Z_{i})

This one proves by showing that ⊕_{i}O_{Z_{i}}^{m_{Z_{i}}Z} and O_{Z} are connected by a chain of sheaves

O_{Z} = F_{0} →^{α_{0}} F_{1} →^{α_{1}} ⋯ → F_{N} = ⊕_{i}O_{Z_{i}}^{m_{Z_{i}}Z}

where each α_{i} has kernel and cokernel supported in dimension smaller than d.

d) The proof of the proposition above also gives a method for computing (L_{1} ⋯ L_{d}·Z). Suppose the section s ∈ Γ(X, L_{d}) is such that

·s: L_{d}^{−1} ⊗ O_{Z} → O_{Z}  (∗)

is injective. In this case the quotient sheaf O_{Z}/sO_{Z} is the structure sheaf of a closed subscheme which we will denote by Z ∩ H_{d}, where H_{d} = {x ∈ X | s(x) = 0}, and we get

(L_{1} ⋯ L_{d} · Z) = (L_{1} ⋯ L_{d−1} · (H_{d} ∩ Z))

Remarks 1) H_{d} is a divisor representing L_{d}, i.e., such that L_{d} = O(H_{d})

2) If L is very ample, then (∗) is injective for general s. Repeating this, we get (L_{1} ⋯ L_{d} · Z) = #(H_{1} ∩ ⋯ ∩ H_{d} ∩ Z) is equal to the number of points in the intersection H_{1} ∩ ⋯ ∩ H_{d} ∩ Z counted with multiplicities, if the divisors H_{1}, …, H_{d} are chosen general enough. By the proof of the proposition above, the local multiplicity of H_{1} ∩ ⋯ ∩ H_{d} ∩ Z at P is:

e_{P} (H_{1}, …, H_{d}; Z) = dim_{k} O_{X,P} / (I_{Z,P} + f_{1}O_{X,P} + ⋯ + f_{d}O_{X,P})

here f_{1}, ⋯, f_{d} are local equations defining H_{1}, …, H_{d}.

e) The number (L^{d} · Z) := (L ⋯ L · Z) (d factors L) can also be computed by the rule:

χ(L^{⊗n} ⊗ O_{Z}) = (L^{d} · Z)(d!)^{−1} · n^{d} + lower order terms

If L is ample we will call (L^{d} · Z) the degree of Z with resperct to L and write deg_{L} Z for it.

Suppose now that X = P^{n} and Z ⊂ P^{n} is an irreducible subvariety of dimension d > 0. By remark d):

deg_{O(1)} Z = # ((general d-plane) ∩ Z) = deg Z,

the classical definition of the degree of Z. Using the fact that any hyperplane in P^{n} intersects Z non-trivially and induction on d we see that we always have that deg Z > 0. This more or less proves one direction of the following theorem. Theorem (Nakai criterion) A line bundle L on X is ample if and only if for every closed subvariety Z ⊂ X we have (L^{d} · Z) > 0. f) The behaviour of the degree under a finite surjective morphism f: X → Y is the following:

deg_{f^{∗}L} Z = deg(f: Z → f(Z)) · deg_{L} f(Z)

Here Z ⊂ X is a closed subvariety and L is an ample line bundle on Y. Sketch of proof and explanation of deg(f: Z → f(Z)). f(Z) is again a closed subvariety. The map f defines an inclusion f^{∗}: k(f(Z)) ↪ k(Z) and this is then a finite field extension. Put n := deg(f: Z → f(Z)) := [k(Z) : k(f(Z))]. If this field extension is separable, then generically for p ∈ f(Z) we have f^{−1}(p) = {q_{1}, …, q_{n}} ⊂ Z. Hence, if H_{1} ∩ ⋯ ∩ H_{d} ∩ f(Z) = {p_{1}, …, p_{n}} then f^{∗}H_{1} ∩ ⋯ ∩ f^{∗}H_{d} ∩ Z = {q_{11}, …, q_{1n}, …, q_{n1}, …, q_{nn}}. The result follows. □

g) Suppose that k = C and that X is smooth. Then X(C) also has the structure of a complex manifold, which we will denote by X^{an}. A line bundle L on X gives rise to a line bundle L^{an} on X^{an} which is determined by an element of H^{1}(X^{an}, O_{an}^{∗}). The exponential sequence:

0 → 2πiZ → O_{an} →^{exp} O_{an}^{∗} → 0

gives us the first Chern class c_{1}(L) ∈ H^{2}(X^{an}, Z). On the other hand, an irreducible subvariety Z ⊂ X will give rise to an analytic subvariety Z^{an} ⊂ X^{an}. After triangulizing it, we see that it gives rise to a class [Z] = [Z^{an}] ∈ H_{2d}(X^{an}, Z). For a general closed subscheme Z we put

[Z] := ∑_{i} m_{Z_{i}}Z · [Z_{i}^{an}] ∈ H_{2d}(X^{an}, Z)

The connection between intersection numbers and cup product in cohomology is now:

(L_{1} ⋯ L_{d} · Z) = ⟨c_{1}(L_{1}) ∧ ⋯ ∧ c_{1}(L_{d}), [Z]⟩

where ⟨ , ⟩: H^{2d}(X^{an}, Z) × H_{2d}(X^{an}, Z) → Z is the canonical pairing.

4. Numerical equivalence and ample line bundles.

We write F_{1}(X) = {∑ n_{i}C_{i} | C_{i} ⊂ X is a closed subvariety of dimension 1} for the free abelian group generated by the set of all curves in X. By abuse of language we call an element of F_{1}(X) a curve on X. A curve ∑ n_{i}C_{i} is effective if n_{i} ≥ 0 for all i. Our intersection number gives us a bilinear pairing

Pic(X) × F_{1}(X) → Z (L, ∑ n_{i}C_{i}) ↦ ∑ n_{i}(L · C_{i})

We say that two curves C_{1}, C_{2} (resp. line bundles L_{1}, L_{2}) are numerically equivalent if (L·C_{1}) = (L · C_{2}) for all L ∈ Pic(X) (resp. (L_{1} · C) = (L_{2} · C) for all C ∈ F_{1}(X)). Notation: C_{1} ≡ C_{2} (resp. L_{1} ≡ L_{2}). Definition A^{1}(X) := (Pic(X)/ ≡) ⊗_{Z} R and A_{1}(X) := (F_{1}(X)/ ≡) ⊗_{Z} R. (By definition these are dual R-vector spaces.) Remark If k = C and X is smooth then by Remark 3g) we know that the intersection products (L·C) only depend on the first Chern class c_{1}(L) ∈ H^{2}(X^{an}, Z). Hence we conclude that A^{1}(X) is a subquotient of H^{2}(X^{an}, R) (it is actually a subspace!). This proves the following general theorem in this special case.

Theorem A^{1}(X) is finite dimensional. In general one reduces to the case where X is a smooth surface and then the theorem is a consequence of the Mordell-Weil theorem for abelian varieties over function fields. Important remark. If L_{1} ≡ L′_{1} then (L_{1} · L_{2} ⋯ L_{d} · Z) = (L′_{1} · L_{2} ⋯ L_{d} · Z) always. Hence, by the Nakai criterion, whether or not L is ample depends only on the numerical equivalence class of L; i.e., it depends only the point in A^{1}(X) determined by L. Let A_{1}^{+}(X) ⊂ A_{1}(X) be the cone generated by the classes of effective curves. We know that if L is ample then (L · C) > 0 for all C ∈ A_{1}^{+}(X). Hence the cone P^{o} spanned by the classes of ample line bundles is contained in the pseudo-ample cone

P = {D ∈ A^{1}(X) | D · C ≥ 0 ∀C ∈ A_{1}^{+}(X)}

We will have to use the following basic theorem once: Theorem P^{o} is the interior of P. (The ample cone is the interior of the pseudo-ample cone.)

5. Some lemmas to be used in the proof of the main theorem of Faltings.

Suppose X is an abelian variety over k and L is an ample line bundle on X. We denote by [n] the morphism given by multiplication by n on X: [n]: X → X, x ↦ nx. Lemma [n]^{∗}L ≡ L^{⊗n^{2}} Proof for k = C. We can write X^{an} = C^{g}/L, L ≅ Z^{2g}. The map [n]^{∗}: H^{2}(X^{an}, Z) → H^{2}(X^{an}, Z) is equal to multiplication by n^{2}, as is seen from the identification H^{2}(X^{an}, Z) ≅ ⋀^{2} H^{1}(X^{an}, Z) ≅ ⋀^{2} L^{∗}. □ Suppose we have a product situation X = X_{1}×X_{2}. Let us denote by p_{1}: X → X_{1} the projection onto X_{1}. Further, we assume given line bundles L_{1}, …, L_{k} on X_{1} and line bundles L_{k+1}, …, L_{d} on X. Finally, Z ⊂ X is a closed subvariety of dimension d. Lemma If dim p_{1}(Z) < k then (p_{1}^{∗}(L_{1}) ⋯ p_{k}^{∗}(L_{k}) · L_{k+1} ⋯ L_{d} · Z) = 0. Proof By linearity of the intersection product and of p_{1}^{∗} we may assume L_{1}, …, L_{k} are very ample. By our assumption we can find divisors H_{1}, …, H_{k} (divisors of sections of L_{1}, …, L_{k}) such that H_{1}∩⋯∩H_{k}∩p_{1}(Z) = (recall that dim p_{1}(Z) < k). Hence also p_{1}^{∗}H_{1}∩⋯∩p_{1}^{∗}H_{k}∩Z =. This implies by Remark 3d) that

(p_{1}^{∗}(L_{1}) ⋯ L_{d} · Z) = (L_{k+1} ⋯ L_{d} · (p_{1}^{∗}H_{1} ∩ ⋯ ∩ p_{1}^{∗}H_{k} ∩ Z)) = (L_{k+1} ⋯ L_{d}·) = 0  □

Put P := P^{n_{1}} × ⋯ × P^{n_{m}}. On it we have the line bundles L_{i} := pr_{i}^{∗}(O_{P^{n_{i}}}(1)). Exercises: a) Sections F ∈ Γ(P, L_{1}^{⊗d_{1}} ⊗ ⋯ ⊗ L_{m}^{⊗d_{m}}) correspond with multihomogeneous polynomials of multidegree (d_{1}, …, d_{m}).

b) Pic(P) = Z[L_{1}] ⊕ ⋯ ⊕ Z[L_{m}] and A^{1}(P) = R^{m} (same basis). The pseudo-ample cone is {(x_{1}, …, x_{m}) ∈ R^{m} | ∀i: x_{i} ≥ 0}, the ample cone is {(x_{1}, …, x_{m}) ∈ R^{m} | ∀i: x_{i} > 0}. If Z ⊂ P is an irreducible subvariety, then we get degrees indexed by e_{1}, …, e_{m} with ∑ e_{i} = dim(Z): the numbers (L_{1}^{e_{1}} · L_{2}^{e_{2}} ⋯ L_{m}^{e_{m}} · Z). These numbers are ≥ 0; use induction on dim Z and the fact that one can always find a section in L_{i} “transversal” to Z.

Lemma 2.3 of Faltings. Suppose X ⊂ P is a closed subscheme which is an intersection of hypersurfaces of multidegree (d_{1}, …, d_{m}). If the X_{j} are irreducible components of X with multiplicities m_{j}, and of the same codimension t, then

∑ m_{j}(L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · X_{j}) ≤ (L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · (L_{1}^{⊗d_{1}} ⊗ ⋯ ⊗ L_{m}^{⊗d_{m}})^{t} · P )

Proof The difficult point in this lemma is the fact that the X_{j} do not need to be the irreducible components of X of the maximal dimension (!); in a picture:

The proof is by induction on t (t = 0 is trivial). First we choose t polynomials F_{1}, …, F_{t} of multidegree (d_{1}, …, d_{m}) in the ideal of X, such that each X_{j} is an irreducible component of their set of common zeros. (Having chosen F_{1}, …, F_{k} one simply chooses F_{k+1} such polynomial which is non-zero on all the components of V (F_{1}) ∩ ⋯ ∩ V (F_{k}) which contain an X_{j}.) Then we might as well replace X by the closed subscheme defined by F_{1}, …, F_{t} (this will only make the m_{j} bigger) and enlarge the set of X_{j} to include all the components of X of codimension t.

Let us denote by Y the closed subscheme defined by F_{1}, …, F_{t−1} and by Y_{i} the components of Y of codimension t−1 containing some X_{j}. The multiplicity of Y_{i} in Y is n_{i}. By our choice of F_{t} we have: the irreducible components of (F_{t} = 0) ∩ Y_{i} are the X_{j} say with multiplicities k_{ij}. The formula m_{j} = ∑_{i} k_{ij}n_{i} is a consequence of the fact that the sequence F_{1}, …, F_{t} is a regular sequence in the local ring O_{P,x_{j}} (x_{j} ∈ X_{j} a generic point). Hence we get:

∑_{j} m_{j}(L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · X_{j}) = ∑_{i,j} k_{ij}n_{i}(L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · X_{j}) = ∑_{i} n_{i}(L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · (F_{t} = 0) ∩ Y_{i})
= ∑_{i} n_{i}(L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · L · Y_{i}) ≤ (induction) ≤ (L_{1}^{e_{1}} ⋯ L_{m}^{e_{m}} · L · L^{t−1})

here L = L_{1}^{⊗d_{1}} ⊗ ⋯ ⊗ L_{m}^{⊗d_{m}}. □

References

[1] R. Hartshorne. Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics 155, Springer-Verlag, 1970.