Ample Vector Bundles and Branched Coverings, II

Thomas Peternell  Andrew J. Sommese

November 5, 2002

Introduction

To a degree d covering f : X → Y of projective manifolds there is naturally attached a holomorphic rank (d−1)−vector bundle E_{f}, just by dualizing f_{∗}(O_{X}) and dividing by the trivial factor. This bundle reflects interesting properties of f or X. It has certain positivity properties: it is nef on the generic curve, as observed in [PS00], the first part to this paper, and even more, it is nef on every curve not contained in the branch locus of f (Lazarsfeld). Often it has a lot of sections ([PS00]). For special Y much more is true: if Y = ℙ_{n}, then E is ample and spanned (Lazarsfeld); the same being true for some rational homogeneous manifolds (Kim, Manivel) and conjecturally for all (as long as b_{2} = 1). In the first part to this paper we investigated Fano manifolds and proved nefness in certain cases (del Pezzo manifolds of higher degrees). We suspected that we might have ampleness as well. The first result in this paper is that E is not always ample (2.1): we provide a triple covering over the Fano threefold V_{5}, appearing as a linear section of the Grassmannian G(2, 5) ⊂ ℙ_{9}, for which E is not ample. However the only curves on which E fails to be ample, are lines (not necessarily in the branch locus). Then (2.2) we give a criterion for ampleness, and as a consequence we see that given a covering X → V_{5}, then E is ample if and only if E is ample on every line. Moreover we show that E is always 1−ample. The second part of the paper is devoted to the study of low degree coverings (of Fano manifolds, usually). Again, a result of Lazarsfeld is the starting point: if f : X → ℙ_{n} has degree d, then the restriction maps

H^{i}(ℙ_{n}) → H^{i}(X)

are surjective for i ≤ n + 1 − d. We give various generalizations, e.g., we prove the following: Let f : X → Y have degree 3. If Y is simply connected and h^{1,1}(Y) = 1, then q(X) = q(Y) = 0. Actually, even if Y is not simply connected, we have q(X) = q(Y) unless f is étale. We also show that if say E is spanned and the map defined by the global sections of O(1) on ℙ(E) is “semi-small”, then we have the same topological conclusion as if E were ample. The decisive role here is played by the top Chern class c_{d−1}(E), in connection with the hard Lefschetz theorem. It is an interesting question whether the positivity of c_{d−1}(E), in a differential-geometric or algebraic sense, suffices to guarantee the hard Lefschetz theorem (cupping with c_{d−1}(E)). In sect.4 we apply the positivity of E_{f} to study the ramification divisor R ⊂ X. E.g. we prove that R is ample if Y is a Grassmannian, the cases of projective space and quadrics (of dimension at least 3) being known since the 80’s. We also show that the ramification divisor of a covering over the Fano threefold V_{5} is always ample unless the covering arises by base change from the Example 2.1 (in that case R is nef but not ample).

Contents

1 Preliminaries 2

2 Coverings of the del Pezzo 3−fold V_{5}. 3

3 Topology of low degree coverings of Fano manifolds 9

4 The structure of the ramification divisor 15

5 Further problems and remarks 18

1 Preliminaries

Notation 1.1 Let X and Y be projective manifolds of dimension n and let f : X → Y be a finite covering of degree d. Then we set

E = [f_{∗}(O_{X})/O_{Y}]^{∗},

a vector bundle of rank d − 1 on Y, the vector bundle associated to f.

For details and many properties we refer to [PS00] and the references given there. The bundle E has certain positivity properties which depend on the geometry of Y. For the convenience of the reader here are some results relevant for this paper.

Theorem 1.2 Let f : X → Y be a covering of projective manifolds.

1. ([La00]) E is nef modulo the branch locus B ⊂ Y, i.e., E|C is nef for every curve C ⊄ B.

2. ([PS00]) Let Y be a del Pezzo manifold of degree at least 5. Then E is spanned by global sections, in particular E is nef.

Notation 1.3 Let X be a projective manifold and L a line bundle on X. L is semi-ample, if some positive multiple L^{⊗m} is spanned by global sections. The Stein factorization ϕ : X → Z of the map associated to the linear system of L^{m} is called the map associated to L. So by definition, Z is normal and ϕ has connected fibers.

Notation 1.4 A line L is said to be k−ample, if L is semi-ample and the associated map has at most k−dimensional fibers. A vector bundle E is k−ample, if O_{ℙ(E)}(1) is k−ample on ℙ(E).

Notation 1.5 Let X be a projective manifold. Then H^{k}(X) will always denote cohomology with complex coefficients: H^{k}(X) = H^{k}(X, ℂ).

2 Coverings of the del Pezzo 3−fold V_{5}.

In the paper [PS00] we suspected that the vector bundle E associated to a branched covering f : X → Y over a Fano manifold Y with b_{2}(Y) = 1 is always ample. Here we present an example that this is not quite true, however the deviation from being ample is very small.

Example 2.1 (1) Let Y be a Fano 3-fold of index 2 and of type V_{5}. By definition, Y is a submanifold of the Grassmannian G(2, 5) ⊂ ℙ_{9}, given by three linear sections. If H is the ample generator of Pic(Y) = ℤ, then H^{3} = 5. We consider the family of lines in Y, the geometry of which being studied in [FN89]. The parameter space T of this family is just the projective plane, so that the graph p : G → Y is a ℙ_{1}−bundle q : G → ℙ_{2} = T. By [FN89], p is finite of degree 3; this will be our example: set X = G and f = p. Let R ⊂ X be the branch locus of f; then

c_{1}(E) = \frac{f_{∗}(R)}{2}

by [PS00](1.5). Since the ramification divisor B ⊂ Y is an element of | − K_{Y} |, we conclude from [FN89](2.3) that f_{∗}(R) = B, hence

c_{1}(E) = H.

Since E is spanned by [PS00], the rank 2-bundle E splits on every line l ⊂ Y as

E = O ⊕ O(1).

Hence E cannot be ample.

(2) Let us have a further look at the geometry of the bundle E. First notice that because of the nefness of E, the manifold ℙ(E) is Fano. Let

ψ : ℙ(E) → Z

be the second extremal contraction of ℙ(E) besides the projection π : ℙ(E) → Y. Then ψ is given - up to Stein factorization - by the linear system |O_{ℙ(E)}(1)|. Being careful, let ϕ : ℙ(E) → Z′ be the map defined by |O_{ℙ(E)}(1)|. In order to analyze ϕ, we compute c_{2}(E). Notice first that H^{4}(Y, ℤ) is generated by [l]. Let W ∈ |H| be general and set W̃ = f^{∗}(W). Since W · l = 1, the smooth surface W̃ is generically a section of q : X → T, hence W̃ (and W) are rational. Let g = f|W̃ . By [PS00](1.15) we have

c_{2}(E_{g}) = 3χ(O_{W}) − χ(OW̃) + \frac{(K_{W} + H_{W})}{2} · H_{W}^{2} ,

hence c_{2}(E_{g}) = 2. Thus c_{2}(E) · H = c_{2}(E_{g}) = 2, and consequently

c_{2}(E) = 2l.

Having in mind c_{1}^{2}(E) = 5l, we obtain

c_{1}(O_{ℙ(E)}(1))^{4} = c_{1}^{3}(E) − 2c_{1}(E)c_{2}(E) = 1 > 0.

Thus O_{ℙ(E)}(1) is big, and therefore the map ϕ is generically finite. Now Riemann-Roch gives χ(Y, E) = 5. Since H^{q}(Y, E) = 0 for q ≥ 1, [PS00], we have h^{0}(E) = 5. Therefore Z′ ⊂ ℙ_{4} and since ϕ is generically finite, we have Z′ = ℙ_{4}. Moreover by c_{1}(O_{ℙ(E)}(1))^{4} = 1, we see that ϕ is generically 1 : 1. Zariski’s Main Theorem now implies that ϕ = ψ and Z = Z′. Since Z is smooth, ψ cannot be a small contraction; so let D ⊂ ℙ(E) be the irreducible divisor contracted by the Mori contraction ψ. We are going to show that ψ(D) is a 2-dimensional submanifold of ℙ_{4} = Z and ψ is the blow-up of ψ(D). We first show that dim ψ(D) = 2. In fact, if dim ψ(D) = 0, then Y would be dominated by ℙ_{3} and if dim ψ(D) = 1, then Y would be covered by linear images of ℙ_{2}, which is absurd. Thus dim ψ(D) = 2. In order to show that ψ(D) is smooth and ψ is the blow-up, it is - by Ando’s theorem [An85] - sufficient to verify that ψ|D is equidimensional, i.e., has only 1-dimensional fibers. Notice that the exceptional sections in ℙ(E|l) = O_{l} ⊕ O_{l}(1), l any line in Y, are contracted, so that every line l ⊂ Y has a unique lifting l_{0} which is contracted by ψ. Therefore the map X → Y lifts to a map X → ℙ(E) whose image is just D. Then the projection X → ℙ_{2} yields a map D → ℙ_{2} which must be ψ|D. Therefore ψ|D has only 1-dimensional fibers so that Ando’s theorem applies. Moreover we have seen that ψ(D) = ℙ_{2} and the exceptional sections l_{0} are the only curves contracted by ψ. Therefore the only curves C ⊂ Y on which E is not ample are lines. We also note that E is 1−ample.

(3) As already mentioned, χ(Y, E) = 5. Since H^{q}(Y, E) = 0 for q ≥ 1, [PS00], we have h^{0}(E) = 5. Since E is spanned, we obtain a holomorphic map

τ : Y → G(2, 5).

Since b_{2}(Y) = 1, this map is of course finite onto its image; we are going to prove that τ is actually an embedding and that the image is a linear section of the Plücker embedding G = G(2, 5) ⊂ ℙ_{9}. If V denotes the tautological rank 2−bundle on G, we have E ≃ τ^{∗}(V). Denoting by L the ample generator of Pic(G), then we obtain by taking determinants that H = τ^{∗}(L). Since H^{3} = 5 and since L^{6} = 5, the map τ must be generically finite and τ(Y) ⊂ G is a linear section, i.e., τ(Y) = G ∩ L_{1} ∩ L_{2} ∩ L_{3}, where L_{i} are hyperplane sections of the Plücker embedding G ⊂ ℙ_{9}. By Zariski’s Main Theorem, it only remains to show that τ(Y) cannot be non-normal. This is done as follows. First observe that S̃ = τ^{−1}(S) is a general linear section of Y and therefore a smooth del Pezzo surface with K2 ; S̃ = 5. On the other hand, by the classification of non-normal del Pezzo surfaces, Reid [Re95], S̃ must be ℙ_{2} or a ruled surface. This is a contradiction.

Recall that a vector bundle E is very ample by definition if the line bundle O_{ℙ(E)}(1) is very ample. A line in a Fano manifold with b_{2} = 1 is a rational curve l such that H · l = 1 where H denotes the ample generator. Towards positive results we first prove:

Theorem 2.2 Let f : X → Y be a covering from a projective manifold to the Fano manifold Y. Suppose b_{2}(Y) = 1 and that Y has index at least 2. Let H be the ample generator of Pic(Y). Suppose furthermore that

1. E = E_{f} is spanned

2. there exists S ∈ |H| such that E_{S} is very ample

3. for all lines l ⊂ Y, the restriction E_{l} is ample.

Then E is ample.

Proof. Let ψ : ℙ = ℙ(E) → Z be the map defined by |O_{ℙ(E)}(1)|. Assume that E is not ample, i.e., O_{ℙ(E)}(1) is not ample. Then there exists a curve C ⊂ ℙ with dim ψ(C) = 0. We claim that the restriction map

r : H^{0}(ℙ, O_{ℙ(E)}(1)) → H^{0}(ℙ(E_{S}), O_{ℙ(E_{S})}(1))

is surjective. In fact, this is guaranteed by

H^{1}(E(−H)) = 0.

Now

H^{1}(Y, E(−H)) = H^{n−1}(Y, E^{∗} ⊗ H ⊗ K_{Y}) = H^{n−1}(Y, f_{∗}(O_{X}) ⊗ H ⊗ K_{Y})

since H ⊗ K_{Y} is negative by our assumption on the index. Now the last group is just

H^{n−1}(X, f^{∗}(K_{Y} ⊗ H))

which is zero by Kodaira vanishing. Using the surjectivity of r and the very ampleness of E_{S}, we conclude

0 ≤ C · ℙ(E_{S}) ≤ 1.

Since dim π(C) = 1 (π denoting the projection ℙ → Y), we conclude first

0 ≤ π(C) · H ≤ 1

and thus that π(C) must be a line in Y. But by the choice of C, the bundle E|π(C) is not ample, contradicting our assumption (3). Q.E.D.

Note 2.3 We observe the following (and restrict ourselves to dimension 2 for simplicity). Let S be a smooth projective surface and E a vector bundle on S. Then E is ample, if the following holds.

(*) For all x, y ∈ S there exists a curve C passing through x and y such that E|C is very ample and such that

H^{0}(S, E) → H^{0}(C, E|C)

is surjective.

In fact, (*) first implies that O_{ℙ(E)}(1) is spanned; let ϕ : ℙ(E) → Z be the associated map. We proven even that ϕ is 1 : 1, implying in particular ampleness of O_{ℙ(E)}(1), hence of E. So let x, y ∈ ℙ(E). We may assume that π(x) ≠ π(y), otherwise clearly ϕ(x) ≠ ϕ(y). Let C ⊂ X be a curve containing π(x) and π(y) such that (*) holds for C. Then ϕ|ℙ(E) is an embedding, hence ϕ(x) ≠ ϕ(y).

In the special case that Y = V_{5}, the last theorem can be improved:

Theorem 2.4 Let f : X → Y be a covering from the smooth 3-fold X to the Fano 3-fold Y of type V_{5}. Suppose that E_{f} = E is ample on every line l ⊂ Y. Then E is ample.

Proof. Since E is spanned by [PS00],(4.2), it suffices by Theorem 2.2 to prove that E_{S} is very ample for the general S ∈ |H|. We first follow the arguments of [PS00](3.2). Since K_{S}^{2} = 5, we can realize S as a blow-up p : S → ℙ_{2} of the plane in 4 points in general position. Let H̃ ⊂ ℙ_{2} be a hyperplane and set H = p^{∗}(H̃). Then we can write

K_{S} = −3H + ∑_{i=1}^{4} E_{i}.

Let H_{i} = H − E_{i}. Then every H_{i} is semi-ample and defines a conic bundle structure ϕ_{i} : S → ℙ_{1} with exactly 4 singular fibers. We are going to apply Note 2.3 to the fibers and the fiber components of the maps ϕ_{i}. All fibers resp. fiber components C_{i} of ϕ are smooth rational, whence ampleness and very ampleness is the same. (I) If p(x) = p(y), we argue as follows. There exists some i such that x, y ∈ E_{i}; denote for simplicity l = E_{i}. Then l is a line in Y, hence E|l is ample and thus very ample. Thus we only need to show that

H^{0}(S, E) → H^{0}(l, E_{l})

is onto. This is certainly guaranteed by

H^{1}(S, E(−l)) = 0.

Consider the conic bundle ϕ_{i} which contracts l. Let l′ be the (−1)−curve in S such that l + l′ is a fiber of ϕ_{i}. Then

H^{1}(E_{S}(−F)) = 0

since −K_{S} − F is big and nef (apply [PS00,3.1]). By

0 → E(−F) → E(−l) → E_{l′}(−1) → 0

we conclude

H^{1}(E(−l)) = 0,

settling the case p(x) = p(y). (II) So we shall assume p(x) ≠ p(y). Let x_{i} = p(E_{i}). Choose a possible singular conic C̃ = C̃(x, y) passing through p(x), p(y), x_{1}, x_{2} and x_{3} (including the infinitesimal case p(x) = x_{i} or p(y) = x_{j}). Then we obtain a possible singular curve

C ∈ |2H − E_{1} − E_{2} − E_{3}|

passing through x and y. We are going to prove that

H^{1}(E(−C)) = 0.  (1)

Let S̃ = f^{−1}(S), which is a smooth surface in X, since S is chosen generic. The covering map S̃ → S is again called f. Since C is big and nef, Kawamata-Viehweg vanishing gives

H^{1}(E(−C)) = H^{1}(E^{∗} ⊗ C ⊗ K_{S}) =
= H^{1}(f_{∗}(OS̃) ⊗ C ⊗ K_{S}) = H^{1}(f^{∗}(K_{S} ⊗ C)) = H^{1}(f^{∗}(−H ⊗ E_{4})).

Now consider the map g := ϕ_{4} ∘ f : S̃ → ℙ_{1}. We claim that g_{4} has connected fibers. In fact, otherwise Stein factorization gives a factorization S̃ → B → ℙ_{1} with a finite map B → ℙ_{1}. Hence the ramification divisor R_{S} ⊂ S of f contains some fibers of ϕ_{4}. On the other hand, the ramification divisor R ⊂ Y of the full map f is an ample divisor. Moreover in the decomposition R = ∑ r_{i}R_{i} every R_{i} is an ample irreducible hypersurface and therefore R_{i} ∩ S is an irreducible ample divisor for general S, contradiction. Therefore g has connected fibers. Let F be a general fiber of g. Then I_{F} = g^{∗}(O(−1)) = f^{∗}(−H +E_{4}). Now the Leray spectral sequence yields

h^{1}(S̃, f^{∗}(−H + E_{4})) ≤ h^{1}(ℙ_{1}, O(−1)) + h^{0}(ℙ_{1}, R^{1}g_{∗}(OS̃)).

Since R^{1}g_{∗}(OS̃) is a semi-negative vector bundle, our claim (1) follows. In order to verify 2.3(1), it remains to show that E_{C} is ample. Note that C̃ cannot be a double line; otherwise the 3 points x_{1}, x_{2}, x_{3} would be on a line, which is impossible since −K_{S} is ample. If C̃ = l̃_{1} + l̃_{2} is a sum of two different line, we are already done, since E|l_{i} is ample by assumption, hence E_{l_{i}} is very ample, and then E_{C} is very ample, too. It remains to show that E_{C} is ample in case C̃ is a smooth conic.

Suppose to the contrary that E_{C} is not ample. Then by [PS00,1.3], X_{C} = f^{−1}(C) is disconnected. For this argument we need to observe that C ⊄ R. This follows from the above observation that otherwise C would be an ample curve on S (again we use the genericity of S). Now C is big and nef but not ample: let l be the strict transform of the line joining say x_{1} and x_{2}; then C · l = 0. Now consider the family of “conics” (C̃_{t})_{t∈T} in Y. Say C = C_{t_{0}}. Then of course f^{−1}(C_{t}) is disconnected for general t ∈ T and we claim that this has to be true for all t. In fact, this follows, e.g., from the semi-continuity of h^{0}(O_{X_{C_{t}}}) or from the openness of ampleness in families (again applying [PS00,1.3]). Now consider t_{1} ∈ T such that C_{t_{1}} splits into 2 lines l_{i}. Then say E_{l_{1}} is not ample, contradicting our assumption. Therefore (*) is verified. Q.E.D.

Lemma 2.5 Let f : X → Y be a covering of projective varieties. Let D be an effective nef and big divisor on Y. Then E_{f}|D does not have a trivial direct summand.

Proof. Suppose that

E_{f}|D = O_{D} ⊕ V.

Then h^{0}(E_{f}^{∗}|D) ≠ 0 and therefore h^{0}(f_{∗}(O_{X})|D) ≥ 2. But then

h^{1}(f_{∗}(O_{X}) ⊗ O_{Y} (−D)) = h^{1}(f^{∗}(O(−D)) ≠ 0.

This contradicts the Kawamata-Viehweg vanishing theorem, D being big and nef. Q.E.D.

Corollary 2.6 Let f : X → Y be a covering of smooth n−dimensional projective varieties. Suppose that Pic(Y) = ℤ and that E is spanned. Then E_{f} is (n−2)−ample.

Proof. Let ψ : ℙ = ℙ(E) → Z be the map defined by |O_{ℙ(E)}(1)|. Notice that ψ|π^{−1}(y) is biholomorphic, hence every fiber F of ψ meets π^{−1}(y) in at most one point. Hence π|F is biholomorphic onto its image. Suppose E is not (n−2)−ample. Then we can find a fiber F of ψ with dim F ≥ n−1. Choose a general irreducible variety S ⊂ F of dim S = n − 1 and let S′ = π(S). Then S is a section of ℙ(E_{S′}), given by a rank 1-quotient

E_{S′} → L → 0.

Since E is spanned, clearly L = O_{S′} and moreover

E_{S′} = O_{S′} ⊕ V

with some bundle V. Since Pic(Y) = ℤ, the divisor S′ is ample and Lemma 2.5 provides a contradiction. Q.E.D.

Corollary 2.7 Let f : X → Y be a covering from the smooth 3−fold X to the Fano 3−fold Y of type V_{5}. Then E = E_{f} is 1−ample.

Proof. E is spanned by [PS00,4.2]. Q.E.D.

3 Topology of low degree coverings of Fano manifolds

In [La80] R.Lazarsfeld proved a remarkable theorem on the topology of branched covering of projective space:

Theorem 3.1 Let f : X → ℙ_{n} be a branched cover of degree d. Then the pull-back maps

H^{i}(ℙ_{n}) → H^{i}(X)

are surjective for i ≤ n + 1 − d.

Here as always in this section H^{q}(X) denotes cohomology with values say in ℂ. We will fix for this section a finite cover f : X → Y of projective manifolds of degree d. The associated bundle E_{f} is simply denoted E. We address the question to what extent this theorem holds more generally, say for Fano manifolds instead of ℙ_{n}. We assume that π_{1}(Y) = 0; since otherwise we can produce counterexamples via factorization over finite étale covers. Also we shall assume h^{1,1}(Y) = 1 in order to avoid maps, e.g., to curves which immediately produce counterexamples, too. So we are reduced to Fano manifolds with b_{2} = 1, to Calabi-Yau manifolds or symplectic manifolds with h^{1,1} = 1 and to simply connected manifolds with ample canonical bundles (and h^{1,1} = 1).

We first note that the proof of Lazarsfeld actually gives

Proposition 3.2 The canonical maps

H^{i}(Y) → H^{i}(X)

are surjective (i ≤ n + 1 − d), if

∧c_{d−1}(E) : H^{n+1−d+l}(X) → H^{n−1+d+l}(X)

is surjective for all l ≥ 0. Actually, fixing l ≥ 0,

H^{n+1−d−l}(Y) → H^{n−1+d−l}(X)

is surjective if

∧c_{d−1}(E) : H^{n+1−d+l}(X) → H^{n−1+d+l}(X)

is surjective.

Corollary 3.3 Suppose that E is ample. Then H^{i}(Y) → H^{i}(X) is surjective for i ≤ n + 1 − d. More generally, if E is k−ample, then H^{i}(Y) → H^{i}(X) is surjective for i ≤ n + 1 − d − k.

Proof. If E is k−ample, then ⋀ c_{d−1}(E) is surjective in the given range by [So78]. Q.E.D.

Proposition 3.4 Suppose that h^{d−1,d−1}(Y) = 1 and that c_{d−1}(E) ≠ 0. Then H^{i}(Y) → H^{i}(X) is surjective for i ≤ n + 1 − d.

Proof. By the assumption h^{d−1,d−1}(Y) = 1 we can write

c_{d−1}(E) = c[ω^{d−1}]

for a fixed Kähler form ω and some real number r. Now c ≠ 0 by our assumption and thus the Hard Lefschetz theorem gives the result via (3.2). Q.E.D.

To obtain H^{1}(Y) → H^{1}(X) surjective, it is actually sufficient to assume that h^{1,1}(Y) = 1 (and c_{d−1}(E) ≠ 0 of course). To see this, reduce yourself to d = n by taking hyperplane sections and apply 3.4.

Corollary 3.5 Suppose that Y is a Fano 3−fold of type V_{5}. Then H^{i}(Y) → H^{i}(X) is surjective for i ≤ 4 − d.

Essentially this says that triple covers X → V_{5} have q(X) = 0.

Proof. If E is ample, this follows from (3.3). In general, we only know that E is 1−ample, so we want to apply (3.4) and need to show that c_{2}(E) ≠ 0 (we are left with d = 3). Suppose c_{2}(E) = 0. Since E is spanned, E will have a section without zeroes. So we obtain a sequence

0 → O_{Y} → E → L → 0

with L ample. By Kodaira vanishing H^{1}(L^{∗}) = 0, hence E = O_{Y} ⊕ L. This contradicts h^{0}(E^{∗}) = 0. Q.E.D.

Using stronger tools, we can prove much more:

Corollary 3.6 Let d = 3 and dim Y ≥ 3. Suppose Pic(Y) = ℤ. Then q(X) = q(Y) = 0.

Proof. By taking hyperplane sections of Y we reduce ourselves to dim Y = 3. Then by (3.4) we may assume that c_{2}(E) = 0. Since E might not be spanned, the arguments of (3.5) have to be modified. Since c_{1}(E) > 0, the Bogomolov-Yau inequality is violated and therefore E is H−unstable for a polarization H. Let L ⊂ E be the maximal H−destabilizing subsheaf. Then we obtain a sequence

0 → L → E → I_{Z} ⊗ L′ → 0  (∗)

with Z ⊂ Y of codimension 2 or empty and L′ another line bundle on Y. The destabilizing property

c_{1}(L) · H^{2} > \frac{1}{2}c_{1}(E) · H^{2}

together with h^{1,1}(Y) = 1 implies that L is ample. Moreover E is generically nef (see [PS00]), thus L′ is generically nef and therefore nef. Now by (*)

0 = c_{2}(E) = c_{1}(L) · c_{1}(L′) + [Z],

hence Z = ∅ and c_{1}(L′) = 0. Since Pic(Y) = ℤ, the bundle L′ is trivial. Dualizing (*) we therefore obtain a section in E^{∗}, which is absurd. Q.E.D.

Note that Pic(Y) = ℤ is satisfied as soon as π_{1}(Y) = 0 and h^{1,1}(Y) = 1.

If d ≥ 4, then E has rank d − 1 > 2, and it is much harder to make use of the vanishing c_{d−1}(E) = 0. Nevertheless it seems reasonable to expect that c_{d−1}(E) ≠ 0 if Y is Fano; and maybe much more should be true (see the end of this section and sect.4).

In 3.5/3.6 flat (generic) quotients of E appeared and led to contradictions. This leads to the following considerations. In [PS00] (2.1 and its proof) we have seen that if dim Y = 1 and if E_{f} is not ample, then f factors via an étale cover. This can be restated as follows: if E_{f} has a numerically flat quotient, then f factors. One may ask whether this is true also in higher dimensions.

Problem 3.7 Let f : X → Y be a finite cover of projective manifolds. Suppose that there exists an epimorphism

E_{f} → V → 0

with a numerically flat vector bundle V. Does there exist an étale cover h : Z → Y and a finite cover g : X → Z such that f = h ∘ g, together with an epimorphism E_{h} → V → 0?

We prove that this problem has a positive answer even in a slightly more general context.

Proposition 3.8 Suppose that there is an epimorphism

E_{f} → I_{Z} ⊗ V → 0

with a numerically flat vector bundle V and an analytic set Z of codimension at least 2. Suppose that f is not étale. Then

1. for a general complete intersection curve C ⊂ Y there exists a factorization X_{C} → Z(C) → C with h_{C} : Z(C) → C étale and an epimorphism E_{h_{C}} → V_{C};

2. deg f is not prime;

3. f factors via an étale map h : Z → Y such that V is a quotient of E_{h}.

Proof. (1) follows from [PS00,2.1] (including the proof) by remarking that X_{C} is connected (Bertini).

(2) is trivial from (1).

(3) First we construct h outside a set W ⊂ Y of codimension at least 2. Fix an ample divisor H on Y. Let C ⊂ Y be a general complete intersection curve cut out by n − 1 elements in |mH|, where m is sufficiently large. Let G be the kernel of E → I_{Z} ⊗ V. By our assumption, G_{C} is a H_{C}-destabilizing subsheaf of E_{C}. Let F_{C} be the corresponding maximal destabilizing subsheaf; then by Mehta-Ramanathan, F_{C} extends to a torsion free coherent subsheaf F ⊂ E; we may assume F ⊂ E is saturated. Let T = E/F, a torsion free sheaf. By throwing away a subset of codimension 2, we may assume that T is locally free (of course now X need not be compact any longer) and also that Z = ∅. We claim that - analogously to [PS00,2.1] - O_{Y} ⊕T^{∗} is a subring of f_{∗}(O_{X}) = O_{Y} ⊕E^{∗} outside a set of codimension 2. In fact, consider the multiplication map

σ : (O_{Y} ⊕ T^{∗}) ⊗ (O_{Y} ⊕ T^{∗}) → O_{Y} ⊕ E^{∗}.

In combination with the projection E^{∗} → F^{∗}, we obtain a map

τ : (O_{Y} ⊕ T^{∗}) ⊗ (O_{Y} ⊕ T^{∗}) → F^{∗}.

Since F_{C} is the unique maximal ample subbundle of E_{C} (see [PS00,2.3]) and since T_{C} is numerically flat, we conclude that τ|C = 0. Thus τ = 0 generically, hence τ = 0. Therefore σ maps into O_{Y} ⊕ T^{∗}, so that O_{Y} ⊕ T^{∗} is a subring of f_{∗}(O_{X}). As in the proof of [PS00,2.1], we put Z = Specan(O_{Y} ⊕ T^{∗}), a normal complex space. Removing a set of codimension 2, we may assume that Z is smooth, hence Z → Y is flat. Since Z_{C} → C is étale, it follows that Z → Y is unramified, hence étale. By construction, V^{∗} is a subsheaf of T^{∗}, and since V_{C}^{∗} is flat, V^{∗} must be a subbundle of T^{∗} outside codimension 2. Coming to the original notations, we have now constructed a finite étale map h : Z → Y \W with W in Y being of codimension at least 2. Since π_{1}(Y \W) = π_{1}(Y), clearly h extends to a covering h : Z̃ → Y with B̃ compactifying B. Furthermore the map g : X \ f^{−1}(W) → B extends to a map g : X → Z̃ such that f = h ∘ g everywhere. Finally by construction, V is a quotient of E_{h}.

Q.E.D.

Remark 3.9 Actually the conclusions of (3.8) hold in a more general setting. Instead of assuming the existence of V, it suffices to assume that E_{C} is not ample for general C cut out by general elements in |m_{i}H_{i}|, where m_{i} ≫ 0, so that the theorem of Mehta-Ramanathan applies.

The following generalization of 3.6 now holds:

Corollary 3.10 Let d = 3, dim Y ≥ 3 and h^{1,1}(Y) = 1. Then q(X) = q(Y) unless f is étale.

Proof. We start as in the proof of 3.6, but we can only conclude L′ ≡ 0. Now we apply 3.8(2) to conclude. Q.E.D.

Let us come back to the general context. We introduce the following notation.

Definition 3.11 Let Y_{n} be a projective manifold. Then we set (for k ≤ n and l ≤ n − k):

Lef^{k,l}(Y) = {α ∈ H^{k,k}(Y)| ∧ α : H^{n−k+l}(Y) → H^{n+k+l}(Y) is onto}.

By the Lefschetz theorem [ω^{k}] ∈ Lef^{k,l}(Y) for all Kähler classes [ω] and for all k, l. Moreover it is clear that Lef^{k,l}(Y) is non-empty and Zariski open in H^{k,k}(Y). Let E_{r} be a vector bundle on the projective manifold X_{n}. Then c_{i}(E) > 0 is to say that c_{i}(E) · Z > 0 for all irreducible subvarieties Z ⊂ X such that dim Z = n − i. If c_{r}(E) is represented by a positive form, then of course c_{r}(E) > 0, but the converse is not true.

Problem 3.12 Describe the structure of Lef^{k,l}(Y). If α is a class represented by a positive (k, k)−form, is it true that α ∈ Lef^{k,l}(Y)?

In the notation of (3.11), 3.2 can be restated as follows.

Proposition 3.13 Suppose that c_{d−1}(E_{f}) ∈ Lef^{d−1,l}(Y). Then H^{i}(Y) → H^{i}(X) is onto for i ≤ n + 1 − d − l.

There is a class of non-ample vector bundles whose top Chern classes are in Lef^{k,l}(Y). These bundles are provided by the theory developed in [dCM00]. First we set up some definitions.

Definition 3.14 Let X_{n} be a projective manifold and L a line bundle on X.

1. L satisfies the condition (HL)_{k} (HL for hard Lefschetz) if and only if

∧c_{1}(L)^{k} : H^{n−k}(X) → H^{n+k}(X)

is injective (isomorphic, equivalently).

2. L is HL-ample if L fulfills (HL)_{k} for all k.

HL-ample bundles are called “lef” in [dCM00]. That paper establishes the following result:

Theorem 3.15 (de Cataldo, Migliorini) Suppose that L is semi-ample with associated morphism ϕ. Then L is HL-ample if and only if ϕ is semi-small.

Recall that ϕ : W → Z is semi-small if

dim Z^{k} + 2k ≤ dim W

for all k ≥ 0, where Z^{k} is the set of all z ∈ Z such that dim ϕ^{−1}(z) has dimension k. Given a vector bundle E, we let

ζ = c_{1}(O_{ℙ(E)}(1)).

Definition 3.16 A vector bundle E is HL-ample if and only if O_{ℙ(E)}(1) is HL-ample, i.e if and only if

∧ζ^{k} : H^{dim ℙ−k}(ℙ(E)) → H^{dim ℙ+k}(ℙ(E))

is injective for all k.

Now (3.15) yields:

Corollary 3.17 Let E be a vector bundle such that O_{ℙ(E)}(1) is semi-ample. Then E is HL-ample if and only if the canonical morphism ℙ(E) → Z associated to O_{ℙ(E)}(1) is semi-small.

The connection to Problem 3.12 is provided by

Proposition 3.18 Let X_{n} be a projective manifold and E a rank r−bundle over X. Let l ≥ 0.

∧c_{r}(E) : H^{n−r−l}(X) → H^{n+r−l}(X)

is injective if

∧ζ : H^{n+r−2−l}(ℙ(E)) → H^{n+r−l}(ℙ(E))

is injective.

Proof. [BG71,1.1] Q.E.D.

Corollary 3.19 Let E be a vector bundle of rank r such that O_{ℙ(E)}(1) is semi-ample and that the associated map ℙ(E) → Z is semi-small, i.e., E is HL−ample. Then ∧c_{r}(E) : H^{n−r−l}(X) → H^{n+r−l}(X) is an isomorphism for all l ≥ 0.

Proof. We verify the injectivity of ζ : H^{n+r−2−l}(ℙ(E)) → H^{n+r−l}(ℙ(E)) by realizing ζ as part of the map

(∧ζ)^{l+2} : H^{n+r−2−l}(ℙ(E)) → H^{n+r−l}(ℙ(E))

which is an isomorphism by our assumption. Q.E.D.

Corollary 3.20 Let f : X → Y be a d−sheeted covering of n−dimensional projective manifolds. Suppose that E_{f} is HL-ample. Then

H^{i}(Y) → H^{i}(X)

is surjective for i ≤ n + 1 − d.

Proof. We verify the condition given in (3.2) by proceeding as in [So78,1.17] by virtue of (3.18); notice that it is of course irrelevant whether we shift by −l or by l.

Q.E.D.

Observe that the bundle E from example 2.1 is HL-ample! Thus (3.2) gives a new proof of (2.7).

Remark 3.21 It is also very interesting to consider only

∧c_{r}(E) : H^{n−r}(X) → H^{n+r}(X)

and also to see what the positivity of c_{r}(E) gives. In order to see wether this map is injective we do not need the full strength of (HL) for ζ; we just need (HL)_{1} - besides spannedness - apart from the fact that the injectivity of ∧c_{r}(E) is weaker than the injectivity of ∧ζ. Also note that the injectivity of ∧c_{r}(E) is indeed weaker than the semi-smallness of ϕ_{ζ}; consider e.g., E = T_{ℙ_{n}}(−1). A necessary condition is given by the next proposition.

Proposition 3.22 Suppose that c_{r}(E_{r}) > 0 and that O_{ℙ(E)}(1) is semi-ample. Let ϕ : ℙ(E) → Z be the associated map. Then 2 dim T − dim ℙ(E) ≤ dim ϕ(T) for all T with dim T ≤ n.

Proof. First observe that c_{r}(E) > 0 implies dim ϕ^{−1}(z) ≤ r − 1 for all z ∈ Z. In fact, otherwise there exists an irreducible S ⊂ ℙ(E) of dimension r such that dim ϕ(S) = 0. Now let π : ℙ(E) → X be the projection; then π|S is finite, so that S is a multi-section over S′ = π(S). After a base change h : S → S′, the variety S will be a section. Now h^{∗}(E|S′) must a trivial quotient, hence c_{r}(E|S′) = 0. This is only possible if dim S′ < r, contradiction. So dim ϕ^{−1}(z) − dim ϕ(T) ≤ r − 1 and thus dim T − dim ϕ(T) ≤ r − 1. Now use dim T ≤ n to verify that 2 dim T − dim ℙ(E) ≤ dim ϕ(T). Q.E.D.

4 The structure of the ramification divisor

Again let us fix a covering f : X → Y of projective manifolds. Let n = dim X = dim Y and d be the degree of f. Let R ⊂ X be the ramification divisor of f so that ω_{X} = f^{∗}(ω_{Y}) ⊗ O_{X}(R). We assume always R ≠ ∅. Finally let B ⊂ Y be the branch locus. A standard technique to obtain information on R in case Y is Fano of index m is to use adjunction theory, applied to (now we use the language of divisors) K_{X} + mL, where L = f^{∗}(H) where −K_{Y} = mH with m the index of Y. In particular:

Theorem 4.1 Let Y be projective space or a quadric (of dimension at least 3). Then R is (very) ample.

Proof. [Ei82] and [LPS89]. Q.E.D.

If Y is del Pezzo, then R = K_{X} + (n − 1)L. Now adjunction theory says (see [BS95]) that K_{X} + (n − 1)L is big and nef unless X is in a small list (X is a quadric fibration over a curve or a scroll over a surface), the maps always given by a multiple of K_{X} + (n − 1)L. These special cases need to be considered separately. Apart from that K_{X} + (n − 1)L is big and nef, hence a multiple is spanned giving a birational map which is well understood: the result is the so-called first reduction X′ which needs to be studied further; actually the first reduction only contracts ℙ_{n−1} with normal bundle O(−1). In all other cases R is automatically ample. It is interesting to notice that in Example 2.1, R is not ample. In fact, we have a ℙ_{1}−bundle q : X → ℙ_{2} and R = q^{∗}(O(2)). We will come back to this situation in (4.4)

Here we propose another method, namely we use the surjectivity of the canonical map

κ : f^{∗}f_{∗}(ω_{X/Y}) → ω_{X/Y} = O_{X}(R)

Since f^{∗}f_{∗}(ω_{X/Y}) = O_{X} ⊕ f^{∗}(E), we can use properties of E to get information on R.

Theorem 4.2 1. If E is spanned resp. nef, then R is spanned resp. nef.

2. If E is ample, then R is big and nef. If moreover R = f^{−1}(B) set-theoretically, or if −K_{Y} is nef with b_{2}(Y) = 1 or if some symmetric power S^{p}E is spanned with b_{2}(Y) = 1, then R is ample.

Proof. (1) is clear by what we said before. In order to prove (2) we observe that the map O_{X} → O_{X}(R) induced by κ is the canonical inclusion. Therefore we obtain a surjective map

f^{∗}(E)|R → O_{R}(R)

so that O_{R}(R) is ample. Hence R itself is big and nef and the only potential curves C preventing R from being ample are curves C disjoint from R with R · C = 0. If now R = f^{−1}(B), then B · f(C) = 0. However B is ample by [PS00,1.15], since det E is ample. So suppose now that −K_{Y} is nef. We are going to apply the base point free theorem in order to prove that some multiple mR is spanned. To do so, we will check that rR−K_{X} is nef for some positive integer r. And indeed, rR−K_{X} = (r−1)R−f^{∗}K_{Y}. Now let ϕ : X → W be the associated morphism. Then the exceptional locus E lies completely in the “étale part” of f and therefore ϕ induces a morphism ψ : Y → Z contracting in particular f(C). This contradicts b_{2}(Y) = 1. The last statement follows in the same way, since now R is semi-ample from the beginning. Q.E.D.

From (4.2) we can again derive (4.1), but much more can be said:

Corollary 4.3 1. If Y is a Grassmannian, then R is ample and spanned.

2. If Y is abelian or a product of an abelian variety and a projective space (say), then K_{X} = R is nef.

3. If Y has K_{Y} ≡ 0 and if E_{f} is nef, then K_{X} = R is nef. If E is ample, then K_{X} is big and nef and even ample, if b_{2}(Y) = 1.

In fact, E is ample in case (1) by [Ma97] and nef in case (2) by [PS00]. Notice that if A is abelian, then K_{X} = R so that the corollary says that K_{X} is nef. Of course this follows also from Mori theory: if K_{X} were not nef, then there is a rational curve C ⊂ X (with K_{X} · C < 0), so that X cannot be a covering of an abelian variety.

Corollary 4.4 Let Y be the Fano 3-fold V_{5} (example 2.1). Then R is ample unless X arises by base change from Example 2.1. To be more precise, let f_{0} : X_{0} → Y be the triple cover from (2.1) with projection p : X_{0} → T = ℙ_{2}. Then there exists a finite cover g : S → T such that X = X_{0} ×_{T} S and f = f_{0} ∘ q with q : X → X_{0} the projection.

Proof. Since E is spanned, R is spanned. Moreover we deduce from 2.4 and 4.2 (proof) that R.C > 0 for all curves with the following (possible) exceptions: f(C) is a line or C ∩ R = ∅. Let L = f^{∗}(H), with H the ample generator on Y. Then we have K_{X} + 2L = R. Suppose that R is not ample. Then K_{X} + 2L is nef but not ample, hence by adjunction theory (and R ≠ ∅) we are in one of the following case [BS95]:

1. K_{X} + 2L is big and defines a birational map ϕ : X → X′

2. K_{X} + 2L defines a quadric fibration over a curve

3. K_{X} + 2L defines a scroll X → S.

Let us rule out (1) and (2). In case (1) ϕ is just the blow-up of smooth points, so that ϕ contracts E_{i} = ℙ_{2} with normal bundle O(−1). If none of the E_{i} meets R, then we obtain a contradiction to b_{2}(Y) = 1 as in the proof of (4.2). So suppose that E_{1} ∩ R ≠ ∅. Then we conclude that every curve in E_{1} must be mapped to a line in Y which is absurd. So (1) is ruled out. (2) is done in exactly the same way (consider a general fiber). So we end up with a scroll r : X → S. Then L · r^{−1}(s) = 1 and f has degree 1 on every r−fiber. In particular f maps the r−fibers to lines in Y. Since T is the parameter space of lines in Y and X is the total space of the family, we obtain a finite map S → T and X arises from X_{0} by base change. Q.E.D.

It is also interesting to ask, what consequence the ampleness of R for the bundle E might have. At least one can state

Proposition 4.5 If R is ample, then det E is ample.

Proof. By [PS00,1.15] we have 2 det E = f_{∗}(R) (as divisors). Introduce a norm N on H^{2n−2}(X). Via the injective map f^{∗}(Y) : H^{2n−2}(Y) → H^{2n−2}(X), we obtain a norm on H^{2n−2}(Y). Then there is a positive constant ϵ such that |R · B| ≥ ϵN(B) for every effective curve B ⊂ X since R is positive on the closure of effective curves. Via the projection formula we obtain for every irreducible curve C ⊂ Y that

|2c_{1}(E) · C| = |R · f^{∗}(C)| ≥ ϵN(C)

Kleiman’s criterion now gives the ampleness of det E. Q.E.D.

5 Further problems and remarks

In the following we collect some problems on coverings f : X → Y of projective manifolds of dimension n. Mostly Y will be Fano with b_{2} = 1. We do not touch any more problems concerning the topology of low degree coverings, this being intensively studied in section 3.

Problem 5.1 Suppose that Y is Fano with b_{2}(Y) = 1 or even just a projective manifold with b_{2}(Y) = 1. Is then E_{f} nef? If H denotes the ample generator, is then E spanned outside the base locus of |H|? What can be said about positivity of Chern classes of E_{f}?

Note that one cannot expect in general E to be spanned: let Y be a del Pezzo 3-fold of degree 1, so that H^{3} = 1. Then it is known that |H| has exactly one base point y_{0} and |2H| is spanned. Now take a smooth member B ∈ |2H| and let f : X → Y be the cyclic cover of degree 2 branched along B. Then E_{f} = H and therefore E is not spanned at y_{0}.

Problem 5.2 Let Y be a Fano manifold with b_{2}(Y) = 1. Suppose that E is ample on all rational curves with minimal degree or ample on all covering families of rational curves of minimal degree among covering families or on all rational curves C with −K_{X} · C ≤ dim Y + 1. Is then E ample?

The only evidence for a positive answer is provided by (2.1) and the general philosophy that the geometry of Y is very much dictated by the minimal (covering) rational curves. A slightly weaker formulation would be:

Problem 5.3 Is E_{f} generically ample in the sense that E_{f} is ample say on sufficiently general complete intersections given by very ample divisors?

Problem 5.4 Fix the Fano manifold Y and assume that E_{f} is ample for all coverings f : X → Y. Is then Y rational homogeneous?

Note that necessarily b_{2}(Y) = 1, since otherwise by Mori theory Y admits a non=trivial map to a normal projective variety. Again the only support for a positive answer is provided by (2.1). Even for threefolds the answer is not clear. However here an explicit list exists and one can treat the problem case by case, e.g., the 3-dimensional cubic in ℙ_{4}. It might be interesting to investigate also coverings over other manifolds such as K3 surfaces or Calabi-Yau manifolds.

Problem 5.5 Let Y be a K3 surface or a Calabi-Yau manifold. Is E always nef?

Note that if f : X → Y is a cyclic cover over the K3 surface Y, then E is spanned. In fact, let B ⊂ Y be the smooth branch locus; we may assume D connected, hence D is a smooth irreducible curve. Since D has a root as line bundle, D cannot be a (−2)−curve. Hence g(D) ≥ 1 and if g(D) = 1, then D is a fiber of an elliptic fibration (apply Riemann-Roch to see h^{0}(O_{Y} (D)) ≥ 2). Finally if g(D) ≥ 2, equivalently D^{2} > 0, then the normal bundle sequence reads

0 → O_{Y} → O_{Y} (D) → N_{D} = K_{D} → 0

and therefore H^{1}(O_{Y}) = 0 yields the spannedness of D. Since E_{f} = O_{Y} (D) ⊕ … ⊕ O((d − 1)D), this bundle is spanned, too.

Remark 5.6 There is an analogy with positive-dimensional fibrations which we would like to mention. One standard reference here is [Mo87]. So let f : X → Y be an algebraic fiber space, say with X and Y smooth (projective). One says f has snc-branching if the set (in Y) of singular fibers is contained is contained in a divisor in Y with simple normal crossings. If f has snc-branching, then all sheaves R^{j}f_{∗}(ω_{X/Y}) are locally free, and if moreover f is semi-stable in codimension 1, which is to say that in codimension 1 all fiber are reduced with only normal crossings, then these sheaves are even nef (Viehweg). If in addition L is a very ample line bundle on X, then Kollár has shown that L^{k} ⊗ ω_{Y} ⊗ f_{∗}(ω_{X/Y}) is spanned for k > m := dim Y. In particular f_{∗}(ω_{X/ℙ_{m}}) is spanned. Using Mumford regularity, one can e.g., prove for Y = ℙ_{m} and all j :

(1) R^{j}f_{∗}(ω_{X/Y}) is spanned outside the singular set of this sheaf. Using semi-stable reduction it also can be seen that

(2) R^{j}f_{∗}(ω_{X/Y})(−1) is spanned outside the singular set, unless H^{0}(R^{k−j}(O_{X})^{∗∗}) ≠ 0, where k = dim X − dim Y. In general, R^{j}f_{∗}(ω_{X/Y}) is nef on every curve which is not in the closure of the set of all y ∈ Y such that the fiber X_{y} is singular, and not reduced or not with normal crossings.This is the analogue to Lazarsfeld’s theorem that E_{f} is nef outside the branch locus for coverings f.

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Thomas Peternell  Andrew J. Sommese
Mathematisches Institut  Department of Mathematics
Universität Bayreuth  University of Notre Dame
D-95440 Bayreuth, Germany  Notre Dame, Indiana 46556-4618, U.S,A,
fax: Germany + 921–552785  fax: U.S.A. + 574–631-6579
thomas.peternell@uni-bayreuth.de  sommese@nd.edu

                                 URL: www.nd.edu/∼sommese