CHARACTERIZATION OF PROJECTIVE SPACES AND $\mathbb{P}^{r}$-BUNDLES AS AMPLE DIVISORS

Abstract

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_{X}$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^{n}$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^{r}$-bundle as an ample divisor by the recent work of Litt.

1 Introduction

Projective spaces are the simplest algebraic varieties. They can be characterized in many ways. A very famous one is given by the Hartshorne’s conjecture, which was proved by Mori.

Theorem A. [25, Theorem 8]

Let $X$ be a projective manifold defined over an algebraically closed field $k$ of characteristic ${\geqslant}0$ . Then $X$ is a projective space if and only if $T_{X}$ is ample.

This result has been generalized, over the field of complex number, by several authors (see [1, 12, 27]).

Theorem B. [1, Theorem]

Let $X$ be a projective manifold of dimension $n$ . If $T_{X}$ contains an ample locally free subsheaf ${\mathcal{E}}$ of rank $r$ , then $X\cong \mathbb{P}^{n}$ and ${\mathcal{E}}\cong {\mathcal{O}}_{\mathbb{P}^{n}}(1)^{\oplus r}$ or ${\mathcal{E}}\cong T_{\mathbb{P}^{n}}$ .

This theorem was successively proved for $r=1$ by Wahl [27] and later for $r\geqslant n-2$ by Campana and Peternell [12]. The proof was finally completed by Andreatta and Wiśniewski [1]. The main aim of the present article is to prove the following generalization.

Theorem 1.1. Let $X$ be a projective manifold of dimension $n$ . Suppose that $T_{X}$ contains an ample subsheaf ${\mathcal{F}}$ of positive rank $r$ , then $(X,{\mathcal{F}})$ is isomorphic to $(\mathbb{P}^{n},T_{\mathbb{P}^{n}})$ or $(\mathbb{P}^{n},{\mathcal{O}}_{\mathbb{P}^{n}}(1)^{\oplus r})$ .

We refer to Section 2.1 for the basic definition and properties of ample sheaves. Comparing with Theorem B, we do not require a priori the locally freeness of the subsheaf ${\mathcal{F}}$ in Theorem 1.1. In the case where the Picard number of $X$ is one, Theorem 1.1 is proved in [2]. In fact, in [2], it was shown that the subsheaf ${\mathcal{F}}$ must be locally free under the additional assumption $\unicode[STIX]{x1D70C}(X)=1$ , and then Theorem B immediately implies Theorem 1.1. In particular, to prove Theorem 1.1, it suffices to show that $X$ is isomorphic to some projective space if its tangent bundle contains an ample subsheaf ${\mathcal{F}}$ ; then, the locally freeness of ${\mathcal{F}}$ follows from [2]. An interesting and important special case of Theorem 1.1 is when the subsheaf ${\mathcal{F}}$ comes from the image of an ample vector bundle $E$ over $X$ . This confirms a conjecture of Litt [23, Conjecture 2].

Corollary 1.2. Let $X$ be a projective manifold of dimension $n$ , and let $E$ be an ample vector bundle on $X$ . If there exists a nonzero map $E\rightarrow T_{X}$ , then $X\cong \mathbb{P}^{n}$ .

As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^{r}$ -bundle as an ample divisor. This problem has attracted a great deal of interest over the past few decades (see [6–8, 13, 26], etc.). Recently, in [23, Corollary 7], Litt proved that it can be reduced to Corollary 1.2. To be more precise, we have the following classification theorem.

Theorem 1.3. Let $X$ be a projective manifold of dimension $n\geqslant 3$ , and let $A$ be an ample divisor on $X$ . Assume that $A$ is a $\mathbb{P}^{r}$ -bundle, $p:A\rightarrow B$ , over a manifold $B$ of dimension $b>0$ . Then one of the following holds.

1. (i) $(X,A)=(\mathbb{P}(E),H)$ for some ample vector bundle $E$ over $B$ such that $H\in |{\mathcal{O}}_{\mathbb{P}(E)}(1)|$ . $p$ is equal to the restriction to $A$ of the induced projection $\mathbb{P}(E)\rightarrow B$ .

2. (ii) $(X,A)=(\mathbb{P}(E),H)$ for some ample vector bundle $E$ over $\mathbb{P}^{1}$ such that $H\in |{\mathcal{O}}_{\mathbb{P}(E)}(1)|$ . $H=\mathbb{P}^{1}\times \mathbb{P}^{n-2}$ , and $p$ is the projection to the second factor.

3. (iii) $(X,A)=(Q^{3},H)$ , where $Q^{3}$ is a smooth quadric threefold and $H$ is a smooth quadric surface with $H\in |{\mathcal{O}}_{Q^{3}}(1)|$ . $p$ is the projection to one of the factors of $H\cong \mathbb{P}^{1}\times \mathbb{P}^{1}$ .

4. (iv) $(X,A)=(\mathbb{P}^{3},H)$ . $H$ is a smooth quadric surface and $H\in |{\mathcal{O}}_{\mathbb{P}^{3}}(2)|$ , and $p$ is again a projection to one of the factors of $H\cong \mathbb{P}^{1}\times \mathbb{P}^{1}$ .

Convention.

Throughout, we work over the field $\mathbb{C}$ of complex numbers unless otherwise stated. Varieties are always assumed to be integral separated schemes of finite type over $\mathbb{C}$ . If $D$ is a Weil divisor on a projective normal variety $X$ , we denote by ${\mathcal{O}}_{X}(D)$ the reflexive sheaf associated to $D$ . Given a coherent sheaf ${\mathcal{F}}$ on a variety $X$ of generic rank $r$ , then we denote by ${\mathcal{F}}^{\vee }$ the sheaf ${\mathcal{H}}om_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{O}}_{X})$ , and by $\det ({\mathcal{F}})$ the sheaf $(\wedge ^{r}{\mathcal{F}})^{\vee \vee }$ . We denote by ${\mathcal{F}}(x)={\mathcal{F}}_{x}\otimes _{{\mathcal{O}}_{X,x}}k(x)$ the fiber of ${\mathcal{F}}$ at $x\in X$ . If ${\mathcal{F}}$ is a coherent sheaf on a variety $X$ , we denote by $\mathbb{P}({\mathcal{F}})$ the Grothendieck projectivization $\text{Proj}(\bigoplus _{m\geqslant 0}\text{Sym}^{m}{\mathcal{F}})$ . If $f:X\rightarrow Y$ is a morphism between projective normal varieties, we denote by $\unicode[STIX]{x1D6FA}_{X/Y}^{1}$ the relative differential sheaf. Moreover, if $Y$ is smooth, we denote by $K_{X/Y}$ the relative canonical divisor $K_{X}-f^{\ast }K_{Y}$ , and by $\unicode[STIX]{x1D714}_{X/Y}$ the reflexive sheaf $\unicode[STIX]{x1D714}_{X}\otimes f^{\ast }\unicode[STIX]{x1D714}_{Y}^{\vee }$ .

2 Ample sheaves and rational curves

Let $X$ be a projective manifold. In this section, we gather some results about the behavior of an ample subsheaf ${\mathcal{F}}\subset T_{X}$ with respect to a family of minimal rational curves on $X$ .

2.1 Ample sheaves

Recall that an invertible sheaf ${\mathcal{L}}$ on a quasi-projective variety $X$ is said to be ample if for every coherent sheaf ${\mathcal{G}}$ on $X$ , there is an integer $n_{0}>0$ such that for every $n\geqslant n_{0}$ , the sheaf ${\mathcal{G}}\otimes {\mathcal{L}}^{n}$ is generated by its global sections (see [18, Section II.7]). In general, a coherent sheaf ${\mathcal{F}}$ on a quasi-projective variety $X$ is said to be ample if the invertible sheaf ${\mathcal{O}}_{\mathbb{P}({\mathcal{F}})}(1)$ is ample on $\mathbb{P}({\mathcal{F}})$  [22].

Well-known properties of ampleness of locally free sheaves still hold in this general setting.

1. (i) A sheaf ${\mathcal{F}}$ on a quasi-projective variety $X$ is ample if and only if, for any coherent sheaf ${\mathcal{G}}$ on $X$ , ${\mathcal{G}}\otimes \text{Sym}^{m}{\mathcal{F}}$ is globally generated for $m\gg 1$  (see [22, Theorem 1]).

2. (ii) If $i:Y\rightarrow X$ is an immersion, and ${\mathcal{F}}$ is an ample sheaf on $X$ , then $i^{\ast }{\mathcal{F}}$ is an ample sheaf on $Y$  (see [22, Proposition 6]).

3. (iii) If $\unicode[STIX]{x1D70B}:Y\rightarrow X$ is a finite morphism with $X$ and $Y$ quasi-projective varieties, and ${\mathcal{F}}$ is a coherent sheaf on $X$ , then ${\mathcal{F}}$ is ample if and only if $\unicode[STIX]{x1D70B}^{\ast }{\mathcal{F}}$ is ample. Note that $\mathbb{P}(\unicode[STIX]{x1D70B}^{\ast }{\mathcal{F}})=\mathbb{P}({\mathcal{F}})\times _{X}Y$ , and ${\mathcal{O}}_{\mathbb{P}({\mathcal{F}})}(1)$ pulls back, by a finite morphism, to ${\mathcal{O}}_{\mathbb{P}(\unicode[STIX]{x1D70B}^{\ast }{\mathcal{F}})}(1)$ .

4. (iv) Any quotient of an ample sheaf is ample (see [22, Proposition 1]). In particular, the image of an ample sheaf under a nonzero map is also ample.

5. (v) If ${\mathcal{F}}$ is a locally free ample sheaf of rank $r$ , then the $s$ th exterior power $\wedge ^{s}{\mathcal{F}}$ is ample for any $1\leqslant s\leqslant r$  (see [17, Corollary 5.3]).

6. (vi) If ${\mathcal{L}}$ is an ample invertible sheaf on a quasi-projective variety $X$ , then ${\mathcal{L}}^{m}$ is very ample for some $m>0$ ; that is, there is an immersion $i:X\rightarrow \mathbb{P}^{n}$ for some $n$ such that ${\mathcal{L}}^{m}=i^{\ast }{\mathcal{O}}_{\mathbb{P}^{n}}(1)$ (see [18, II, Theorem 7.6]).

2.2 Minimal rational curves

Let $X$ be a normal projective variety. By $\text{Hom}(\mathbb{P}^{1},X)$ we denote the open subscheme $\subset \text{Hilb}(\mathbb{P}^{1}\times X)$ of morphisms from $\mathbb{P}^{1}$ to $X$ . Let $\text{Hom}_{1}(\mathbb{P}^{1},X)\subset \text{Hom}(\mathbb{P}^{1},X)$ be the open subscheme corresponding to those morphisms $f:\mathbb{P}^{1}\rightarrow X$ that are birational onto their image. The group $\text{Aut}(\mathbb{P}^{1})$ acts on $\text{Hom}_{1}(\mathbb{P}^{1},X)$ and its quotient “really parametrizes” morphisms from $\mathbb{P}^{1}$ into $X$ . It can be proved that the quotient exists, and its normalization is denoted $\text{RatCurves}^{n}(X)$ and called the space of rational curve on $X$ . For more details we refer to [19].

Let ${\mathcal{V}}$ be an irreducible component of $\text{RatCurves}^{n}(X)$ . ${\mathcal{V}}$ is said to be a covering family of rational curves on $X$ if the corresponding universal family dominates $X$ . A covering family ${\mathcal{V}}$ of rational curves on $X$ is called minimal if its general members have minimal anticanonical degree. If $X$ is a uniruled projective manifold, then $X$ carries a minimal covering family of rational curves. We fix such a family ${\mathcal{V}}$ , and let $[\ell ]\in {\mathcal{V}}$ be a general point. Then, the tangent bundle $T_{X}$ can be decomposed on the normalization of $\ell$ as ${\mathcal{O}}_{\mathbb{P}^{1}}(2)\oplus {\mathcal{O}}_{\mathbb{P}^{1}}(1)^{\oplus d}\oplus {\mathcal{O}}_{\mathbb{P}^{1}}^{\oplus (n-d-1)}$ , where $d+2=\det (T_{X})\cdot \ell \geqslant 2$ is the anticanonical degree of ${\mathcal{V}}$ .

Let $\bar{{\mathcal{V}}}$ be the normalization of the closure of ${\mathcal{V}}$ in $\text{Chow}(X)$ . We define the following equivalence relation on $X$ . Two points $x$ , $y\in X$ are $\bar{{\mathcal{V}}}$ -equivalent if they can be connected by a chain of $1$ -cycle from $\bar{{\mathcal{V}}}$ . By [10] (see also [21]), there exists a proper surjective morphism $\unicode[STIX]{x1D711}_{0}:X_{0}\rightarrow T_{0}$ from an open subset of $X$ onto a normal variety $T_{0}$ whose fibers are $\bar{{\mathcal{V}}}$ -equivalence classes. We call this map the $\bar{{\mathcal{V}}}$ -rationally connected quotient of $X$ .

The first step toward Theorem 1.1 is the following result, which was essentially proved in [3].

Theorem 2.1. [5, Proposition 2.7]

Let $X$ be a projective uniruled manifold, and let ${\mathcal{V}}$ be a minimal covering family of rational curves on $X$ . If $T_{X}$ contains a subsheaf ${\mathcal{F}}$ of rank $r$ such that ${\mathcal{F}}|_{\ell }$ is an ample vector bundle for a general member $[\ell ]\in {\mathcal{V}}$ , then there exists a dense open subset $X_{0}$ of $X$ and a $\mathbb{P}^{d+1}$ -bundle $\unicode[STIX]{x1D711}_{0}:X_{0}\rightarrow T_{0}$ such that any curve on $X$ parametrized by ${\mathcal{V}}$ and meeting $X_{0}$ is a line on a fiber of $\unicode[STIX]{x1D711}_{0}$ . In particular, $\unicode[STIX]{x1D711}_{0}$ is the $\bar{{\mathcal{V}}}$ -rationally connected quotient of $X$ .

Recall that the singular locus $\text{Sing}({\mathcal{S}})$ of a coherent sheaf ${\mathcal{S}}$ over $X$ is the set of all points of $X$ where ${\mathcal{S}}$ is not locally free.

Remark 2.2. The hypothesis in Theorem 2.1 that ${\mathcal{F}}$ is locally free over a general member of ${\mathcal{V}}$ is automatically satisfied. In fact, since ${\mathcal{F}}$ is torsion-free and $X$ is smooth, ${\mathcal{F}}$ is locally free in codimension one. By [19, II, Proposition 3.7], a general member of ${\mathcal{V}}$ is disjoint from $\text{Sing}({\mathcal{F}})$ ; hence, ${\mathcal{F}}$ is locally free over a general member of ${\mathcal{V}}$ .

As an immediate application of Theorem 2.1, we can derive a weak version of [2, Theorem 4.2].

Corollary 2.3. Let $X$ be a projective uniruled manifold with $\unicode[STIX]{x1D70C}(X)=1$ , and let ${\mathcal{V}}$ be a minimal covering family of rational curves on $X$ . If $T_{X}$ contains a subsheaf ${\mathcal{F}}$ of rank $r$ such that ${\mathcal{F}}|_{\ell }$ is ample for a general member $[\ell ]\in {\mathcal{V}}$ , then $X\cong \mathbb{P}^{n}$ .

Corollary 2.4. [2, Corollary 4.3]

Let $X$ be a projective manifold with $\unicode[STIX]{x1D70C}(X)=1$ . Assume that $T_{X}$ contains an ample subsheaf, then $X\cong \mathbb{P}^{n}$ .

Proof. Since the tangent bundle $T_{X}$ contains an ample subsheaf ${\mathcal{F}}$ , $X$ is uniruled (see [24, Corollary 8.6]), and it carries a minimal covering family ${\mathcal{V}}$ of rational curves. Note that the restriction ${\mathcal{F}}|_{C}$ is ample for any curve $C\subset X$ ; thus, we can deduce the result from Corollary 2.3.☐

Remark 2.5. Our approach above is quite different from that in [2]. The proof in [2] is based on a careful analysis of the singular locus of ${\mathcal{F}}$ , and the locally freeness of ${\mathcal{F}}$ has been proved. Even though our argument does not tell anything about the singular locus of ${\mathcal{F}}$ , it has the advantage of giving a rough description of the geometric structure of projective manifolds whose tangent bundle contains a “positive” subsheaf.

3 Foliations and Pfaff fields

Let ${\mathcal{S}}$ be a subsheaf of $T_{X}$ on a quasi-projective manifold $X$ . We denote by ${\mathcal{S}}^{\text{reg}}$ the largest open subset of $X$ such that ${\mathcal{S}}$ is a subbundle of $T_{X}$ over ${\mathcal{S}}^{\text{reg}}$ . Note that, in general, $\text{Sing}({\mathcal{S}})$ is a proper subset of $X\setminus {\mathcal{S}}^{\text{reg}}$ .

Definition 3.1. Let $X$ be a quasi-projective manifold, and let ${\mathcal{S}}\subsetneq T_{X}$ be a coherent subsheaf of positive rank. ${\mathcal{S}}$ is called a foliation if it satisfies the following conditions.

1. (i) ${\mathcal{S}}$ is saturated in $T_{X}$ ; that is, $T_{X}/{\mathcal{S}}$ is torsion-free.

2. (ii) The sheaf ${\mathcal{S}}$ is closed under the Lie bracket.

In addition, ${\mathcal{S}}$ is called an algebraically integrable foliation if the following holds.

1. (iii) For a general point $x\in X$ , there exists a projective subvariety $F_{x}$ passing through $x$ such that

   $$\begin{eqnarray}{\mathcal{S}}|_{F_{x}\cap {\mathcal{S}}^{\text{reg}}}=T_{F_{x}}|_{F_{x}\cap {\mathcal{S}}^{\text{reg}}}\subset T_{X}|_{F_{x}\cap {\mathcal{S}}^{\text{reg}}}.\end{eqnarray}$$
   We call $F_{x}$ the ${\mathcal{S}}$ -leaf through $x$ .

Remark 3.2. Let $X$ be a projective manifold, and let ${\mathcal{S}}$ be a saturated subsheaf of $T_{X}$ . To show that ${\mathcal{S}}$ is an algebraically integrable foliation, it is sufficient to show that it is an algebraically integrable foliation over a Zariski open subset of $X$ .

Example 3.3. Let $X\rightarrow Y$ be a fibration with $X$ and $Y$ projective manifolds. Then $T_{X/Y}\subset T_{X}$ defines an algebraically integrable foliation on $X$ such that the general leaves are the fibers.

Example 3.4. [4, 4.1]

Let ${\mathcal{F}}$ be a subsheaf ${\mathcal{O}}_{\mathbb{P}^{n}}(1)^{\oplus r}$ of $T_{\mathbb{P}^{n}}$ on $\mathbb{P}^{n}$ . Then ${\mathcal{F}}$ is an algebraically integrable foliation and it is defined by a linear projection $\mathbb{P}^{n}{\dashrightarrow}\mathbb{P}^{n-r}$ . The set of points of indeterminacy $S$ of this rational map is an ( $r-1$ )-dimensional linear subspace. Let $x\not \in S$ be a point. Then the leaf passing through $x$ is the $r$ -dimensional linear subspace $L$ of $\mathbb{P}^{n}$ containing both $x$ and $S$ .

Definition 3.5. Let $X$ be a projective variety, and $r$ a positive integer. A Pfaff field of rank $r$ on $X$ is a nonzero map $\unicode[STIX]{x2202}:\unicode[STIX]{x1D6FA}_{X}^{r}\rightarrow {\mathcal{L}}$ , where ${\mathcal{L}}$ is an invertible sheaf on $X$ .

Lemma 3.6. [5, Proposition 4.5]

Let $X$ be a projective variety, and let $n:\widetilde{X}\rightarrow X$ be its normalization. Let ${\mathcal{L}}$ be an invertible sheaf on $X$ , let $r$ be a positive integer, and let $\unicode[STIX]{x2202}:\unicode[STIX]{x1D6FA}_{X}^{r}\rightarrow {\mathcal{L}}$ be a Pfaff field. Then $\unicode[STIX]{x2202}$ can be extended uniquely to a Pfaff field $\widetilde{\unicode[STIX]{x2202}}:\unicode[STIX]{x1D6FA}_{\widetilde{X}}^{r}\rightarrow n^{\ast }{\mathcal{L}}$ .

Let $X$ be a projective manifold, and let ${\mathcal{S}}\subset T_{X}$ be a subsheaf with positive rank $r$ . We denote by $K_{{\mathcal{S}}}$ the canonical class $-c_{1}(\det ({\mathcal{S}}))$ of ${\mathcal{S}}$ . Then there is a natural associated Pfaff field of rank $r$ :

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{X}^{r}=\wedge ^{r}(\unicode[STIX]{x1D6FA}_{X}^{1})=\wedge ^{r}(T_{X}^{\vee })=(\wedge ^{r}T_{X})^{\vee }\rightarrow {\mathcal{O}}_{X}(K_{{\mathcal{S}}}).\end{eqnarray}$$

Lemma 3.7. [4, Lemma 3.2]

Let $X$ be a projective manifold, and let ${\mathcal{S}}$ be an algebraically integrable foliation on $X$ . Then there is a unique irreducible projective subvariety $W$ of $\text{Chow}(X)$ whose general point parametrizes a general leaf of ${\mathcal{S}}$ .

Remark 3.8. Let $X$ be a projective manifold, and let ${\mathcal{S}}$ be an algebraically integrable foliation of rank $r$ on $X$ . Let $W$ be the subvariety of $\text{Chow}(X)$ provided in Lemma 3.7. Let $Z\subset W$ be a general closed subvariety of $W$ , and let $U\subset Z\times X$ be the universal cycle over $Z$ . Let $\widetilde{Z}$ and $\widetilde{U}$ be the normalizations of $Z$ and $U$ , respectively. We claim that the Pfaff field $\unicode[STIX]{x1D6FA}_{X}^{r}\rightarrow {\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ can be extended to a Pfaff field $\unicode[STIX]{x1D6FA}_{\widetilde{U}/\widetilde{Z}}^{r}\rightarrow n^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ .

Let $V$ be the universal cycle over $W$ with $v:V\rightarrow X$ . From the proof of [4, Lemma 3.2], we know that the Pfaff field $\unicode[STIX]{x1D6FA}_{X}^{r}\rightarrow {\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ extends to be a Pfaff field $\unicode[STIX]{x1D6FA}_{V}^{r}\rightarrow v^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ . It induces a Pfaff field $\unicode[STIX]{x1D6FA}_{U}^{r}\rightarrow p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ . Note that $U$ is irreducible since $Z$ is a general subvariety. By Lemma 3.6, it can be uniquely extended to a Pfaff field $\unicode[STIX]{x1D6FA}_{\widetilde{U}}^{r}\rightarrow n^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ .

Let ${\mathcal{K}}$ be the kernel of the morphism $\unicode[STIX]{x1D6FA}_{\widetilde{U}}^{r}{\twoheadrightarrow}\unicode[STIX]{x1D6FA}_{\widetilde{U}/\widetilde{Z}}^{r}$ . Let $F$ be a general fiber of $\widetilde{q}$ such that its image under $p\circ n$ is an ${\mathcal{S}}$ -leaf and the morphism $p\circ n$ restricted on $F$ is finite and birational. Let $x\in F$ be a point such that $F$ is smooth at $x$ and $p\circ n$ is an isomorphism at a neighborhood of $x$ . Then the composite map $\unicode[STIX]{x1D6FA}_{\widetilde{U}}^{r}|_{F}{\twoheadrightarrow}\unicode[STIX]{x1D6FA}_{\widetilde{U}/\widetilde{Z}}^{r}|_{F}{\twoheadrightarrow}\unicode[STIX]{x1D6FA}_{F}^{r}$ implies that the composite map

$$\begin{eqnarray}{\mathcal{K}}\rightarrow \unicode[STIX]{x1D6FA}_{\widetilde{U}}^{r}\rightarrow n^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})\end{eqnarray}$$

vanishes in a neighborhood of $x$ ; hence, it vanishes generically over $\widetilde{U}$ . Since the sheaf $n^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ is torsion-free, it vanishes identically and finally yields a Pfaff field $\unicode[STIX]{x1D6FA}_{\widetilde{U}/\widetilde{Z}}^{r}\rightarrow n^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{S}}})$ .

Let $X$ be a projective manifold, and let ${\mathcal{S}}\subset T_{X}$ be a subsheaf. We define its saturation $\overline{{\mathcal{S}}}$ as the kernel of the natural surjection $T_{X}{\twoheadrightarrow}(T_{X}/{\mathcal{S}})/(torsion)$ . Then $\overline{{\mathcal{S}}}$ is obviously saturated.

Theorem 3.9. Let $X$ be a projective manifold. Assume that $T_{X}$ contains an ample subsheaf ${\mathcal{F}}$ of rank $r<\dim (X)$ . Then its saturation $\overline{{\mathcal{F}}}$ defines an algebraically integrable foliation on $X$ , and the $\overline{{\mathcal{F}}}$ -leaf passing through a general point is isomorphic to $\mathbb{P}^{r}$ .

Proof. Let $\unicode[STIX]{x1D711}_{0}:X_{0}\rightarrow T_{0}$ be as the morphism provided in Theorem 2.1. Since ${\mathcal{F}}$ is locally free in codimension one, we may assume that no fiber of $\unicode[STIX]{x1D711}_{0}$ is completely contained in $\text{Sing}({\mathcal{F}})$ .

The first step is to show that ${\mathcal{F}}|_{X_{0}}\subset T_{X_{0}/T_{0}}$ . Since $\unicode[STIX]{x1D711}_{0}:X_{0}\rightarrow T_{0}$ is smooth, we get a short exact sequence of vector bundles,

$$\begin{eqnarray}0\rightarrow T_{X_{0}/T_{0}}\rightarrow T_{X}|_{X_{0}}\rightarrow \unicode[STIX]{x1D711}_{0}^{\ast }T_{T_{0}}\rightarrow 0.\end{eqnarray}$$

The composite map ${\mathcal{F}}|_{X_{0}}\rightarrow T_{X}|_{X_{0}}\rightarrow \unicode[STIX]{x1D711}_{0}^{\ast }T_{T_{0}}$ vanishes on a Zariski open subset of every fiber. Since $\unicode[STIX]{x1D711}_{0}^{\ast }T_{T_{0}}$ is torsion-free, it vanishes identically, and it follows that ${\mathcal{F}}|_{X_{0}}\subset T_{X_{0}/T_{0}}$ .

Next, we show that, after shrinking $X_{0}$ and $T_{0}$ if necessary, ${\mathcal{F}}$ is actually locally free over $X_{0}$ . By the generic flatness theorem [15, Théorème 6.9.1], after shrinking $T_{0}$ , we can suppose that $(T_{X}/{\mathcal{F}})|_{X_{0}}$ is flat over $T_{0}$ . Let $F\cong \mathbb{P}^{d+1}$ be an arbitrary fiber of $\unicode[STIX]{x1D711}_{0}$ . The short exact sequence of sheaves

$$\begin{eqnarray}0\rightarrow {\mathcal{F}}|_{X_{0}}\rightarrow T_{X}|_{X_{0}}\rightarrow (T_{X}/{\mathcal{F}})|_{X_{0}}\rightarrow 0\end{eqnarray}$$

induces a long exact sequence of sheaves

$$\begin{eqnarray}{\mathcal{T}}or((T_{X}/{\mathcal{F}})|_{X_{0}},{\mathcal{O}}_{F})\rightarrow {\mathcal{F}}|_{F}\rightarrow T_{X}|_{F}\rightarrow (T_{X}/{\mathcal{F}})|_{F}\rightarrow 0.\end{eqnarray}$$

Since $(T_{X}/{\mathcal{F}})|_{X_{0}}$ is flat over $T_{0}$ , it follows that ${\mathcal{F}}|_{F}$ is a subsheaf of $T_{X}|_{F}$ ; in particular, ${\mathcal{F}}|_{F}$ is torsion-free. Without loss of generality, we may assume that the restrictions of ${\mathcal{F}}$ on all fibers of $\unicode[STIX]{x1D711}_{0}$ are torsion-free. By Remark 2.5, the restrictions of ${\mathcal{F}}$ on all fibers of $\unicode[STIX]{x1D711}_{0}$ are locally free. This yields, in particular, that the dimension of the fibers of ${\mathcal{F}}$ is constant on every fiber of $\unicode[STIX]{x1D711}_{0}$ due to ${\mathcal{F}}(x)=({\mathcal{F}}|_{F})(x)$ . Note that no fiber of $\unicode[STIX]{x1D711}_{0}$ is contained in $\text{Sing}({\mathcal{F}})$ . We conclude that the dimension of the fibers ${\mathcal{F}}(x)$ of ${\mathcal{F}}$ is constant over $X_{0}$ . Hence, ${\mathcal{F}}$ is locally free over $X_{0}$ .

Now, we claim that $\overline{{\mathcal{F}}}$ actually defines an algebraically integrable foliation on $X_{0}$ . Let $F\cong \mathbb{P}^{d+1}$ be an arbitrary fiber of $\unicode[STIX]{x1D711}_{0}$ . We know that $(F,{\mathcal{F}}|_{F})$ is isomorphic to $(\mathbb{P}^{d+1},T_{\mathbb{P}^{d+1}})$ or $(\mathbb{P}^{d+1},{\mathcal{O}}_{\mathbb{P}^{d+1}}(1)^{\oplus r})$ (cf. Theorem B); therefore, ${\mathcal{F}}$ defines an algebraically integrable foliation over $X_{0}$ (cf. Example 3.4). Note that we have ${\mathcal{F}}|_{X_{0}}=\overline{{\mathcal{F}}}|_{X_{0}}$ , since ${\mathcal{F}}|_{X_{0}}$ is saturated in $T_{X_{0}}$ . Hence, $\overline{{\mathcal{F}}}$ also defines an algebraically integrable foliation over $X$ (cf. Remark 3.2).☐

Remark 3.10. Since ${\mathcal{F}}$ is locally free on $X_{0}$ , it follows that ${\mathcal{O}}_{X}(-K_{{\mathcal{F}}})|_{X_{0}}$ is isomorphic to $\wedge ^{r}({\mathcal{F}}|_{X_{0}})$ and the invertible sheaf ${\mathcal{O}}_{X}(-K_{{\mathcal{F}}})$ is ample over $X_{0}$ . Moreover, as ${\mathcal{F}}$ is locally free in codimension one, there exists an open subset $X^{\prime }\subset X$ containing $X_{0}$ such that $\text{codim}(X\setminus X^{\prime })\geqslant 2$ and ${\mathcal{O}}_{X}(-K_{{\mathcal{F}}})$ is ample on $X^{\prime }$ .

4 Proof of main theorem

The aim of this section is to prove Theorem 1.1. Let $X$ be a normal projective variety, and let $X\rightarrow C$ be a surjective morphism with connected fibers onto a smooth curve. Let $\unicode[STIX]{x1D6E5}$ be an effective Weil divisor on $X$ such that $(X,\unicode[STIX]{x1D6E5})$ is log-canonical over the generic point of $C$ . In [4, Theorem 5.1], it was proved that $-(K_{X/C}+\unicode[STIX]{x1D6E5})$ cannot be ample. In the next theorem, we give a variant of this result which is the key ingredient in our proof of Theorem 1.1.

Theorem 4.1. Let $X$ be a normal projective variety, and let $f:X\rightarrow C$ be a surjective morphism with connected fibers onto a smooth curve. Let $\unicode[STIX]{x1D6E5}$ be a Weil divisor on $X$ such that $K_{X}+\unicode[STIX]{x1D6E5}$ is Cartier and $\unicode[STIX]{x1D6E5}^{\text{hor}}$ is reduced. Assume that there exists an open subset $C_{0}$ such that the pair $(X,\unicode[STIX]{x1D6E5}^{\text{hor}})$ is snc over $X_{0}=f^{-1}(C_{0})$ . If $X^{\prime }\subset X$ is an open subset such that no fiber of $f$ is completely contained in $X\setminus X^{\prime }$ and $X_{0}\subset X^{\prime }$ , then the invertible sheaf ${\mathcal{O}}_{X}(-K_{X/C}-\unicode[STIX]{x1D6E5})$ is not ample over $X^{\prime }$ .

Proof. To prove the theorem, we assume, to the contrary, that the invertible sheaf ${\mathcal{O}}_{X}(-K_{X/C}-\unicode[STIX]{x1D6E5})$ is ample over $X^{\prime }$ . Let $A$ be an ample divisor supported on $C_{0}$ . Then, for some $m\gg 1$ , the sheaf ${\mathcal{O}}_{X}(-m(K_{X/C}+\unicode[STIX]{x1D6E5})-f^{\ast }A)$ is very ample over $X^{\prime }$  (see [14, Corollaire 4.5.11]). It follows that there exists a prime divisor $D^{\prime }$ on $X^{\prime }$ such that the pair $(X^{\prime },\unicode[STIX]{x1D6E5}^{\text{hor}}|_{X^{\prime }}+D^{\prime })$ is snc over $X_{0}$ and

$$\begin{eqnarray}D^{\prime }\sim (-m(K_{X/C}+\unicode[STIX]{x1D6E5})-f^{\ast }A)|_{X^{\prime }}.\end{eqnarray}$$

This implies that there exists a rational function $h\in K(X^{\prime })=K(X)$ such that the restriction of the Cartier divisor $D=\text{div}(h)-m(K_{X/C}+\unicode[STIX]{x1D6E5})-f^{\ast }A$ on $X^{\prime }$ is $D^{\prime }$ , and $D^{\text{hor}}$ is the closure of $D^{\prime }$ in $X$ . Note that we can write $D=D_{+}-D_{-}$ for some effective divisors $D_{+}$ and $D_{-}$ with no common components. Then we have $\operatorname{Supp}(D_{-})\subset X\setminus X^{\prime }$ . In particular, no fiber of $f$ is supported on $D_{-}$ . By [20, Theorem 4.15], there exists a log-resolution $\unicode[STIX]{x1D707}:\widetilde{X}\rightarrow X$ such that we have the following.

1. (i) The induced morphism $\widetilde{f}=f\circ \unicode[STIX]{x1D707}:\widetilde{X}\rightarrow C$ is prepared (cf. [11, Section 4.3]).

2. (ii) The birational morphism $\unicode[STIX]{x1D707}$ is an isomorphism over $X_{0}$ .

3. (iii) $\unicode[STIX]{x1D707}_{\ast }^{-1}\unicode[STIX]{x1D6E5}^{\text{hor}}+\unicode[STIX]{x1D707}_{\ast }^{-1}D^{\text{hor}}$ is a snc divisor.

Let $E$ be the exceptional divisor of $\unicode[STIX]{x1D707}$ . Note that we have $\widetilde{f}_{\ast }(E)\not =C$ . Moreover, we also have

$$\begin{eqnarray}K_{\widetilde{X}}+\unicode[STIX]{x1D707}_{\ast }^{-1}\unicode[STIX]{x1D6E5}+\frac{1}{m}\unicode[STIX]{x1D707}_{\ast }^{-1}D_{+}=\unicode[STIX]{x1D707}^{\ast }\biggl(K_{X}+\unicode[STIX]{x1D6E5}+\frac{1}{m}D\biggr)+\frac{1}{m}\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}+E_{+}-E_{-},\end{eqnarray}$$

where $E_{+}$ and $E_{-}$ are effective $\unicode[STIX]{x1D707}$ -exceptional divisors with no common components.

Set $\widetilde{D}=m\unicode[STIX]{x1D707}_{\ast }^{-1}\unicode[STIX]{x1D6E5}+\unicode[STIX]{x1D707}_{\ast }^{-1}D_{+}+mE_{-}$ . Then $\widetilde{D}^{\text{hor}}=m\unicode[STIX]{x1D707}_{\ast }^{-1}\unicode[STIX]{x1D6E5}^{\text{hor}}+\unicode[STIX]{x1D707}_{\ast }^{-1}D^{\text{hor}}$ is an snc effective divisor with coefficients ${\leqslant}m$ . Since $D$ is linearly equivalent to $-m(K_{X/C}+\unicode[STIX]{x1D6E5})-f^{\ast }A$ , we can write

$$\begin{eqnarray}K_{\widetilde{X}/C}+\frac{1}{m}\widetilde{D}{\sim}_{\mathbb{Q}}-\frac{1}{m}\widetilde{f}^{\ast }A+\frac{1}{m}\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}+E_{+}.\end{eqnarray}$$

After multiplying by some positive $l$ divisible by the denominators of the coefficients of $E_{+}$ and $E_{-}$ , we may assume that $\text{lm}\,E_{+}$ and $\text{lm}\,E_{-}$ are integer coefficients. By replacing $\widetilde{D}$ by $l\widetilde{D}$ , the weak positivity theorem [11, Theorem 4.13] implies that the direct image sheaf

$$\begin{eqnarray}\displaystyle \widetilde{f}_{\ast }(\unicode[STIX]{x1D714}_{\widetilde{X}/C}^{\text{lm}}\otimes {\mathcal{O}}_{\widetilde{X}}(\widetilde{D})) & \simeq & \displaystyle \widetilde{f}_{\ast }({\mathcal{O}}_{\widetilde{X}}(-l\widetilde{f}^{\ast }A+\text{lm}\,E_{+}+l\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}))\nonumber\\ \displaystyle & \simeq & \displaystyle {\mathcal{O}}_{C}(-lA)\otimes \widetilde{f}_{\ast }{\mathcal{O}}_{\widetilde{X}}(\text{lm}\,E_{+}+l\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-})\nonumber\end{eqnarray}$$

is weakly positive.

Observe that $\widetilde{f}_{\ast }({\mathcal{O}}_{\widetilde{X}}(\text{lm}\,E_{+}+l\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}))={\mathcal{O}}_{C}$ . Indeed, $E_{+}$ is a $\unicode[STIX]{x1D707}$ -exceptional divisor. It follows that $\unicode[STIX]{x1D707}_{\ast }({\mathcal{O}}_{\widetilde{X}}(\text{lm}\,E_{+}+l\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}))={\mathcal{O}}_{X}(lD_{-})$ . Note that we have $f_{\ast }({\mathcal{O}}_{X}(lD_{-}))={\mathcal{O}}_{C}(P)$ for some effective divisor $P$ on $C$ such that $\operatorname{Supp}(P)\subset f(\operatorname{Supp}(D_{-}))$ . Let $V$ be an open subset of $C$ , and let $\unicode[STIX]{x1D706}\in H^{0}(V,{\mathcal{O}}_{C}(P))$ . That is, $\unicode[STIX]{x1D706}$ is a rational function on $C$ such that $\text{div}(\unicode[STIX]{x1D706})+P\geqslant 0$ over $V$ . It follows that $\text{div}(\unicode[STIX]{x1D706}\circ f)+lD_{-}\geqslant 0$ over $f^{-1}(V)$ . Since there is no fiber of $f$ completely supported on $D_{-}$ , the rational function $\unicode[STIX]{x1D706}\circ f$ is regular over $f^{-1}(V)$ . Consequently, the rational function $\unicode[STIX]{x1D706}$ is regular over $V$ . This implies that the natural inclusion ${\mathcal{O}}_{C}\rightarrow {\mathcal{O}}_{C}(P)$ is surjective, which yields $\widetilde{f}_{\ast }({\mathcal{O}}_{\widetilde{X}}(\text{lm}\,E_{+}+l\unicode[STIX]{x1D707}_{\ast }^{-1}D_{-}))={\mathcal{O}}_{C}$ . However, this shows that ${\mathcal{O}}_{C}(-lA)$ is weakly positive, which is a contradiction. Hence, ${\mathcal{O}}_{X}(-K_{X/C}-\unicode[STIX]{x1D6E5})$ is not ample over $X^{\prime }$ .☐

Lemma 4.2. Let $X$ be a normal projective variety, and let $f:X\rightarrow C$ be a surjective morphism with reduced and connected fibers onto a smooth curve $C$ . Let $D$ be a Cartier divisor on $X$ . If there exists a nonzero morphism $\unicode[STIX]{x1D6FA}_{X/C}^{r}\rightarrow {\mathcal{O}}_{X}(D)$ , where $r$ is the relative dimension of $f$ , then there exists an effective Weil divisor $\unicode[STIX]{x1D6E5}$ on $X$ such that $K_{X/C}+\unicode[STIX]{x1D6E5}=D$ .

Proof. Since all of the fibers of $f$ are reduced, the sheaf $\unicode[STIX]{x1D6FA}_{X/C}^{r}$ is locally free in codimension one. Hence, the reflexive hull of $\unicode[STIX]{x1D6FA}_{X/C}^{r}$ is $\unicode[STIX]{x1D714}_{X/C}\simeq {\mathcal{O}}_{X}(K_{X/C})$ . Note that ${\mathcal{O}}_{X}(D)$ is reflexive; the nonzero morphism $\unicode[STIX]{x1D6FA}_{X/C}^{r}\rightarrow {\mathcal{O}}_{X}(D)$ induces a nonzero morphism $\unicode[STIX]{x1D714}_{X/C}\rightarrow {\mathcal{O}}_{X}(D)$ . This shows that there exists an effective divisor $\unicode[STIX]{x1D6E5}$ on $X$ such that $K_{X/C}+\unicode[STIX]{x1D6E5}=D$ .☐

As an application of Theorem 4.1, we derive a special property about foliations defined by an ample subsheaf of $T_{X}$ . A similar result was established for Fano foliations with mild singularities in the work of Araujo and Druel (see [4, Proposition 5.3]), and we follow the same strategy.

Proposition 4.3. Let $X$ be a projective manifold. If ${\mathcal{F}}\subset T_{X}$ is an ample subsheaf of rank $r<n=\dim (X)$ , then there is a common point in the closure of general leaves of $\overline{{\mathcal{F}}}$ .

Proof. Since ${\mathcal{F}}$ is torsion-free and $X$ is smooth, ${\mathcal{F}}$ is locally free over an open subset $X^{\prime }\subset X$ such that $\text{codim}(X\setminus X^{\prime })\geqslant 2$ . In particular, ${\mathcal{O}}_{X}(-K_{{\mathcal{F}}})$ is ample over $X^{\prime }$ . By Theorem 2.1, there exist an open subset $X_{0}\subset X$ and a $\mathbb{P}^{d+1}$ -bundle $\unicode[STIX]{x1D711}_{0}:X_{0}\rightarrow T_{0}$ . Moreover, from the proof of Theorem 3.9, the saturation $\overline{{\mathcal{F}}}$ defines an algebraically integrable foliation on $X$ , and we may assume that ${\mathcal{F}}$ is locally free over $X_{0}$ . In particular, we have $X_{0}\subset X^{\prime }$ . In view of Lemma 3.7, we denote by $W$ the subvariety of $\text{Chow}(X)$ parametrizing the general leaves of $\overline{{\mathcal{F}}}$ , and by $V$ the normalization of the universal cycle over $W$ . Let $p:V\rightarrow X$ and $\unicode[STIX]{x1D70B}:V\rightarrow W$ be the natural projections. Note that there exists an open subset $W_{0}$ of $W$ such that $p(\unicode[STIX]{x1D70B}^{-1}(W_{0}))\subset X_{0}$ .

To prove our proposition, we assume to the contrary that there is no common point in the general leaves of $\overline{{\mathcal{F}}}$ .

First, we show that there exists a smooth curve $C$ with a finite morphism $n:C\rightarrow n(C)\subset W$ such that we have the following.

1. (i) Let $U$ be the normalization of the fiber product $V\times _{W}C$ with projection $\unicode[STIX]{x1D70B}:U\rightarrow C$ . Then the induced morphism $\widetilde{p}:U\rightarrow X$ is finite onto its image.

2. (ii) There exists an open subset $C_{0}$ of $C$ such that the image of $U_{0}$ under $p$ is contained in $X_{0}$ . In particular, $U_{0}=\unicode[STIX]{x1D70B}^{-1}(C_{0})$ is a $\mathbb{P}^{r}$ -bundle over $C_{0}$ .

3. (iii) For any point $c\in C$ , the image of the fiber $\unicode[STIX]{x1D70B}^{-1}(c)$ under $\widetilde{p}$ is not contained in $X\setminus X^{\prime }$ .

4. (iv) All of the fibers of $\unicode[STIX]{x1D70B}$ are reduced.

Note that we have $X\setminus X^{\prime }=\operatorname{Sing}({\mathcal{F}})$ and $\text{codim}(\operatorname{Sing}({\mathcal{F}}))\geqslant 2$ . We consider the subset

$$\begin{eqnarray}Z=\{w\in W~|~\unicode[STIX]{x1D70B}^{-1}(w)\subset p^{-1}(\operatorname{Sing}({\mathcal{F}}))\}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D70B}$ is equidimensional, it is a surjective universally open morphism (see [16, Théorème 14.4.4]). Therefore, the subset $Z$ is closed. Note that the general fiber of $\unicode[STIX]{x1D70B}$ is disjoint from $p^{-1}(\operatorname{Sing}({\mathcal{F}}))$ , so $\text{codim}(Z)\geqslant 1$ . Moreover, by the definition of $Z$ , we have $p(\unicode[STIX]{x1D70B}^{-1}(Z))\subset \operatorname{Sing}({\mathcal{F}})$ and $\text{codim}(\operatorname{Sing}({\mathcal{F}}))\geqslant 2$ . Hence, we can choose some very ample divisors $H_{i}$ $(1\leqslant i\leqslant n)$ on $X$ such that the curve $B$ defined by the complete intersection $\widetilde{p}^{\ast }H_{1}\cap \cdots \cap \widetilde{p}^{\ast }H_{n}$ satisfies the following conditions.

(i ^′ )

There is no common point in the closure of the general fibers of $\unicode[STIX]{x1D70B}$ over $\unicode[STIX]{x1D70B}(B)$ .

(ii ^′ )

$\unicode[STIX]{x1D70B}(B)\cap W_{0}\not =\emptyset$ .

(iii ^′ )

$\unicode[STIX]{x1D70B}(B)\subset W\setminus Z$ .

Let $B^{\prime }\rightarrow B$ be the normalization, and let $V_{B^{\prime }}$ be the normalization of the fiber product $V\times _{B}B^{\prime }$ . The induced morphism $V_{B^{\prime }}\rightarrow V$ is denoted by $\unicode[STIX]{x1D707}$ . Then it is easy to check that $B^{\prime }$ satisfies (i), (ii) and (iii). By [9, Theorem 2.1], there exists a finite morphism $C\rightarrow B^{\prime }$ such that all of the fibers of $U\rightarrow C$ are reduced, where $U$ is the normalization of $U\times _{B^{\prime }}C$ . Then we see at once that $C$ is the desired curve.

The next step is to get a contradiction by applying Theorem 4.1. From Remark 3.8, we see that the Pfaff field $\unicode[STIX]{x1D6FA}_{X}^{r}\rightarrow {\mathcal{O}}_{X}(K_{\overline{{\mathcal{F}}}})$ extends to a Pfaff field $\unicode[STIX]{x1D6FA}_{V_{B^{\prime }}/B^{\prime }}^{r}\rightarrow \unicode[STIX]{x1D707}^{\ast }p^{\ast }{\mathcal{O}}_{X}(K_{\overline{{\mathcal{F}}}})$ , and it induces a Pfaff field $\unicode[STIX]{x1D6FA}_{U/C}^{r}\rightarrow \widetilde{p}^{\ast }{\mathcal{O}}_{X}(K_{\overline{{\mathcal{F}}}})$ . The natural inclusion ${\mathcal{F}}{\hookrightarrow}\overline{{\mathcal{F}}}$ induces a morphism ${\mathcal{O}}_{X}(K_{\overline{{\mathcal{F}}}})\rightarrow {\mathcal{O}}_{X}(K_{{\mathcal{F}}})$ . This implies that we have a Pfaff field $\unicode[STIX]{x1D6FA}_{U/C}^{r}\rightarrow \widetilde{p}^{\ast }{\mathcal{O}}_{X}(K_{{\mathcal{F}}})$ . By Lemma 4.2, there exists an effective Weil divisor $\unicode[STIX]{x1D6E5}$ on $U$ such that $K_{U/C}+\unicode[STIX]{x1D6E5}=\widetilde{p}^{\ast }K_{{\mathcal{F}}}$ .

Let $\unicode[STIX]{x1D6E5}^{\text{hor}}$ be the $\unicode[STIX]{x1D70B}$ -horizontal part of $\unicode[STIX]{x1D6E5}$ . After shrinking $C_{0}$ , we may assume that $\unicode[STIX]{x1D6E5}|_{U_{0}}=\unicode[STIX]{x1D6E5}^{\text{hor}}|_{U_{0}}$ . According to the proof of Theorem 3.9, for any fiber $F\cong \mathbb{P}^{r}$ over $C_{0}$ , we have $(\widetilde{p}^{\ast }K_{{\mathcal{F}}})|_{F}-K_{F}=0$ or $H$ , where $H\in |{\mathcal{O}}_{\mathbb{P}^{r}}(1)|$ . This shows that either $\unicode[STIX]{x1D6E5}^{\text{hor}}$ is zero or $\unicode[STIX]{x1D6E5}^{\text{hor}}$ is a prime divisor such that $\unicode[STIX]{x1D6E5}|_{U_{0}}=\unicode[STIX]{x1D6E5}^{\text{hor}}|_{U_{0}}\in |{\mathcal{O}}_{U_{0}}(1)|$ . In particular, the pair $(U,\unicode[STIX]{x1D6E5}^{\text{hor}})$ is snc over $U_{0}$ and $\unicode[STIX]{x1D6E5}^{\text{hor}}$ is reduced. Note that $\widetilde{p}:U\rightarrow \widetilde{p}(U)$ is a finite morphism, so the invertible sheaf $\widetilde{p}^{\ast }{\mathcal{O}}_{X}(-K_{{\mathcal{F}}})$ is ample over $U^{\prime }=U\cap \widetilde{p}^{-1}(X^{\prime })$ . That is, the sheaf ${\mathcal{O}}_{U}(-K_{U/C}-\unicode[STIX]{x1D6E5})$ is ample over $U^{\prime }$ , which contradicts Theorem 4.1.☐

Now, our main result immediately follows.

Proof of Theorem 1.1.

Theorem 2.1 implies that there exist an open subset $X_{0}\subset X$ and a normal variety $T_{0}$ such that $X_{0}\rightarrow T_{0}$ is a $\mathbb{P}^{d+1}$ -bundle and $d+1\geqslant r$ . Without loss of generality, we may assume that $r<\dim (X)$ . By Theorem 3.9 followed by Proposition 4.3, $\overline{{\mathcal{F}}}$ defines an algebraically integrable foliation over $X$ such that there is a common point in the closure of general leaves of $\overline{{\mathcal{F}}}$ . However, this cannot happen if $\dim (T_{0})\geqslant 1$ . Hence, we have $\dim T_{0}=0$ and $X\cong \mathbb{P}^{n}$ .☐

5 $\mathbb{P}^{r}$ -bundles as ample divisors

As an application of Theorem 1.1, we classify projective manifolds $X$ containing $\mathbb{P}^{r}$ -bundles as ample divisors. This was originally conjectured by Beltrametti and Sommese (see [8, Conjecture 5.5.1]). In the remainder of this section, we follow the same notation and assumptions as in Theorem 1.3.

The case $r\geqslant 2$ follows from Sommese’s extension theorem [26, Proposition III] (see also [8, Theorem 5.5.2]). For $r=1$ and $b=1$ , it is due to Bădescu [6, Theorem D] (see also [8, Theorem 5.5.3]). For $r=1$ and $b=2$ , it is due to the work of several authors (see [7, Theorem 7.4]). As mentioned in the introduction, Litt proved the following result, by which we can deduce Theorem 1.3 from Corollary 1.2.

Proposition 5.1. [23, Lemma 4]

Let $X$ be a projective manifold of dimension ${\geqslant}3$ , and let $A$ be an ample divisor. Assume that $p:A\rightarrow B$ is a $\mathbb{P}^{1}$ -bundle, then either $p$ extends to a morphism $\widehat{p}:X\rightarrow B$ , or there exists an ample vector bundle $E$ on $B$ and a nonzero map $E\rightarrow T_{B}$ .

For the reader’s convenience, we outline the argument of Litt that reduces Theorem 1.3 to Corollary 1.2.

Proof of Theorem 1.3.

Since the case $r\geqslant 2$ is already known, we can assume that $r=1$ ; that is, $p:A\rightarrow B$ is a $\mathbb{P}^{1}$ -bundle.

If $p$ extends to a morphism $\widehat{p}:X\rightarrow B$ , then the result follows from [7, Theorem 5.5] and we are in case (i) of the theorem.

If $p$ does not extend to a morphism $X\rightarrow B$ , by Proposition 5.1, there exists an ample vector bundle $E$ over $B$ with a nonzero map $E\rightarrow T_{B}$ . Due to Corollary 1.2, we have $B\cong \mathbb{P}^{b}$ . As the case $b\leqslant 2$ is also known, we may assume that $b\geqslant 3$ . In this case, by [13, Theorem 2.1], we conclude that $X$ is a $\mathbb{P}^{n-1}$ -bundle over $\mathbb{P}^{1}$ and we are in case (ii) of the theorem.☐