A NOTE ON FANO MANIFOLDS WHOSE SECOND CHERN CHARACTER IS POSITIVE

A. J. DE JONG AND JASON MICHAEL STARR

Abstract. This note outlines some first steps in the classification of Fano manifolds for which c_{1}^{2} − 2c_{2} is positive or nef.

1. Introduction

This note lists the few known examples of Fano manifolds X for which the second graded piece of the Chern character is positive, ch_{2}(T_{X}) = (C_{1}^{2}−2C_{2})(T_{X})/2. There are also many non-examples. Presumably there are many more positive examples. They do not seem easy to find.

Theorem 1.1. In the following cases X is Fano and ch_{2}(T_{X}) is ample, positive or nef.

(1) For every projective and weighted projective space, ch_{2}(T_{X}) is ample.

(2) For a Grassmannian X = Grass(k, n) of k-dimensional subspaces of a fixed n-dimensional space with 2k ≤ n, ch_{2}(T_{X}) is ample if k = 1, positive if n = 2k or n = 2k + 1, and nef if n = 2k + 2.

(3) A complete intersection Y = D_{1} ∩ ⋯ ∩ D_{r} in X is Fano if (C_{1}(T_{X}) − ([D_{1}] + ⋯ + [D_{r}]))|_{Y} is ample. And ch_{2}(T_{Y}) is ample, resp. positive, weakly positive, nef, if (ch_{2}(T_{X}) − 1/2([D_{1}]^{2} + ⋯ + [D_{r}]))|_{Y} is ample, resp. positive, weakly positive, nef.

(4) In particular, for a complete intersection of type (d_{1}, …, d_{r}) in a n-dimensional weighted projective space, ch_{2}(T_{X}) is ample, resp. nef, if d_{1}^{2}+⋯+d_{r}^{2} < n+1, resp. ≤ n + 1.

(5) A product X × Y of Fano manifolds X and Y is Fano, and ch_{2}(T_{X×Y}) is nef if ch_{2}(T_{X}) and ch_{2}(T_{Y}) are nef.

(6) A projective bundle Y = ℙ(E) over a Fano manifold X associated to an extension E of O_{X} by an invertible sheaf L, is Fano if c_{1}(T_{X}) − c_{1}(L) is ample. And ch_{2}(T_{Y}) is nef if ch_{2}(T_{X}) + C_{1}(L)^{2}/2 is nef.

(7) In particular, let (n, d, a) be integers such that d ≥ 1, n ≥ (d^{2} + d + 1)/2, and n − d ≥ a ≥ ⌈\sqrt{max(0, d^{2} − n − 1)}⌉. Let X be a degree d hypersurface in ℙ^{n}, and let E = (O_{ℙ^{n}}(−a) ⊕ O_{ℙ^{n}})|_{X}. Then Y = ℙ(E) is Fano and ch_{2}(T_{Y}) is nef.

Theorem 1.2. In the following cases ch_{2}(T_{X}) is not ample.

(1) For a Grassmannian Grass(k, n) with 2k ≤ n, ch_{2}(T_{X}) is not ample if k > 1, and it is not nef if (n − 2)/2 > k > 1.

(2) For a product X × Y of positive-dimensional Fano manifolds, ch_{2}(T_{X×Y}) is not weakly positive.

(3) For a projective bundle Y = ℙ(E) over a positive-dimensional Fano manifold X, ch_{2}(T_{Y}) is not weakly positive. Moreover, if rk(E) > 2, then Y is nef only if the restriction to every curve in X is semistable.

(4) For a blowing up Y of ℙ^{n} in a nonempty, codimension 2 center, ch_{2}(T_{Y}) is not nef.

Following are the definitions of nef, weakly positive, positive and ample for cycles of codimension greater than one.

Notation 1.3. Let X be a projective variety over an algebraically closed field. For every integer k ≥ 0, denote by N_{k}(X) the finitely-generated free Abelian group of k-cycles modulo numerical equivalence, and denote by N^{k}(X) the k^{th} graded piece of the quotient algebra A^{∗}(X)/Num^{∗}(X), cf. [Ful84, Example 19.3.9]. For every ℤ-module B, denote N_{k}(X)_{B} := N_{k}(X)⊗B, resp. N^{k}(X)_{B} := N^{k}(X)⊗B. Denote by NE_{k}(X) ⊂ N_{k}(X) the semigroup generated by nonzero, effective k-cycles. For B a subring of ℝ, denote by NE_{k}(X)_{B} the B_{>0}-semigroup in N_{k}(X)_{B} generated by NE_{k}(X).

Definition 1.4. A class in N^{k}(X)_{ℝ} is nef if it pairs nonnegatively with every element in \overline{NE}_{k}(X). The corresponding cone is denote Nef^{k}(X). A class is weakly positive if it pairs positively with every element in NE_{k}(X). The corresponding cone is denoted WPos^{k}(X). A class is positive if it is contained in the interior of Nef^{k}(X); the interior of Nef^{k}(X) is denoted Pos^{k}(X). The ample cone is the ℝ_{>0}- semigroup generated by the image of the cup-product map, (Pos^{1}(X))^{k} → N^{k}(X)_{ℝ}. It is denoted Ample^{k}(X), and its elements are ample classes.

Remark 1.5. There are obvious inclusions,

Ample^{k}(X) ⊂ Pos^{k}(X) ⊂ WPos^{k}(X) ⊂ Nef^{k}(X).

For k = 1, Ample^{1}(X) = Pos^{1}(X) by definition. Moreover, by Kleiman’s criterion, this is the ℝ_{>0}-semigroup generated by first Chern classes of ample invertible sheaves. For k > 1, it can happen that Ample^{k}(X) ≠ Pos^{k}(X); for instance, because (N^{1}(X))^{⊗k} → N^{k}(X) is not surjective. There are also examples where Pos^{k}(X) ≠ WPos^{k}(X) and WPos^{k}(X) ≠ Nef^{k}(X).

2. Projective spaces, Grassmannians, products and complete intersections

2.1. Projective spaces. The simplest example is ℙ^{n} for n ≥ 2. Denote by h ∈ N^{1}(ℙ^{n}) the first Chern class of O_{ℙ^{n}}(1). Using the Euler sequence,

0 ⟶ O_{ℙ^{n}} ⟶ O_{ℙ^{n}}(1)^{⊕(n+1)} ⟶ T_{ℙ^{n}} ⟶ 0,

the Chern character of T_{ℙ^{n}} is (n + 1)e^{h} − 1. In particular, ch_{k}(T_{X}) = (n + 1)h^{k}/k! for every k = 1, …, n. So ch_{k}(T_{X}) is ample for k = 1, …, n.

Weighted projective spaces work the same way provided we consider the space as a smooth Deligne-Mumford stack.

2.2. Grassmannians. Let X be the Grassmannian Grass(k, n) of k-dimensional subspaces of a fixed n-dimensional space. Since Grass(k, n) ≅ Grass(n − k, n), assume 2k ≤ n without loss of generality. Denote by O_{X}^{⊕n} → S_{k}^{∨} the universal rank k quotient. There is an analogue of the Euler sequence,

0 ⟶ Hom_{O_{X}} (S_{k}^{∨}, S_{k}^{∨}) ⟶ (S_{k}^{∨})^{⊕n} ⟶ T_{X} ⟶ 0.

The Chern classes of S_{k}^{∨} are the Schubert classes,

C_{m}(S_{k}^{∨}) = σ_{1^{m}} = σ_{1,...,1}.

Therefore, by standard Chern class computations

ch(T_{X}) = (n − k)k + nσ_{1} + [\frac{n + 2 − 2k}{2} σ_{2} − \frac{n − 2 − 2k}{2} σ_{1,1}] .

In particular, if n > 2k+2, then ch_{2}(T_{X}) has negative intersection with the effective Schubert cycle dual to σ_{1,1}. If n = 2k + 2, ch_{2}(T_{X}) has intersection number 0. If n = 2k or n = 2k + 1, ch_{2}(T_{X}) has positive intersection number with every irreducible surface in X. But it is not a multiple of σ_{1}^{2}, thus it is not ample.

2.3. Products. For a product X × Y, there is an isomorphism

T_{X×Y} ≅ pr_{X}^{∗}T_{X} ⊕ pr_{Y}^{∗} T_{Y} .

Therefore there is an equation

ch(T_{X×Y}) = pr_{X}^{∗}ch(T_{X}) + pr_{Y}^{∗} ch(T_{Y}).

In particular C_{1}(T_{X×Y}) = pr_{X}^{∗}C_{1}(T_{X})+pr_{Y}^{∗} C_{1}(T_{Y}) is ample if C_{1}(T_{X}) and C_{1}(T_{Y}) are ample. Similarly, ch_{2}(T_{X×Y}) is nef if ch_{2}(T_{X}) and ch_{2}(T_{Y}) are nef. However, for every curve A ⊂ X and every curve B ⊂ Y, the intersection number of ch_{2}(T_{X×Y}) with A × B is 0. Therefore ch_{2}(T_{X×Y}) is not weakly positive.

2.4. Complete intersections. Let Y be a smooth complete intersection of divisors D_{1}, …, D_{r} in X. There is an exact sequence

0 ⟶ T_{Y} ⟶ T_{X}|_{Y} ⟶ ⊕_{i=1}^{r}O_{X}(D_{i})|_{Y} ⟶ 0.

Therefore there is an equation

ch(T_{Y}) = [ch(T_{X}) − ∑_{i=1}^{r} e^{[D_{i}]}] |_{Y} .

In other words, for every integer m,

ch_{m}(T_{Y}) = [ch_{m}(T_{X}) − \frac{1}{m!} ∑_{i=1}^{r}[D_{i}]^{m}] |_{Y} .

Therefore Y is Fano if (C_{1}(T_{X}) − ([D_{1}] + ⋯ + [D_{r}]))|_{Y} is ample. And ch_{2}(T_{Y}) is ample, resp. positive, weakly positive, nef, if (ch_{2}(T_{X}) − 1/2([D_{1}]^{2} + ⋯ + [D_{r}]))|_{Y} is ample, resp. positive, weakly positive, nef.

In particular, taking X to be an n-dimensional weighted projective spaces, and taking [D_{i}] = d_{i}h for each i = 1, …, r, the Chern character of T_{Y} is (n + 1)e^{h} − 1 − ∑_{i=1}^{r} e^{d_{i}h}. Thus ch_{k}(T_{Y}) = 1/k!(n + 1 − (d_{1}^{k} + ⋯ + d_{r}^{k}))h^{k} for k = 1, …, n − r. In particular, if d_{1}^{2} + ⋯ + d_{r}^{k} < n + 1, resp. ≤ n + 1, then ch_{2}(T_{Y}) is ample, resp. nef.

3. Projective bundles

One way to produce new examples of Fano manifolds is to form the projective bundle of a vector bundle of “low degree” over a given Fano manifold.

Lemma 3.1. Let E be a vector bundle on X of rank r. Denote by π : ℙE → X the associated projective bundle. The graded pieces of the Chern character of T_{ℙE} are, c_{1}(T_{ℙE}) = rζ+π^{∗}(c_{1}(T_{X})+c_{1}(E)) and ch_{2}(T_{ℙE}) = rζ^{2}/2+π^{∗}c_{1}(E)ζ+π^{∗}(ch_{2}(T_{X})+ ch_{2}(E)), where ζ equals c_{1}(O_{ℙE}(1)).

Proof. There is an Euler sequence,

0 ⟶ O_{ℙE} ⟶ π^{∗}E ⊗ O_{ℙE}(1) ⟶ T_{ℙE/X} ⟶ 0.

Therefore ch(T_{ℙE/X}) = π^{∗}ch(E)e^{ζ} − 1, i.e.,

\begin{matrix}(r + π^{∗}c_{1}(E) + π^{∗}ch_{2}(E) + …)(1 + ζ + ζ^{2}/2 + …) − 1 = \\ [r − 1] + [rζ + π^{∗}c_{1}(E)] + [rζ^{2}/2 + π^{∗}c_{1}(E)ζ + π^{∗}ch_{2}(E)] + … \end{matrix}

Using the exact sequence,

0 ⟶ T_{ℙE/X} ⟶ T_{ℙE} ⟶ π^{∗}T_{X} ⟶ 0,

ch(T_{ℙE}) equals ch(T_{ℙE/X})+π^{∗}ch(T_{X}). Thus ch_{1}(T_{ℙE/X}) = rζ +π^{∗}(c_{1}(T_{X})+c_{1}(E)) and,

ch_{2}(T_{ℙE}) = rζ^{2}/2 + π^{∗}c_{1}(E)ζ + π^{∗}(ch_{2}(T_{X}) + ch_{2}(E)).
□

Proposition 3.2. Let X be a smooth Fano manifold and let E be a vector bundle on X of rank r. The projective bundle ℙE is Fano if and only if there exists ϵ > 0 such that for every irreducible curve B ⊂ X,

μ_{B}^{1}(E|_{B}) − μ_{B}(E|_{B}) ≤ (1 − ϵ)deg_{B}(−K_{X})/r,

where μ_{B} and μ_{B}^{1} are the slopes from Definition 5.2, resp. Definition 5.3.

Proof. The invertible sheaf ω_{ℙE}^{∨} is π-relatively ample. By hypothesis, ω_{X}^{∨} is ample. By Lemma 5.4, ω_{ℙE}^{∨} is ample iff there exists a real number ϵ > 0 such that

deg_{B}(g^{∗}ω_{ℙE}^{∨}) ≥ ϵdeg_{B}(g^{∗}π^{∗}ω_{X}^{∨}),

for every finite morphism g : B → ℙE of a smooth, connected curve to X for which π ∘ g is also finite. Using the universal property of ℙE, this holds iff for every finite morphism f : B → X and every invertible quotient f^{∗}E^{∨} → L^{∨},

deg_{B}(g^{∗}ω_{ℙE}^{∨}) ≥ ϵdeg_{B}(g^{∗}π^{∗}ω_{X}^{∨}),

where g : B → ℙE is the associated morphism. By Lemma 3.1, deg_{B}(ω_{ℙE}^{∨}) equals rc_{1}(L^{∨}) + c_{1}(f^{∗}E) + c_{1}(f^{∗}T_{X}), i.e.,

r[c_{1}(f^{∗}T_{X})/r − (μ_{B}(L) − μ_{B}(f^{∗}E))].

So, finally, ω_{ℙE}^{∨} is ample iff there exists ϵ > 0 such that for every finite morphism f : B → X and every invertible quotient f^{∗}E^{∨} ↠ L^{∨},

μ_{B}(L) − μ_{B}(f^{∗}E) ≤ (1 − ϵ)deg_{B}(f^{∗}c_{1}(T_{X}))/r.

Taking the supremum over covers of B and invertible quotients of the pullback of E, this is,

μ_{B}^{1}(f^{∗}E) − μ_{B}(f^{∗}E) ≤ (1 − ϵ)deg_{B}(−f^{∗}K_{X})/r.

Since every finite morphism f : B → X factors through its image, it suffices to consider only irreducible curves B in X. □

For r = 2, there is a necessary and sufficient condition for ch_{2}(T_{ℙE}) to be nef.

Proposition 3.3. Let E be a vector bundle on X of rank 2. Denoting by π : ℙE → X the projection, ch_{2}(T_{ℙE}) = π^{∗}(ch_{2}(T_{X}) + 1/2(c_{1}^{2} − 4c_{2})(E)). Therefore ch_{2}(T_{ℙE}) is nef iff ch_{2}(T_{X}) + 1/2(c_{1}^{2} − 4c_{2})(E) is nef. If dim(X) > 0, ch_{2}(T_{ℙE}) is not weakly positive.

Proof. By Lemma 3.1, ch_{2}(T_{ℙE}) equals ζ^{2} + π^{∗}c_{1}(E)ζ + π^{∗}(ch_{2}(T_{X}) + ch_{2}(E)). By definition of the Chern classes of E, ζ^{2} + π^{∗}c_{1}(E)ζ + π^{∗}c_{2}(E) equals 0. So the class above is −π^{∗}c_{2}(E) + π^{∗}(ch_{2}(T_{X}) + ch_{2}(E)). Finally, ch_{2}(E) − c_{2}(E) equals 1/2(c_{1}^{2} − 2c_{2})(E) − c_{2}(E) = 1/2(c_{1}^{2} − 4c_{2})(E). □

Applying Proposition 3.2 and Proposition 3.3 to the vector bundle E = L^{∨} ⊕ O_{X} gives Theorem 1.1(6).

Finally, for r > 2, there is a necessary condition for ch_{2}(T_{ℙE}) to be nef.

Proposition 3.4. Let E be a vector bundle of rank r > 2 on X. If ch_{2}(T_{ℙE}) is nef, then the pullback of E to every smooth, projective, connected curve is semistable. Also, ch_{2}(T_{ℙE}) is not weakly positive if dim(X) > 0 and if the pullback of E to some curve is strictly semistable, e.g., if X contains a rational curve.

Proof. If the pullback of E to some smooth, projective, connected curve is not semistable, then by Corollary 5.11, there exists a smooth, projective, connected curve B, a morphism f : B → X, and a rank 2 locally free subsheaf F of f^{∗}E such that f^{∗}E/F is locally free and μ_{B}(F) > μ_{B}(E). There is an induced morphism g : ℙF → ℙE such that π ∘ g = f ∘ π. By Lemma 3.1, g^{∗}ch_{2}(T_{ℙE}) equals rξ^{2}/2 + π^{∗}f^{∗}c_{1}(E)ξ + π^{∗}f^{∗}(ch_{2}(T_{X}) + ch_{2}(E)), where ξ equals c_{1}(O_{ℙF} (1)). Since B is a curve, f^{∗}(ch_{2}(T_{X}) + ch_{2}(E)) equals 0. Also, by definition of the Chern classes of F, ξ^{2} + π^{∗}c_{1}(F)ξ = 0. Substituting in,

g^{∗}ch_{2}(T_{ℙE}) = 1/2π^{∗}(2c_{1}(f^{∗}E) − rc_{1}(F))ξ.

In particular, deg_{ℙF} (g^{∗}ch_{2}(T_{ℙE})) equals 1/2(2deg_{B}(c_{1}(f^{∗}E)) − rdeg_{B}(F)). This equals r(μ_{B}(f^{∗}E)−μ_{B}(F)), which is negative by construction. Therefore ch_{2}(T_{ℙE}) is not nef. □

Remark 3.5. A vector bundle on a product of projective spaces whose restriction to every curve is semistable is of the form L^{⊕r}, where L is an invertible sheaf, [OSS80, Thm. 3.2.1]. In this case, ℙE is also a product of projective spaces.

Corollary 3.6. Let X be a Fano manifold. For every vector bundle E on X of rank r > 1, ch_{2}(T_{ℙE}) is not weakly positive.

4. Blowings up

Let X be a smooth, connected, projective variety, let i : Y ↪ X be the closed immersion of a smooth, connected subvariety of X of codimension c. Denote by ν : X̃ → X the blowing up of X along Y. Denote by π : E → Y the exceptional divisor. Denote by j : E → X̃ the obvious inclusion. Then E = ℙN_{Y/X} and i^{∗}OX̃(E) is canonically isomorphic to O_{ℙN}(−1).

Lemma 4.1. The graded pieces of the Chern character of X̃ are, c_{1}(TX̃) = ν^{∗}c_{1}(T_{X})− (c − 1)[E] and ch_{2}(TX̃) = ν^{∗}ch_{2}(T_{X}) + (c + 1)[E]^{2}/2 − i_{∗}π^{∗}c_{1}(N_{Y/X})

Proof. Using the short exact sequence,

0 ⟶ ν^{∗}Ω_{X} ⟶ ΩX̃ ⟶ j_{∗}Ω_{π} ⟶ 0,

ch(ΩX̃) equals ν^{∗}ch(Ω_{X}) + ch(j_{∗}Ω_{π}). Grothendieck-Riemann-Roch for the morphism j gives,

ch(Rj_{∗}a) = j_{∗}(ch(a))(1 − e^{−[E]})/[E].

Using the Euler sequence for Ω_{π},

0 ⟶ Ω_{π} ⟶ π^{∗}N_{Y/X}^{∨} ⊗ O_{ℙN}(−1) ⟶ O_{E} ⟶ 0,

ch(Ω_{π}) equals π^{∗}ch(N_{Y/X}^{∨})i^{∗}(1 + e^{[E]}) − 1. Putting the pieces together gives the lemma. □

When is X̃ Fano? Denote by C_{1} the collection of finite morphisms g : B → X from a smooth, connected curve to X whose image is not contained in Y. Denote by C_{2} the collection of finite morphisms g : B → Y from a smooth, connected curve to Y. The following result is well-known.

Proposition 4.2. Let h be the first Chern class of an ample invertible sheaf on X, e.g., h = c_{1}(T_{X}) if X is Fano. The blowing up X̃ is Fano iff there exists ϵ > 0 such that,

(i) for every g : B → X in C_{1},

deg_{B}(g^{−1}Y) ≤ \frac{1}{c − 1}(deg_{B}(g^{∗}c_{1}(T_{X})) − ϵdeg_{B}(g^{∗}h)),

and (ii) for every g : B → Y in C_{2},

μ_{B}^{1}(g^{∗}N_{Y/X}) ≤ \frac{1}{c − 1}(deg_{B}(g^{∗}c_{1}(T_{X})) − ϵdeg_{B}(g^{∗}h)).

The proof is similar to the proof of Proposition 3.2. Using an analogue of Proposition 3.3, no blowing-up of ℙ^{n} is a Fano manifold with ch_{2} nef.

5. Theorems about vector bundles on curves

There are two theorems in this section. The first theorem goes back to Shou-Wu Zhang, though possibly it is older. A much more sophisticated arithmetic analogue was also proved by Shou-Wu Zhang in [Zha95, Theorem 1.10]. The second theorem in this section is a variation of the first theorem.

Definition 5.1. Let B be a smooth, projective curve. A cover of B is a finite, flat morphism f : C → B of constant, positive degree. A vector bundle on B is a locally free O_{B}-module of constant rank.

Definition 5.2. Let B be a smooth, projective curve. For every non-zero vector bundle E on B, the slope is,

μ_{B}(E) = deg(E)/rank(E) = χ(B, E)/rank(E) − χ(B, O_{B}).

For every cover f : C → B and every non-zero vector bundle E on C, the B-slope is,

μ_{B}(f, E) := deg(E)/(deg(f)rank(E)) = μ_{B}(f_{∗}E) − μ_{B}(f_{∗}O_{C}).

When there is no chance of confusion, this is denoted simply μ_{B}(E).

For every cover g : C′ → C, f ∘ g : C′ → B is a cover and μ_{B}(f ∘ g, g^{∗}E) equals μ_{B}(f, E).

Definition 5.3. Let B be a smooth, projective curve and let E be a vector bundle on B of rank r > 0. For every integer 1 ≤ k ≤ r, define μ_{B}^{k}(E) to be,

sup{−μ_{B}(f, F^{∨})|f : C → B a cover , f^{∗}E^{∨} → F^{∨} a rank k quotient}
= sup{μ_{B}(f, F)|f : C → B a cover , F ⊂ f^{∗}E a rank k

subbundle whose cokernel is locally free}.

Let f : X → Y be a morphism of projective varieties. Denote by C_{1} the collection of all irreducible curves in X not contained in a fiber of f. Denote by C_{2} the collection of finite morphisms g : C → X occurring as the normalization of an irreducible curve in X not contained in a fiber of f. Finally, denote by C_{3} the collection of all finite morphisms from smooth, connected curves to X whose image is not contained in a fiber of f.

Lemma 5.4. Let f : X → Y be a morphism of projective varieties and let L be an ample invertible O_{Y} -module. An f-ample invertible O_{X}-module M is ample iff there exists a real number ϵ > 0 such that for every morphism g : C → X in C_{1}, resp. C_{2}, C_{3}, deg_{C}(g^{∗}M) ≥ ϵdeg_{C}(g^{∗}f^{∗}L).

Proof. Because M is f-ample and L is ample, there exists an integer n > 0 such that M ⊗ f^{∗}L^{⊗n} is ample. By Kleiman’s criterion, M is ample iff there exists a real number 0 < δ < 1 such that for every irreducible curve C in X,

deg_{C}(M) ≥ δdeg_{C}(M ⊗ f^{∗}L^{⊗n}).

Simplifying, this is equivalent to,

deg_{C}(M) ≥ \frac{nδ}{1 − δ} deg_{C}(f^{∗}L).

As M is f-ample, this holds if C is contained in a fiber of f. So M is ample iff the inequality holds for every curve in C_{1}. Setting ϵ = nδ/(1 − δ), δ = ϵ/(n + ϵ), gives the lemma.

Since C_{2} ⊂ C_{3}, the condition for C_{3} implies the condition for C_{2}. Since degrees on a curve can be computed after pulling back to the normalization, the condition for C_{2} implies the condition for C_{1}. Finally, for every morphism g : C → X in C_{3}, g(C) is in C_{1}. The inequality for g(C) implies the inequality for C. Thus the condition for C_{1} implies the condition for C_{3}. □

Lemma 5.5. Let B be a smooth, connected, projective curve. A nonzero vector bundle E on B is ample iff there exists a positive real number δ such that for every cover f : C → B and every invertible quotient f^{∗}E → L, μ_{B}(L) ≥ δ. In other words, E is ample iff μ_{B}^{1}(L^{∨}) < 0.

Proof. Denote by π : ℙE^{∨} → B the projective bundle associated to E^{∨}, and denote by π^{∗}E → O_{ℙE^{∨}}(1) the tautological invertible quotient. By definition, E is ample iff O_{ℙE^{∨}}(1) is an ample invertible sheaf. Of course O_{ℙE^{∨}}(1) is π-relatively ample. Let M be an invertible O_{B}-module of degree 1. Then M is ample. By Lemma 5.4, O_{ℙE^{∨}}(1) is ample iff there exists ϵ > 0 such that for every smooth, connected curve C and every finite morphism g : C → ℙE^{∨} such that π ∘ g is finite, deg_{C}(g^{∗}O_{ℙE^{∨}}(1)) ≥ ϵdeg_{C}(g^{∗}π^{∗}M). Of course deg_{C}(g^{∗}π^{∗}M) = deg(π ∘ g). Using the universal property of ℙE^{∨}, this holds iff for every cover f : C → B and every invertible quotient f^{∗}E → L,

deg_{C}(L) ≥ ϵdeg(f) ⇔ μ_{B}(L) ≥ ϵ.
□

Lemma 5.6. For every ample vector bundle E on B, there exists a cover f : C → B, invertible O_{C}-modules L_{1}, …, L_{r}, and a morphism of O_{C}-modules, ϕ : f^{∗}E → (L_{1} ⊕ ⋯ ⊕ L_{r}) such that,

(i) the support of coker(ϕ) is a finite set, (ii) for every i = 1, …, r, the projection f^{∗}E → ⊕_{j≠i}L_{j} is surjective, and (iii) for every i = 1, …, r, μ_{B}(L_{i}) = deg_{B}(E).

Proof. Denote r = rank(E). The claim is that for every k = 1, …, r, there exists a cover f_{k} : C_{k} → B, invertible O_{C_{k}}-modules L_{k,1}, …, L_{k,k}, and a morphism of O_{C_{k}}-modules, ϕ_{k} : f^{∗}E → (L_{k,1} ⊕ ⋯ ⊕ L_{k,k}) satisfying (ii) and (iii) above and the following variant of (i): for k < r, ϕ_{k} is surjective and for k = r, the support of coker(ϕ_{k}) is a finite set. The lemma is the case k = r. The claim is proved by induction on k.

The base case is k = 1. Denote by π : ℙE^{∨} → B the projective bundle associated to E^{∨}, and denote by π^{∗}E ↠ O_{ℙE^{∨}}(1) the tautological invertible quotient. By hypothesis, O_{ℙE^{∨}}(1) is ample. By Bertini’s theorem, for d_{1}, …, d_{r−1} ≫ 0, there exist effective Cartier divisors D_{1}, …, D_{r−1} with D_{i} ∈ |O_{ℙE^{∨}}(d_{i})| such that the intersection C_{1} = D_{1} ∩ ⋯ ∩ D_{r} is a smooth, connected curve, cf. [Jou83]. Denote by f_{1} : C_{1} → B the restriction of π. Denote by ϕ_{1} : f^{∗}E → L_{1,1} the restriction of π^{∗}E → O_{ℙE^{∨}}(1). This satisfies (i) because π^{∗}E → O_{ℙE^{∨}}(1) is surjective. It satisfies (ii) trivially. Finally, deg(f) equals d_{1} × ⋯ × d_{r−1}, and deg_{C_{1}}(L_{1,1}) equals d_{1}×⋯×d_{r−1}×[c_{1}(O_{ℙE^{∨}}(1))]^{r}, i.e., d_{1}×⋯×d_{r−1}×deg_{B}(E). Therefore μ_{B}(L_{1,1}) = deg_{B}(E), i.e., this satisfies (iii).

By way of induction, assume the result is known for k < r, and consider the case k + 1. Since ϕ_{k} is surjective, there is an induced closed immersion ℙ(L_{k,1} ⊕ ⋯ ⊕ L_{k,k})^{∨} ↪ ℙ(f_{k}^{∗}E)^{∨}. The image is irreducible and has codimension r − k ≥ 1. For every i = 1, …, k, the image of ℙ(⊕_{j≠i}L_{k,j})^{∨} is irreducible and has codimension r − k + 1 ≥ 2. Associated to the finite morphism f_{k}, there is a finite morphism ℙ(f_{k}^{∗}E)^{∨} → ℙE^{∨}. The pullback of an ample invertible sheaf by a finite morphism is ample; hence O_{ℙ(f_{k}^{∗}E)^{∨}}(1) is ample. By Bertini’s theorem, for d_{1}, …, d_{r−1} ≫ 0, there exist effective Cartier divisors D_{1}, …, D_{r−1} with D_{i} ∈ |O_{ℙ(f_{k}^{∗}E)^{∨}}(d_{i})| such that the intersection C_{k+1} = D_{1} ∩ ⋯ ∩ D_{r−1} is a smooth, connected curve, disjoint from ℙ(⊕_{j≠i}L_{j})^{∨} for every i = 1, …, k, and either disjoint from ℙ(⊕_{i}L_{i})^{∨} if k < r − 1, or else intersecting ℙ(⊕_{i}L_{i})^{∨} in finitely many points if k = r − 1. Define g_{k+1} : C_{k+1} → C_{k} to be the restriction of the projection. Define f_{k+1} = f_{k} ∘ g_{k+1}, define L_{k+1,i} = g_{k+1}^{∗}L_{k,i} for i = 1, …, k, and define L_{k+1,k+1} to be the restriction of O_{ℙ(f_{k}^{∗}E)^{∨}}(1). Define ϕ_{k+1} to be the obvious morphism.

The cokernel of ϕ_{k+1} is supported on the intersection of C_{k+1} with ℙ(L_{k,1} ⊕ ⋯ ⊕ L_{k,k})^{∨}. By construction, this is empty if k < r − 1, and is a a finite set if k = r − 1. Thus ϕ_{k+1} satisfies (i). By the induction hypothesis, f_{k+1}^{∗}E → (L_{k+1,1} ⊕ ⋯ ⊕ L_{k+1,k}), which is the pullback under g_{k+1} of ϕ_{k}, is surjective. For i = 1, …, k, the cokernel of f_{k+1}^{∗}E → ⊕_{j≠i}L_{k+1,j} is supported on the intersection of C_{k+1} with the image of ℙ(⊕_{j≠i}L_{k,j})^{∨}). By construction, this is empty, i.e., f_{k+1}^{∗}E → ⊕_{j≠i}L_{k+1,j} is surjective. Thus ϕ_{k+1} satisfies (ii). Finally, ϕ_{k+1} satisfies (iii) by the same argument as in the base case. The claim is proved by induction on k. □

Theorem 5.7. For every non-zero vector bundle E on B, for every ϵ > 0, there exists a cover f : C → B and a invertible quotient f^{∗}E → L such that μ_{B}(L) < μ_{B}(E) + ϵ. In other words, μ_{B}^{1}(E^{∨}) ≥ μ_{B}(E^{∨}).

Proof. Denote r = rank(E). If r = 1, set f = Id_{B} and L = E. Then L is an invertible quotient of f^{∗}E, and μ_{B}(L) equals μ_{B}(E) which is less than μ_{B}(E) + ϵ. Therefore assume r > 1.

Certainly an effective version of the following argument can be given, but a simpler argument is by contradiction.

Hypothesis 5.8. For every cover f : C → B and every invertible quotient f^{∗}E → L, μ_{B}(L) is ≥ μ_{B}(E) + ϵ, i.e., μ_{B}^{1}(E^{∨}) < μ_{B}(E^{∨}) − ϵ.

By way of contradiction, assume Hypothesis 5.8. Let f : C → B be a connected, smooth cover of degree d. For every a/d ∈ \frac{1}{d}ℤ, there exists an invertible sheaf M on C of degree a, and thus μ_{B}(M) = a/d. In particular, for d sufficiently large, there exists an invertible quotient M such that 0 < μ_{B}(E) − μ_{B}(M) < ϵ/(r − 1). Denote δ = μ_{B}(E) − μ_{B}(M). Denote F = f^{∗}E ⊗ M^{∨}. Then μ_{B}(F) equals δ, and 0 < δ < ϵ/(r − 1).

Let g : C′ → C be any cover and let g^{∗}F → N be any invertible quotient. Then f ∘ g : C′ → B is a cover and (f ∘ g)^{∗}E = g^{∗}F ⊗ g^{∗}M → N ⊗ g^{∗}M is an invertible quotient. By Hypothesis 5.8,

μ_{C}(N) = deg(f)μ_{B}(N) = deg(f)(μ_{B}(N ⊗ g^{∗}M) − μ_{B}(M))

≥ deg(f)((μ_{B}(E) + ϵ) − μ_{B}(M)) > deg(f)ϵ.

By Lemma 5.5, F is an ample vector bundle on C. By Lemma 5.6, there exists a cover g : C′ → C and an invertible quotient g^{∗}F → P such that μ_{B}(P) = rμ_{B}(F) = rδ. Therefore L := g^{∗}M ⊗ P is an invertible quotient of g^{∗}f^{∗}E and,

μ_{B}(L) = μ_{B}(g^{∗}M ⊗ P) = μ_{B}(M) + rδ = μ_{B}(E) + (r − 1)δ.

By hypothesis, (r − 1)δ < ϵ. So μ_{B}(L) < μ_{B}(E) + ϵ, contradicting Hypothesis 5.8. The proposition is proved by contradiction. □

Corollary 5.9. For every non-zero vector bundle E on B, for every ϵ > 0, there exists a cover f : C → B and a sequence of vector bundle quotients,

f^{∗}E = E^{r} ↠ E^{r−1} ↠ ⋯ ↠ E^{1},

such that each E^{k} is a vector bundle of rank k and μ_{B}(E^{k}) < μ_{B}(E) + ϵ.

Proof. The proof is by induction on the rank r of E. If rank(E) = 1, defining f = Id_{B} and E^{1} = E, the result follows. Thus, assume r > 1 and the result is known for smaller values of r. By Theorem 5.7, there exists a cover g : B′ → B and a rank 1 quotient g^{∗}E → L such that μ_{B}(L) < μ_{B}(E) + ϵ. Denote by K the kernel of g^{∗}E → L. Then rank(K) = r − 1 and μ_{B}(K) = (rμ_{B}(E) − μ_{B}(L))/(r − 1). By the induction hypothesis, there exists a cover h : C → B′ and a sequence of vector bundle quotients,

h^{∗}K = K^{r−1} ↠ ⋯ ↠ K^{1},

such that each K^{k} is a vector bundle of rank k, and μ_{B′}(K^{k}) ≤ μ_{B′}(K) + deg(g)ϵ. Of course μ_{B}(F) = μ_{B′}(F)/deg(g) for every F. Thus μ_{B}(K^{k}) ≤ μ_{B}(K) + ϵ.

Define f = h ∘ g, define E^{1} = h^{∗}L, and for every k = 2, …, r, define f^{∗}E ↠ E^{k} to be the unique quotient whose kernel is contained in h^{∗}K and such that h^{∗}K → E^{k} has image K^{k−1}. Then μ_{B}(E^{1}) = μ_{B}(L) ≤ μ_{B}(E) + ϵ, and for k = 2, …, r,

\begin{matrix}μ_{B}(E^{k}) = 1/k(μ_{B}(L) + (k − 1)μ_{B}(K^{k−1})) < 1/k(μ_{B}(L) + (k − 1)μ_{B}(K) + (k − 1)ϵ) = \\ \frac{r(k−1)}{(r−1)k}μ_{B}(E) + \frac{r−k}{(r−1)k} μ_{B}(L) + \frac{(r−1)(k−1)}{(r−1)k}ϵ < μ_{B}(E) + \frac{r−k}{(r−1)k} ϵ + \frac{(r−1)(k−1)}{(r−1)k}ϵ < μ_{B}(E) + ϵ. \end{matrix}

□

For semistable bundles in characteristic zero, there is a more precise result. An arithmetic analogue is also proved by Zhang in [Zha95, Theorem 1.10].

Theorem 5.10 (Zhang). Let B be a smooth, projective curve over an algebraically closed field of characteristic 0. Let E be a semistable vector bundle on B. Let ϵ be a positive real number. There exists a cover f : C → B, invertible sheaves L_{1}, …, L_{r} on C, and a morphism of O_{C}-modules, ϕ : f^{∗}E → (L_{1} ⊕ ⋯ ⊕ L_{r}) such that,

(i) the support of coker(ϕ) is a finite set, (ii) for every i = 1, …, r, the projection f^{∗}E → ⊕_{j≠i}L_{j} is surjective, (iii) for every i = 1, …, r, μ_{B}(L_{i}) ≤ μ_{B}(E) + ϵ.

Proof. Denote r = rank(E). If r equals 1, the theorem is trivial. Thus assume r > 1. As in the proof of Theorem 5.7, there exists a cover g : C′ → B and an invertible sheaf M on C′ such that 0 < μ_{B}(E) − μ_{B}(M) < ϵ/(r − 1). Denote δ = μ_{B}(E) − μ_{B}(M) and denote F = g^{∗}E ⊗ M^{∨}. Then μ_{B}(F) equals δ, and 0 < δ < ϵ/(r − 1).

Let h : C → C′ be any cover and let h^{∗}F → N be an invertible quotient. The composition g ∘ h : C → B is a cover. By Kempf’s theorem, [Kem92], which ultimately relies on the theorem that every stable vector bundle admits a Hermite-Einstein metric, (g∘h)^{∗}E is semistable. (Note, there are counterexamples in positive characteristic.) Therefore h^{∗}F is semistable. So μ_{C}(L) ≥ μ_{C}(h^{∗}F), i.e., μ_{C′}(L) ≥ μ_{C′}(F) = δ. Thus by Lemma 5.5, F is an ample vector bundle on C′. Thus by Lemma 5.6, there exists a cover h : C → C′, invertible O_{C}-modules N_{1}, …, N_{r}, and a morphism of O_{C}-modules ψ : h^{∗}F → (N_{1} ⊕ ⋯ ⊕ N_{r}) satisfying (i), (ii) and (iii) of Lemma 5.6. Define f = g ∘ h, L_{i} = N_{i} ⊗ h^{∗}M and ϕ is the twist of ψ by Id_{h^{∗}M}. Then ϕ satisfies (i) and (ii). And for every i = 1, …, r,

μ_{B}(L_{i}) = μ_{B}(N_{i}) + μ_{B}(M) = μ_{C′}(N_{i})/deg(g) + μ_{B}(E) − δ =
μ_{B}(E) + rδ/deg(g) − δ ≤ μ_{B}(E) + (r − 1)δ/deg(g) < μ_{B}(E) + ϵ.
□

Of course, μ_{B}^{r}(E) equals μ_{B}(E). The other values are more interesting.

Corollary 5.11. The slopes μ_{B}^{k}(E) satisfy μ_{B}^{1}(E) ≥ μ_{B}^{2}(E) ≥ ⋯ ≥ μ_{B}^{r}(E) = μ_{B}(E). For each 1 ≤ k < r, μ_{B}^{k}(E) = μ_{B}(E) iff f^{∗}E is semistable for every cover f : C → B.

Proof. By Corollary 5.9, for every ϵ > 0, there exists a cover f : C → B and a rank k quotient f^{∗}E → E^{k} such that μ_{B}(E^{k}) < μ_{B}(E) + ϵ. Thus μ_{B}^{k}(E) ≥ μ_{B}(E). Applying the same reasoning to rank k − 1 quotients of rank k quotients of f^{∗}E, μ_{B}^{k−1}(E) ≥ μ_{B}^{k}(E).

If f^{∗}E is semistable for every cover f : C → B, then every vector bundle quotient of f^{∗}E has slope ≥ μ_{C}(f^{∗}E), and thus has B-slope ≥ μ_{B}(f^{∗}E). Therefore μ_{B}^{k}(E) ≤ μ_{B}(E), i.e., μ_{B}^{k}(E) = μ_{B}(E).

Conversely, suppose there is a cover f : C → B such that f^{∗}E is not semistable. Then there exists a vector bundle quotient f^{∗}E ↠ F such that μ_{B}(F) < μ_{B}(E). Denote the rank by l. Suppose first that l ≥ k, and define ϵ = deg(f)(μ_{B}(E) − μ_{B}(F)). Then by Corollary 5.9, there exists a cover g : C′ → C and a rank k quotient g^{∗}F ↠ G such that μ_{C}(G) < μ_{C}(F) + ϵ. Therefore g^{∗}f^{∗}E ↠ g^{∗}F ↠ G is a rank k quotient of g^{∗}f^{∗}E and μ_{B}(G) < μ_{C}(F) + (μ_{B}(E) − μ_{B}(F)) = μ_{B}(E). Therefore μ_{B}^{k}(E) > μ_{B}(E).

Next suppose that l < k. Denote by K the kernel of f^{∗}E → F. Then rμ_{B}(E) = lμ_{B}(F) + (r − l)μ_{B}(K). Define,

ϵ = \frac{(r − k)ldeg(f)(μ_{B}(E) − μ_{B}(F))}{(r − l)(k − l)}.

By Corollary 5.9, there exists a cover g : C′ → C and a rank k−l quotient g^{∗}K ↠ G′ such that μ_{C}(G′) < μ_{C}(K) + ϵ. Therefore μ_{B}(G′) < μ_{B}(K) + ϵ/deg(f). Define g^{∗}f^{∗}E → G to be the unique vector bundle whose kernel is contained in g^{∗}K and such that the image of g^{∗}K → G equals G′. Then, kμ_{B}(G) = lμ_{B}(F) + (k − l)μ_{B}(G′) < lμ_{B}(F) + (k − l)μ_{B}(K) + (k − l)ϵ/deg(f) =

lμ_{B}(F) + \frac{k−l}{r−l}(rμ_{B}(E) − lμ_{B}(F)) + \frac{k−l}{deg(f)}ϵ =
kμ_{B}(E) − \frac{(r−k)l}{r−l}(μ_{B}(E) − μ_{B}(F)) + \frac{(r−k)l}{r−l}(μ_{B}(E) − μ_{B}(F)) = kμ_{B}(E).

Thus μ_{B}(G) < μ_{B}(E), and therefore μ_{B}^{k}(E) > μ_{B}(E). □

References

[Ful84] William Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984.

[Jou83] Jean-Pierre Jouanolou. Théorèmes de Bertini et applications, volume 42 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1983.

[Kem92] George R. Kempf. Pulling back bundles. Pacific J. Math., 152(2):319–322, 1992.

[OSS80] Christian Okonek, Michael Schneider, and Heinz Spindler. Vector bundles on complex projective spaces, volume 3 of Progress in Mathematics. Birkhäuser Boston, Mass., 1980.

[Zha95] Shouwu Zhang. Small points and adelic metrics. J. Algebraic Geom., 4(2):281–300, 1995.