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On projective varieties with strictly nef tangent bundles Duo Li, Wenhao Ou and Xiaokui Yang Abstract. In this paper, we study smooth complex projective varieties X such that r some exterior power V TX of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. n 2 If TX is strictly nef, then X isomorphic to the projective space P . If V TX is strictly nef and if X has dimension at least 3, then X is either isomorphic to Pn or a quadric Qn. Contents 1. Introduction 1 2. Basic properties of strictly nef vector bundles 3 3. Strictly nef bundles on curves 4 4. Sections of Albanese morphisms 6 5. Projective manifolds with strictly nef tangent bundles 10 References 12 1. Introduction Throughout this paper, we will study projective varieties defined over C, the field of complex numbers. We recall that a line bundle L over a smooth projective variety X is said to be strictly nef if there is arXiv:1801.09191v2 [math.AG] 28 Nov 2018 L · C > 0 for any curve C ⊂ X, while the Nakai-Moishezon-Kleiman criterion asserts that L is ample if and only if there is Ldim Y · Y > 0 for every positive-dimensional subvariety Y ⊂ X. In particular, ampleness implies strict nefness. However, the converse is not true in general, as shown in an example of Mumford (see [Har70, Section 10, Chapter I]). Nevertheless, one might expect more for the canonical bundle ωX of X. Indeed, on the one hand, since a strictly nef semi- ample line bundle must be ample, the abundance conjecture suggests that if ωX is strictly nef, then it should be ample. On the other hand, Campana and Peternell proposed in [CP91, Problem 11.4] the following conjecture. 1 Li-Ou-Yang On projective varieties with strictly nef tangent bundles −1 Conjecture 1.1. Let X be a smooth projective variety. If ωX is strictly nef, then −1 ωX is ample, that is, X is a Fano variety. This conjecture is only verified by Maeda for surfaces (see [Mae93]) and by Serrano for threefolds (see [Ser95], also [Ueh00] and [CCP08]). In this paper, we prove the following theorem, which provides some evidence for this conjecture in all dimensions. Theorem 1.2. Let X be a smooth projective variety of dimension n, and let TX be r its tangent bundle. If TX is strictly nef for some 1 6 r 6 n, then X is rationally connected. In particular, if ω−1 is strictly nef, then X is rationally connected. V X We recall that a vector bundle E is said to be strictly nef if the tautological line bundle OE(1) on the projective bundle Proj E of hyperplanes is strictly nef. One of the key ingredients for the proof of Theorem 1.2 relies on the recent break- through of Cao and H¨oring on the structure theorems for projective varieties with nef anticanonical bundle (see [Cao16] and [CH17]). Actually, based on their work, we prove the following result, which is essential for Theorem 1.2. Theorem 1.3. Let X be a smooth projective variety with nef anticanonical bundle −1 ωX . Then, up to replacing X with some finite ´etale cover if necessary, the Albanese ∗ morphism f : X → A has a section σ : A → X such that σ ωX is numerically trivial. As an application of Theorem 1.2, we prove the following analogue of Mori’s charac- terization of projective spaces (see [Mor79]). Theorem 1.4. Let X be a smooth projective variety of dimension n. If TX is strictly nef, then X is Pn. In another word, this theorem states that the tangent bundle TX is strictly nef if and only if it is ample. Along the same lines as Theorem 1.4, we obtain the following characterization as well. Theorem 1.5. Let X be a smooth projective variety of dimension n ≥ 3. Suppose 2 n n that TX is strictly nef, then X is isomorphic to P or a quadric Q . ThereV are also other characterizations of projective spaces or quadrics, see, for ex- ample, [KO73], [SY80], [Siu80], [Mok88], [YZ90], [Pet90, Pet91], [CS95], [AW01], [CMSB02], [Miy04], [ADK08], [Hwa13], [DH17] , [Yan17], etc. Acknowledgements. The authors would like to thank Shing-Tung Yau for his comments and suggestions on an earlier version of this paper which clarify and improve the presentations. The authors would also like to thank Yifei Chen, Yi Gu and Xiaotao Sun for some useful discussions. The first author is very grateful to Baohua Fu for his support, encouragement and stimulating discussions over the last few years. The second author would like to thank Meng Chen, Zhi Jiang and Burt Totaro for fruitful conversations. The third author wishes to thank Kefeng Liu, Valentino Tosatti and Xiangyu Zhou for helpful discussions. They would also like to thank the referee for 2 On projective varieties with strictly nef tangent bundles Li-Ou-Yang pointing out a more general version of Lemma 3.2. This work was partially supported by China’s Recruitment Program of Global Experts and NSFC 11688101. 2. Basic properties of strictly nef vector bundles In this section, we collect some basic results on strictly nef vector bundles. The fol- lowing proposition is analogous to the Barton-Kleiman criterion for nef vector bundles (see for example [Laz04, Proposition 6.1.18]). Proposition 2.1. Let E be a vector bundle on a smooth projective variety X. Then the following conditions are equivalent. (1) E is strictly nef. (2) For any smooth projective curve C with a finite morphism ν : C → X, and for any line bundle quotient ν∗(E) → L, one has deg L> 0. Proof. For a fixed smooth projective curve C, we know that any non-constant mor- phism µ : C → Proj E whose image is horizontal over X corresponds one-to-one to a finite morphism ν : C → X with a line bundle quotient ν∗E → L. Moreover, we have the following commutative diagram µ / (2.1) C ❅ Proj E ❅ ✇ ❅❅ ✇✇ ❅❅ ✇✇ ν ❅❅ ✇✇ ✇{ ✇ X ∗ such that L = µ OE(1), where OE(1) is the tautological line bundle of Proj E (see for exmaple [Har77, Proposition II.7.12]). (1) =⇒ (2). Let ν : C → X be a finite morphism, where C is a smooth projective curve. Assume that there is a line bundle quotient ν∗E → L. Let µ : C → Proj E be the induced morphism as in diagram (2.1). Then we have ∗ deg L = deg µ OE(1) = OE(1) · µ(C). Since OE(1) is strictly nef, we obtain that deg L> 0. (2) =⇒ (1). Let B be a curve in Proj E. Let f : B → Proj E be its normalization. If B is vertical over X, then we have OE(1) · B > 0. If B is horizontal over X, then the natural morphism g : B → X is finite. In thise case, there is an induced line ∗ ∗ bundle quotient g E → Q such that Q =∼ f OE(1). We note that e OE(1) · B = deg Q, which is positive by hypothesis. Thus E is strictly nef. We also have the following list of properties of strictly nef vector bundles. 3 Li-Ou-Yang On projective varieties with strictly nef tangent bundles Proposition 2.2. Let E and F be two vector bundles on a smooth projective variety X. Then we have the following assertions. (1) E is a strictly nef if and only if for every smooth projective 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) If E ⊕ F is strictly nef, then both E and F are strictly nef. (4) If the symmetric power Symk E is strictly nef for some k ≥ 1, then E is strictly nef. (5) Let f : Y → X be a finite 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. (7) If E is strictly nef, then h0(X, E∗ ⊗ L) = 0 for any numerically trivial line bundle L. Proof. (1) follows directly from Proposition 2.1. For (2), we note that there is a ∗ natural embedding ι : Proj Q → Proj E such that ι OE(1) =∼ OQ(1). Hence if E is strictly nef, then so is Q. (3) follows from (2). For (4), we note that there is a k ∗ ∼ Veronese embedding v : Proj E → Proj(Sym E) such that v OSymk E(1) = OE(k). This implies (4). For (5), we notice that for any smooth projective curve C with a finite morphism ν : C → X, the composition f ◦ ν : C → X is also finite. The assertion then follows from Proposition 2.1. Now we consider (6). We note that for every curve in X, there is a curve in Y which maps onto it. Hence by (1), we only need to prove the case when X and Y are smooth curves. In this case, there is a natural finite surjective morphism ∗ ∗ ∼ ∗ g : Proj(f E) → Proj E induced by f. Moreover, we have Of E(1) = g OE(1). By assumption, Of ∗E(1) is a strictly nef line bundle. Since g is finite surjective, this implies that OE(1) is also strictly nef. Therefore, E is strictly nef.
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