??. AMPLE LINE BUNDLES ON MODULI SPACES

Sándor J. Kovács and James McKernan

This chapter is devoted to prove the projectivity of the moduli space of surfaces of general type. Indeed we prove a more general result as follows. (For the definition of semi-positivity see §1.) Most of the ideas appearing in this chapter are already present in [Kollár90] where the matter is discussed in more generality. Here we include a somewhat simplified proof restricting to the necessary cases.

??.0.1 Theorem. Let f : X ⟶ Y be a flat morphism of proper schemes and L a ℚ-line bundle on X, such that L is f-ample and f_{∗}(L^{k}) is a semi-positive vector bundle for all k large and sufficiently divisible. Suppose every fibre is isomorphic to only finitely many others, and for a fibre X_{y} there are only finitely many automorphisms of X_{y} fixing L^{k}|_{X_{y}}.

Then det f_{∗}(L^{m}) is ample, for m large and sufficiently divisible. In particular Y is projective.

??.0.0.1 Remark. The same statement is true for algebraic spaces with essentially the same proof.

§??.1. Semi-positive vector bundles

??.1.1 Definition. Let V be a vector bundle over a scheme Y. We say that V is semi-positive, if for every morphism g : C ⟶ Y from a smooth projective curve C and every quotient bundle Q of g^{∗}V, c_{1}(Q) ≥ 0.

??.1.2 Lemma. Let V be a vector bundle over a scheme Y. The following are equivalent.

(1) V is semi-positive.

(2) For every morphism g : C ⟶ Y from a smooth projective curve C and every quotient line bundle L of g^{∗}V, c_{1}(L) ≥ 0.

(3) O_{ℙ(V)}(1) is nef on ℙ(V).

(4) For every morphism g : C ⟶ Y, g^{∗}V ⊗ H is ample, where H is ample on C.

Typeset by AMS-TEX

Proof. Clearly (1) implies (2). Let π : ℙ(V) ⟶ Y be the natural projection. Let C ⊂ ℙ(V) be a curve. Then O_{ℙ(V)}(1)|_{C} is a quotient of π^{∗}V |_{C}. Thus (2) implies (3).

Let g : C ⟶ Y be any morphism. Then we have a commutative diagram:

ℙ(g^{∗}V) ⟶^{g̃} ℙ(V)
π̃↓  ↓^{π}
C ⟶_{g} Y

such that O_{ℙ(g^{∗}V)}(1) = g^{∗}O_{ℙ(V)}(1). Now if O_{ℙ(V)}(1) is nef, then so is O_{ℙ(g^{∗}V)}(1) and then for any ample line bundle H on C, O_{ℙ(g^{∗}V)}(1) ⊗ π̃^{∗}H is ample. This is exactly the ampleness of g^{∗}V ⊗ H (cf. [Hartshorne66, 3.2]). Hence (3) implies (4).

A quotient bundle of an ample vector bundle is ample and has positive degree [Hartshorne66, 2.2 and 2.6]. Now choose H ample of degree one and conclude that any line bundle quotient L of g^{∗}V has degree at least 1 − degH = 0. Thus (4) implies (2).

Let g : C ⟶ Y be a morphism, and Q a quotient bundle of g^{∗}V. Let W = ℙ(Q) be the subscheme of ℙ(g^{∗}V) corresponding to Q. Then c_{1}(Q) = c_{1}(g^{∗}O_{ℙ(V)}(1))^{k}|_{W}, where k is the dimension of W. Thus (3) implies (1). □

??.1.2 Lemma. Let V be a semi-positive vector bundle. Then W = Sym^{d}(V) is semi-positive.

Proof. There are several ways to prove this result. We present a very general one given in [Kollár90].

First assume the characteristic of the groundfield is not zero. By definition of semi-positivity, we may assume Y is a curve. Note that H^{1}(Y, V ⊗ L) vanishes, for c_{1}(L) ≥ 2g − 1, since every map from V ⊗ L to ω_{Y} must be zero. Thus V ⊗ L is globally generated, for c_{1}(L) ≥ 2g + 1 and we have a surjective map:

⊕^{r}_{i=1}L^{−1} → V.

It follows that W is a quotient of the direct sum of line bundles whose degree is bounded from below by the constant N = −d(2g + 1). Note that N depends only on d and on the genus of the curve, but not on V.

Suppose Q is a quotient vector bundle of W, of negative degree. Applying base change by powers of the Frobenius, we obtain V′, W′ and Q′, such that c_{1}(Q′) < N, a contradiction.

To finish the proof in characteristic zero we use the reduction mod p technique (cf. [Kollár94, II.5.10]).

Suppose W is not semi-positive, i.e., it has a quotient bundle of negative degree. This remains true after reducing mod p for an open dense set of primes. After base change we may assume, that W reduced mod p has a quotient bundle of degree less than −d for an open dense set of primes.

Now choose an ample line bundle of degree 1. V ⊗ H is ample, ampleness is an open condition in flat families, so V ⊗ H is ample mod p for an open dense set of primes. Then by the above argument Sym^{d}(V ⊗ H) ≃ W ⊗ H^{⊗d} is semi-positive, so every quotient bundle of W has degree at least −d, a contradiction. □

§??.2. Ampleness Theorem

We now turn to the proof of (??.0.0). We review the general idea of the proof. First choose the integers k and d satisfying the following properties:

(1) L^{k} is f-very ample

(2) R^{i}f_{∗}(L^{j·k}) = 0 for i, j > 0

(3) the multiplication map p : Sym^{d}(f_{∗}(L^{k})) → f_{∗}(L^{d·k}) is surjective

(4) every fibre X_{y}, embedded into the projective space ℙ_{y} via L^{k}|_{X_{y}} is defined (set theoretically) by degree d equations.

Set V = f_{∗}(L^{d}), and W = Sym^{d}(V). Then Q = f_{∗}(L^{m}) is a quotient of W for m = dk. Now Q is semi-positive and so is det Q by (??.1.1). We claim that det Q is ample. It remains to use the fact that f moves in moduli as much as possible.

Suppose V has rank r and let G = GL(r). Clearly W has structure group G.

??.2.1 Definition. Let W be a vector bundle of rank n with structure group G and p : W ⟶ Q a quotient bundle of rank l. Let Gr(l, n)/G. denote the set of G-orbits on the l-dimensional quotients of an n-dimensional vector space. The classifying map is the natural map of sets u : Y ⟶ Gr(l, n)/G.

We will say that u is finite, if

(1) the fibres of u are all finite, and

(2) the fibres of p have finite G-stabilisers.

Note that in our case the classifying map is automatically finite. Indeed, if y is a point of Y, the embedding of X_{y} into ℙ_{y} identifies H^{0}(X_{y}, L^{k}|_{X_{y}}) ≃ H^{0}(ℙ_{y}, O_{ℙ_{y}}(1)) in a natural way. Then p at the point y is the same as the restriction map, i.e., we have the following commutative diagram:

Sym^{d}(H^{0}(X_{y}, L^{k}|_{X_{y}})) ⟶ H^{0}(X_{y}, L^{d·k}|_{X_{y}})
↓^{≃}  ↓^{≃}
H^{0}(ℙ_{y}, O_{ℙ_{y}}(d)) ⟶ H^{0}(X_{y}, O_{ℙ_{y}}(d)|_{X_{y}})

Therefore the kernel of p consists of the degree d polynomials cutting out X_{y} under the embedding given by L^{k}. Let y_{1}, y_{2} ∈ Y be such that u(y_{1}) = u(y_{2}). Then there is an element of G that maps the kernel of p at y_{1} isomorphically to the kernel of p at y_{2}. Hence it induces an isomorphism between X_{y_{1}} and X_{y_{2}}. Since every fibre is isomorphic to only finitely many others this implies that u has finite fibres.

Similarly, the fibres of p have finite G-stabilisers since for every X_{y} there are only finitely many automorphisms of X_{y} fixing L^{k}|_{X_{y}}.

Thus we are reduced to proving the following result.

??.2.2 Theorem. Let V be a semi-positive vector bundle over the scheme Y, and suppose Q is a quotient of W = Sym^{d} V, such that the classifying map is finite.

Then det Q is ample.

To prove (??.2.1) we are going to apply the Nakai-Moishezon criteria (cf. [Nakai 63], [Moishezon67] and [Kleiman66]):

??.2.2 Theorem (Nakai-Moishezon). Let H be a line bundle on the proper scheme Y. Suppose that (c_{1}(H|_{Z}))^{dim Z} > 0, for every closed irreducible subvariety Z of Y.

Then H is ample.

Pick a subvariety Z of Y. By Chow’s Lemma, if we normalise Z, and blow it up, then we may make Z projective, but we lose the fact that the classifying map is finite. Thus we are reduced to proving:

??.2.2 Lemma. Let V be a semi-positive vector bundle over the projective variety Y, and suppose Q is a quotient of W = Sym^{d} V, such that the classifying map is finite over an open subset of Y.

Then c_{1}(Q)^{dim Y} > 0.

Proof. We introduce a projective bundle ℙ over Y, so that we may lift the classifying map to a rational map g : ℙ ⟶ Gr(l, n). Set

ℙ = ℙ( ⊕^{r}_{i=1}V^{∗})

and let π be the natural projection to ℙ ⟶ Y. Consider the natural map of (3(!!).7.7)

t : ⊕^{r}_{i=1}O_{ℙ}(−1) ⟶ π^{∗}V.

It is generically surjective and by composition if induces a generically surjective map

θ : Sym^{d}( ⊕^{r}_{i=1}O_{ℙ}(−1)) ⟶ π^{∗}Q

Replacing ℙ by a blow up b : ℙ′ ⟶ ℙ, we may assume that b^{∗}θ surjects onto a locally free subsheaf B of b^{∗}π^{∗}Q.

Now we consider the following twist of b^{∗}θ

θ′ : ⊕^{r}_{i=1}O_{ℙ′} ≃ b^{∗}(Sym^{d}( ⊕^{r}_{i=1}O_{ℙ}(−1)) ⊗ O_{ℙ}(d)) ⟶ b^{∗}(π^{∗}Q ⊗ O_{ℙ}(d)).

The classifying map of this quotient is a morphism

g : ℙ′ ⟶ Gr(l, n).

which is a lifting of the classifying map u : Y ⟶ Gr(l, n)/G.

For simplicity we assume that G is transitive on the fibres of ℙ over Y. This is the case we need for our application. For the general case see [Kollár90, 3.13]. Now g is generically finite, since the classifying map u is finite over an open subset of Y, and ℙ has structure group G. Let O_{Gr(l,n)}(1) be the very ample line bundle on Gr(l, n) giving the Plücker embedding. Then det(B) ⊗ b^{∗}O_{ℙ}(dl) = g^{∗}O_{Gr(l,n)}(1) is big.

Pick an ample divisor H on Y. For e large enough, g^{∗}O_{Gr(l,n)}(e) ⊗ b^{∗}π^{∗}O_{ℙ}(−H) will have a section. Thus there is a non-trivial map

O_{ℙ′} ⟶ g^{∗}O_{Gr(l,n)}(e) ⊗ b^{∗}π^{∗}O_{Y} (−H) ≃ det(B)^{⊗e} ⊗ b^{∗}(O_{ℙ}(edl) ⊗ π^{∗}O_{Y} (−H))

Composing with the natural map det(B) ⟶ det(Q), pushing forward to ℙ and then to Y yields

O_{ℙ′} ⟶ b^{∗}(det(Q)^{⊗e} ⊗ O_{ℙ}(edl) ⊗ π^{∗}O_{Y} (−H))
O_{ℙ} ⟶ π^{∗} det(Q)^{⊗e} ⊗ O_{ℙ}(edl) ⊗ π^{∗}O_{Y} (−H)
O_{Y} ⟶ det(Q)^{⊗e} ⊗ π^{∗}O_{Y} (−H) ⊗ π_{∗}(O_{ℙ}(edl)).

Finally we have a nontrivial map

ϕ : (π_{∗}(O_{ℙ}(edl)))^{∗} ≃ Sym^{edl}( ⊕^{r}_{i=1}V) ⟶ det(Q)^{⊗e} ⊗ π^{∗}O_{Y (−H)}

Replacing Y by a blow up, we may assume the image of ϕ is a locally free subsheaf G. c_{1}(G) is nef since Sym^{edl}( ⊕^{r}_{i=1}V) is semi-positive. Thus we have

e · c_{1}(Q) = E + c_{1}(G) + H,

where E is an effective divisor, c_{1}(G) is nef, H is ample and c_{1}(Q) is nef. Then

e^{dim Y} · c_{1}(Q)^{dim Y} = H^{dim Y} + ∑_{i=1}^{dim Y} H^{(dim Y −i)} · c_{1}(Q)^{(i−1)} · (E + c_{1}(G)) > 0. □

This completes the proof of (??.2.1) and (??.0.0).

References

[Fujita78] T. Fujita, Kaehler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), 779–794. [Hartshorne66] R. Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29

(1966), 63–94. [Kawamata81] Y. Kawamata, Characterisation of Abelian Varieties, Comp. Math. 43 (1981),

253-276. [Kleiman66] S. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84 (1966),

293-344. [Kollár90] J. Kollár, Projectivity of complete moduli, J. Diff. Geom. 32 (1990), 235-268. [Kollár94] J. Kollár, Rational curves on algebraic varieties, preprint, University of Utah, 1994. [Moishezon67] B. Moishezon, A criterion for projectivity of complete abstract varieties, Amer.

Math. Soc. Transl 63 (1967), 1-50.

[Nakai63] Y. Nakai, A criterion of ample sheaf on a projective scheme, Amer. J. Math. 85 (1963), 14-26.

[Viehweg83a] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension I, Algebraic Varieties and Analytic Varieties, Advanced Studies in Pure Math., vol. 1, Kinokuniya North Holland, 1983, pp. 329–353.

[Viehweg83b] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension II, Classification of Algebraic and Analytic manifolds, Birkhäuser, 1983, pp. 567–590.