Strictly nef vector bundles and characterizations of P n

: In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.


Introduction
Let X be a complex projective manifold. A line bundle L over X is said to be strictly nef if L · C > for each irreducible curve C ⊂ X. This notion is also called "numerically positive" in literatures (e.g. [25]). The Nakai-Moishezon-Kleiman criterion asserts that L is ample if and only if L dim Y · Y > for every positive-dimensional irreducible subvariety Y in X. Hence, ample line bundles are strictly nef. In 1960s, Mumford constructed a number of strictly nef but non-ample line bundles over ruled surfaces (e.g. [25]), and they are tautological line bundles of stable vector bundles of degree zero over smooth curves of genus g ≥ . By using the terminology of Hartshorne ([24]), a vector bundle E → X is called strictly nef (resp. ample) if its tautological line bundle O E ( ) is strictly nef (resp. ample). One can see immediately that the strictly nef vector bundles constructed by Mumford are actually Hermitian-at. Therefore, some functorial properties for ample bundles ( [24]) are not valid for strictly nef bundles. In this note, we give a brief exposition on the di erences and similarities between strict nefness and ampleness, and survey some recent progress on understanding the geometry of projective manifolds endowed with some strictly nef bundles.
Starting in the mid 1960's, several mathematicians-notably Grauert, Gri ths and Hartshorne ( [21][22][23][24])undertook the task of generalizing to vector bundles the theory of positivity for line bundles. One of the goals was to extend to the higher rank setting as many as possible of the beautiful cohomological and topological properties enjoyed by ample divisors. In the past half-century, a number of fundamental results have been established. For this rich topic, we refer to the books [31,32] of Lazarsfeld and the references therein.

.A Abstract strictly nef vector bundles.
Let's recall a criterion for strictly nef vector bundles (see [35,Proposition 2.1]) which is analogous to the Barton-Kleiman criterion for nef vector bundles (e.g. [32,Proposition 6.1.18]). This criterion will be used frequently in the sequel. Proposition 1.1. Let E be a vector bundle over a projective manifold X. Then the following conditions are equivalent.
(2) For any smooth projective curve C with a nite morphism ν : C → X, and for any line bundle quotient ν * (E) → L, one has deg L > .
Recall that for an ample line bundle L, one has the Kodaira vanishing theorem For an ample vector bundle E with rank r ≥ , one can only deduce H (X, E * ) = and the higher cohomology groups H i (X, E * ) (i ≥ ) may not vanish. By using Proposition 1.1, we obtain a similar vanishing theorem for strictly nef vector bundles.

Theorem 1.2. Let E be a strictly nef vector bundle over a projective manifold X. Then
It is worth pointing out that for a strictly nef vector bundle E with rank r ≥ , the cohomology group H (X, Sym k E * ) may not vanish for k ≥ , which is signi cantly di erent from properties of ample vector bundles. Indeed, strict nefness is not closed under tensor product, symmetric product or exterior product of vector bundles. This will be discussed in Section 2.
It is well-known that vector bundles over P split into direct sums of line bundles. By using Proposition 1.1 again, one deduces that strictly nef vector bundles over P are ample. In [35,Theorem 3.1], the following result is obtained.

Corollary 1.3. If E is a strictly nef vector bundle over an elliptic curve C, then E is ample.
As we mentioned before, over smooth curves of genus g ≥ , there are strictly nef vector bundles which are Hermitian-at. There also exist strictly nef but non-ample bundles on some rational surfaces ( [13,30]). It is still a challenge to investigate strictly nef vector bundles over higher dimensional projective manifolds. We propose the following conjecture, which is also the rst step to understand such bundles. Conjecture 1.4. Let E be a strictly nef vector bundle over a projective manifold X. If −K X is nef, then det E is ample.
Although this conjecture is shown to be a consequence of the "generalized abundance conjecture" (e.g. [33,34]), we still expect some other straightforward solutions. Indeed, we get a partial answer to it. Theorem 1.5. Let E be a strictly nef vector bundle over a projective manifold X. If −K X is nef and big, then det E is ample.
We refer to [6,26] for more details on positivity of equivariant vector bundles.
As we pointed out before, strict nefness is a notion of positivity weaker than ampleness. Even though there are signi cant di erences between them, we still expect that strict nefness and ampleness could play similar roles in many situations. The following result, which extends Mori's Theorem, is obtained in [35,Theorem 1.4]. Theorem 1.6. Let X be a projective manifold. If T X is strictly nef, then X is isomorphic to a projective space.
Therefore, T X is ample if and only if it is strictly nef. However, this is not valid for cotangent bundles. Indeed, Shepherd-Barron proved in [48] that there exists a projective surface whose cotangent bundle is strictly nef but not ample (see e.g. Example 4.2).
Let us consider manifolds with strictly nef canonical or anti-canonical bundles. Campana and Peternell proposed in [7,Problem 11.4] the following conjecture, which is still a major problem along this line.
This conjecture has been veri ed for projective manifolds of dimension in [40] and dimension in [47] (see also [52] and the references therein). Recently, some progress has been achieved in [35, Theorem 1.2].
Theorem 1.8. If K − X is strictly nef, then X is rationally connected.
Indeed, we show in [39] that if (X, ∆) is a projective simple normal crossing pair and −(K X + ∆) is strictly nef, then X is rationally connected.
The following dual version of Conjecture 1.7 is actually a consequence of the abundance conjecture.

Conjecture 1.9.
If K X is strictly nef, then K X is ample.
As analogous to the Fujita conjecture, Serrano proposed in [47] the following conjecture, which is a generalization of Conjecture 1.7.

Conjecture 1.10. Let X be a projective manifold. If L is a strictly nef line bundle, then K
This conjecture has been solved for projective surfaces in [47]. For the progress on projective threefolds and higher dimensional manifolds, we refer to [9, Theorem 0.4] and the references therein.
It is also known that the existence of "positive" subsheaves of the tangent bundle can also characterize the ambient manifold. For instance, Andreatta and Wiśniewski established in [1,Theorem] that if the tangent bundle T X of a projective manifold X contains a locally free ample subsheaf F, then X is isomorphic to a projective space. When F is a line bundle, this result is proved by Wahl in [54], and in [8], Campana and Peternell established the cases rank(F) dim X − . It is also shown that the assumption on the local freeness can be dropped ( [2,37]). On the other hand, according to Mumford's construction (see Example 2.5), Andreatta-Wiśniewski's result does not hold if the subsheaf F is assumed to be strictly nef. Indeed, we obtained in [38, Theorem 1.3] the following result. Theorem 1.11. Let X be a projective manifold. Assume that the tangent bundle T X contains a locally free strictly nef subsheaf F of rank r. If X is not isomorphic to a projective space, then X admits a P d -bundle structure φ : X → T for some integer d ≥ r where T is a hyperbolic projective manifold of general type.
Recall that T is said to be hyperbolic if every holomorphic map from C to it is constant. We expect a stronger geometric positivity on the cotangent bundle of the base T in Theorem 1.11. Moreover, we obtain in [38, Theorem 1.4] a characterization of projective spaces. Theorem 1.12. Let X be an n-dimensional complex projective manifold such that T X contains a locally free strictly nef subsheaf F. If π (X) is virtually solvable, then X is isomorphic to P n , and F is isomorphic to either There are many other characterizations of projective spaces for which we refer to [3,4,8,18,19,27,28,36,45,51,55] and the references therein.
Acknowledgements. The authors would like to thank the anonymous referee whose comments have significantly improved this paper.

Basic properties and examples
In this section, we investigate basic properties of strictly nef bundles and discuss some examples. As we mentioned before, Mumford constructed a strictly nef vector bundle which is not ample (see [25, Chapter I, Example 10.6]). We shall describe this example in details. Let E be a rank vector bundle over a smooth curve C of genus g ≥ , X = P(E) be the projectivized bundle and π : P(E) → C be the projection. Let O E ( ) be the tautological line bundle of P(E) and D be the corresponding divisor over X.
For any e ective curve Y in X, we denote by m(Y) the degree of Y over C. Then there is an exact sequence It follows that the divisors on X, modulo numerical equivalence, form a free abelian group of rank , generated by D and F where F is any ber of P(E). Therefore, by formula (2.1) We summarize Mumford's example as following.
Example 2.5. Let C be a smooth curve of genus g ≥ . There exists a rank vector bundle E → C satisfying the following properties: (1) deg(E) = ; (2) Sym k E are stable for all k ≥ ; (3) E is strictly nef but not ample; E * is strictly nef but not ample; (4) E admits a Hermitian-at metric. (5) Let X = P(E), π : X → C be the projection and O E ( ) be the tautological line bundle. Then Although the strict nefness is not closed under tensor products and wedge products, we still have the following properties by using the Barton-Kleiman type criterion (Proposition 1.1).

Proposition 2.6. Let E and F be two vector bundles on a projective manifold X. (1) E is a strictly nef vector bundle if and only if for every smooth curve C and for any non-constant morphism f : C → X, f * E is strictly nef. (2) If E is strictly nef, then any non-zero quotient bundle Q of E is strictly nef. (3) E ⊕ F is strictly nef if and only if both E and F are strictly nef. (4) If the symmetric power Sym k E is strictly nef for some k ≥ , then E is strictly nef. (5) Let f : Y → X be a nite morphism such that Y is a smooth projective variety. If E is strictly nef, then so is f * E. (6) Let f : Y → X be a surjective morphism such that Y is a smooth projective variety. If f * E is strictly nef, then
E is strictly nef.
Example 2.7. Let E be the strictly nef vector bundle in Example 2.5.
(1) Λ E = det E is numerically trivial and it is not strictly nef; (2) the tensor product E ⊗ E * is not strictly nef since H (C, E ⊗ E * ) ∼ = C; (3) E ⊕ E * is strictly nef, but Sym (E ⊕ E * ) is not strictly nef.
The following result is well-known and it will be used frequently in the sequel.

Lemma 2.8. If L is a strictly nef and semi-ample line bundle, then it is ample.
Proof. Since L is semi-ample, mL is globally generated for some large m. Let ϕm : X → Y be the morphism de ned by |mL|. Since L is strictly nef, it is easy to see that ϕm is nite and so L is ample. Recall that for an ample line bundle L, one has the Kodaira vanishing theorem. For an ample vector bundle E, one can only deduce H (X, E * ) = and the higher cohomology groups H i (X, E * ) (i ) may not vanish. For instance, when X = P n with n and E = TP n , one has For strictly nef vector bundles, we have a similar vanishing theorem.

Theorem 2.10. Let E be a strictly nef vector bundle over a projective manifold X. Then
H (X, E * ) = .
Proof. Suppose σ ∈ H (X, E * ) is a nonzero section. Then by [16,Proposition 1.16], σ does not vanish anywhere. This section gives a trivial subbundle of E * and so a trivial quotient bundle of the strictly nef vector bundle E. This contradicts to Proposition 2.6 (2).
Note that the vanishing in (2.2) does not hold for higher rank vector bundles. Indeed, let E be the strictly nef vector bundle in Example 2.5, X = P(E) and π : X → C be the projection. Let F = K ⊗m C for some su ciently large m. Then for all k > . Since Sym k E * is a direct summand of (E * ) ⊗k for some large k , we obtain the non-vanishing H (C, (E * ) ⊗k ⊗ F) ≠ for strictly nef vector bundle E.
Remark 2.11. For a strictly nef vector bundle E with rank r , in general, H (X, Sym k E * ) = dos not hold for k .
We give more examples of strictly nef vector bundle over higher dimensional projective manifolds (see [38,Section 5] for details). A line bundle L over a projective variety X of dimension n is called k-strictly nef if for every irreducible subvariety Y in X with < dim Y ≤ k. Hence, -strictly nef is exactly strictly nef, and an n-strictly nef line bundle is ample.

Theorem 2.12. [50, Lemma 3.2 and Theorem 6.1] Let C be a smooth curve of genus g . Then for any r , there exists a Hermitian at vector bundle E of rank r such that the tautological line bundle O E ( ) is (r− )-strictly nef. In particular, E is strictly nef.
Fix a smooth curve C of genus g . Let r and E be a vector bundle of rank r as in Theorem 2.12.
Example 2.13. Let X = P(E). Then we have the following relative Euler exact sequence where p : X = P(E) → C is the natural projection. It is shown in [38,Example 5.9] that p * E * ⊗ O E ( ) is strictly nef.
Example 2.14. We consider the following extension of vector bundles where Q is a nef vector bundle of positive rank. Since E * is Hermitian at, it is numerically at. In particular, E * is nef and so is G ([16, Proposition 1.15]). Let X = P(G) and p : X = P(G) → C be the natural projection. Then we have the following relative Euler sequence We proved in [38, Example 5.10] that p * E ⊗ O G ( ) is strictly nef and the restriction of F to bers of p is isomorphic to O P d ( ) ⊕r . In particular, F is not a subbundle of T X/C .

Strictly nef vector bundles
In this section, we consider strictly nef vector bundles over higher dimensional projective manifolds. As pointed out in the previous sections, the properties of strictly nef vector bundles are closely related to the geometry of ambient manifolds. When X is P or an elliptic curve, a strictly nef line bundle L over X is ample. It is also known that every strictly nef line bundle over an abelian variety is ample ([47, Proposition 1.4]), and Chaudhuri proved in [13] that every strictly nef homogeneous bundle on a complex ag variety is ample. The following conjecture is proposed for strictly nef vector bundles which is a consequence of Serrano's Conjecture 1.10. Proof. Suppose Conjecture 1.10 is valid. Let Y = P(E), L = O P(E) ( ) and π : Y → X be the projection. For large m, K Y ⊗ L ⊗m is ample. Since −K X is nef, K Y /X ⊗ L ⊗m is ample. We know det π * K Y /X ⊗ K Y /X ⊗ L ⊗m is ample and so is det E.
A special case of Conjecture 3.1 is established and we expect a direct proof of it by adapting similar ideas. Theorem 3.3. Let E be a strictly nef vector bundle over a projective manifold X. If −K X is nef and big, then det E is ample.
Proof. If E is a strictly nef line bundle, then E − K X is nef and big. By Kawamata-Reid-Shokurov base point free theorem, E is semi-ample. Thanks to Lemma 2.8, E is ample. If E has rank r , we consider the projective bundle Y = P(E). Let O E ( ) be the tautological line bundle of the projection π : Y → X. By the projection formula, we have For any m > , the line bundle L = O E (m) ⊗ π * (det E) is strictly nef. Since we deduce L − K Y is strictly nef. On the other hand, L − K Y is big. Indeed, since both O E ( ) and −K X are nef, the top intersection number Therefore, by the base point free theorem again, L is semi-ample and so L is ample. By the positivity of direct image sheaves ( [43]), we deduce that π * (K Y /X ⊗ L ⊗k ) is ample for k large enough. By using the projection formula, one can see that the ample vector bundle π * (K Y /X ⊗ L ⊗k ) is of the form Sym k E ⊗ (det E) ⊗k where k and k are some positive integers. In particular, det E is ample.
The following result is proved in [35,Section 3].

Proposition 3.4. Let E be a strictly nef vector bundle over a projective manifold X. If either of the following holds
(1) the Kodaira dimension κ(X) satis es κ(X) < dim X, then det E is not numerically trivial.

Geometry of projective manifolds with strictly nef bundles
In this section, we describe the geometry related to strictly nef and ample bundles. As analogous to classical results of Mori ([42]), Cho-Miyaoka-Shepherd-Barron ( [14]) and ), we obtained characterizations of P n and quadrics ([35, Theorem 1.3 and Theorem 1.5]). (1) If T X is strictly nef, then X is isomorphic to a projective space.
(2) If n ≥ and T X is strictly nef, then X is isomorphic to P n or a quadric Q n .
Gachet studied in [20] the case when Λ T X is strictly nef.
We have already established in general that the tangent bundle T X is strictly nef if and only if it is ample. However, the same does not hold for cotangent bundles.

Example 4.2.
Let X be a bidisk quotient, ∆ × ∆/Γ, with Γ an irreducible torsion-free cocompact lattice. Let E = T * X and L be its tautological line bundle. It is proved in [48] that L is strictly nef and big, but it is not semi-ample.
We propose the following problem which is analogous to the classical result of Kobayashi that projective manifolds with ample cotangent bundle are hyperbolic.  Theorem 4.5. Let X be a projective manifold of dimension n and ≤ r ≤ n. If Λ r T X is strictly nef, then X is rationally connected. In particular, if K − X is strictly nef, then X is rationally connected.
Let f : X → Y be a smooth surjective morphism between two projective manifolds. It is well-known that if K − X is ample, then so is K − Y ( [29], see also [5] for semi-ampleness). It is natural to propose the following conjecture.
Conjecture 4.6. Let f : X → Y be a smooth surjective morphism between two projective manifolds. If K − X is strictly nef, then so is K − Y .
Indeed, this conjecture can be regarded as a consequence of Conjecture 4.4. Thanks to Theorem 4.5, one obtains a partial answer to Conjecture 4.6.
Corollary 4.7. Let f : X → Y be a smooth surjective morphism between two projective manifolds. If K − X is strictly nef, then Y is rationally connected. Example 4.8. Let f : X → Y be a smooth surjective morphism between two projective manifolds. It is wellknown that K − X/Y cannot be ample ( [29]). However, it can be strictly nef.
We also propose the following general conjecture concerning strictly nef bundles. (1) If Λ r T X is strictly nef for some r > , then K − X is ample; (2) If Λ r T * X is strictly nef for some r > , then K X is ample.
The case hen T * X is strictly nef is of particular interest and it is also related to the Kobayashi-Lang conjecture on hyperbolicity.
Let us consider the geometry of projective manifolds whose tangent bundle contains a "positive" subsheaf. Recall that, Andreatta and Wiśniewski obtained in [1,Theorem] the following characterization of projective spaces. Theorem 4.10. Let X be an n-dimensional projective manifold. Assume that the tangent bundle T X contains a locally free ample subsheaf F of rank r. Then X ∼ = P n and either F ∼ = T P n or F ∼ = O P n ( ) ⊕r .
According to Example 2.5, this does not hold if the subsheaf F is assumed to be strictly nef. Indeed, we obtained in [38, Theorem 1.3] the following structure theorem for projective manifolds whose tangent bundle contains a strictly nef subsheaf.
Theorem 4.11. Let X be a projective manifold. Assume that the tangent bundle T X contains a locally free strictly nef subsheaf F of rank r. Then X admits a P d -bundle structure φ : X → T for some d ≥ r. Moreover, if T is not a single point, then T is a hyperbolic projective manifold of general type.
Actually, we obtained in [38,Theorem 8.1] a concrete description of the structure of the subsheaf F and it is exactly one of the following: (1) F ∼ = T X/T and X is isomorphic to a at projective bundle over T; (2) F is a numerically projectively at vector bundle and its restriction on every ber of φ is isomorphic to O P d ( ) ⊕r .
When dim T > , we established in [38, Corollary 1.5] the existence of non-zero symmetric di erentials.
Corollary 4.12. Let X be a projective manifold whose tangent bundle contains a locally free strictly nef subsheaf. If X is not isomorphic to a projective space, then X has a non-zero symmetric di erential, i.e. H (X, Sym i Ω X ) ≠ for some i > .