Proof. This follows easily from [Reference Demailly, Peternell and Schneider22, Theorem 1.18] and [Reference Simpson54] together (see [Reference Druel26, Theorem 2.9]).

Proof. This follows easily from [Reference Jahnke and Radloff42, Theorem 1.1] and [Reference Jahnke and Radloff42, Proposition 1.1] together (see also [Reference Druel26, Theorem 2.13]).

Proof. Let F be any fiber of f. Because the function $y \mapsto h^0(X_y,\mathscr {E}|_{X_y}\!)$ is upper semicontinuous in the Zariski topology on Y (see [Reference Hartshorne36, Theorem 12.8]), we have $h^0(F,\mathscr {E}|_F) \geqslant r$ . By [Reference Demailly, Peternell and Schneider22, Proposition 1.16], the evaluation map $H^0(F,\mathscr {E}|_F)\otimes \mathscr {O}_F \to \mathscr {E}|_F$ is an injective morphism of vector bundles. This implies that $\mathscr {E}|_F \cong \mathscr {O}_F^{\oplus r}$ . By [Reference Hartshorne36, Corollary 12.9] together with the base change theorem [Reference Hartshorne36, Theorem 12.11], the sheaf $\mathscr {G}:=f_*\mathscr {E}$ is locally free of rank r and the formation of $f_*\mathscr {E}$ commutes with arbitrary base change. This easily implies that $\mathscr {E} \cong f^*\mathscr {G}$ .

Proof. The first part of the statement is obvious. For the second part, note that $K_{f^{-1}\mathscr {F}}=f^*K_{\mathscr {F}}$ by (1). Then we have $\kappa (K_{f^{-1}\mathscr {F}})=\kappa (K_{\mathscr {F}})$ by [Reference Ueno58, Theorem 5.13] and $\nu (K_{f^{-1}\mathscr {F}})=\nu (K_{\mathscr {F}})$ by [Reference Nakayama52, Proposition V.2.22].

Setup 2.19. Let X be a complex projective manifold of dimension $n\geqslant 3$ , and let $\mathscr {F} \subset T_X$ be a regular codimension $1$ foliation with normal bundle $\mathscr {N}$ . Suppose that $\mathscr {F}$ is numerically projectively flat. Let $\rho \colon \pi _1(X) \to \text {PGL}(n-1,\mathbb {C})$ denote a representation that defines the projectively flat structure on $\mathscr {F}$ .

Proof. In order to prove the lemma, it suffices to show that $K_{\mathscr {F}}$ is nef since the vector bundle $\mathscr {F}^*|_C$ is semistable for any morphism $C \to X$ from a smooth complete curve C by Theorem 2.11. We argue by contradiction and assume that $K_{\mathscr {F}}$ is not nef. Then there exists a smooth complete curve $C \to X$ such that $\deg _C K_{\mathscr {F}} <0$ . This implies that the vector bundle $\mathscr {F}|_C$ is ample since it is semistable with positive slope by our current assumption. By [Reference Bogomolov and McQuillan9, Main Theorem] (see also [Reference Kebekus, Conde and Toma50, Theorem 1]), the leaf F of $\mathscr {F}$ through a general point in C is algebraic and rationally connected. It follows that $\mathscr {F}$ is induced by a morphism with connected fibers $f \colon X \to B$ onto a smooth complete curve B. By [Reference Kollár49, Theorem IV.2.10] applied to a general fiber F of f, there exists a minimal free rational curve $\mathbb {P}^1 \to F$ which means that there exists an integer $0 \leqslant p \leqslant n-2$ such that

$$ \begin{align*}\mathscr{F}|_{\mathbb{P}^1}\cong T_F|_{\mathbb{P}^1}\cong \mathscr{O}_{\mathbb{P}^1}(2)\oplus\mathscr{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathscr{O}_{\mathbb{P}^1}^{\oplus n-2-p}.\end{align*} $$

Since $\mathscr {F}|_{\mathbb {P}^1}$ is projectively flat by assumption, we must have $p=0$ and $n=2$ , yielding a contradiction. This finishes the proof of the lemma.

Proof. Item (1) follows from an easy (local) computation. The arguments of [Reference Baum and Bott3, Proof of Corollary 3.4] give a proof of Item (2).

Proof. The second assertion follows immediately from the first using the Hodge index theorem.

Set $c:=c_1(\mathscr {L}_1)=\cdots =c_1(\mathscr {L}_{n-1})\in H^1(X,\Omega _X^1)$ . Notice that $\mathscr {L}_i \subset T_X$ defines a regular foliation for any index $i\in I$ . Denote by $\mathscr {N}_i$ its normal bundle. Then Lemma 3.1 shows that $c_1(\mathscr {N}_i)=c_1(\mathscr {N})+(n-2)c\in H^1(X,\Omega ^1_X)$ lies in the image of the natural map

$$ \begin{align*}H^1(X,\mathscr{N}^*_i) \to H^1(X,\Omega^1_X).\end{align*} $$

On the other hand, we have

$$ \begin{align*} \bigcap_{1 \leqslant i \leqslant n-1} \text{Im}\left (H^1(X,\mathscr{N}^*_i) \to H^1(X,\Omega^1_X)\right) & = \bigcap_{1 \leqslant i \leqslant n-1} \text{Ker}\left (H^1(X,\Omega^1_X) \to H^1(X,\mathscr{L}^*_i) \right) \\ & = \text{Ker}\left ( H^1(X,\Omega^1_X) \to \bigoplus_{1 \leqslant i \leqslant n-1} H^1(X,\mathscr{L}^*_i) \right)\\ & = \text{Ker}\left ( H^1(X,\Omega^1_X) \to H^1(X,\mathscr{F}^*) \right) \\ & = \text{Im}\left (H^1(X,\mathscr{N}^*) \to H^1(X,\Omega^1_X)\right),\\ \end{align*} $$

and hence $c_1(\mathscr {N})+(n-2)c\in H^1(X,\Omega ^1_X)$ lies in the image of the natural map $H^1(X,\mathscr {N}^*) \to H^1(X,\Omega ^1_X)$ . By Lemma 3.1 again, $c_1(\mathscr {N})\in H^1(X,\Omega ^1_X)$ lies in the image of the natural map

$$ \begin{align*}H^1(X,\mathscr{N}^*) \to H^1(X,\Omega^1_X).\end{align*} $$

Now, we have a commutative diagram

The first assertion follows easily since $\mathscr {N}$ has rank $1$ by assumption.

Proof. By [Reference Kollár and Mori45, Proposition 5.67] and [Reference Kollár and Mori45, Proposition 5.68] together, we have $\nu _*\omega _{\widetilde {X}}\cong \mathscr {C} \otimes \omega _X$ , where $\mathscr {C}:=\mathscr {H} om_{\mathscr {O}_X}(\nu _*\mathscr {O}_{\widetilde {X}},\mathscr {O}_X)\subseteq \mathscr {O}_X$ is the conductor ideal of $\widetilde {X}$ in X. This gives $\omega _{\widetilde {X}}\cong \widetilde {\mathscr {C}}\otimes \nu ^*\omega _X$ , where $\widetilde {\mathscr {C}}:=\mathscr {C}\cdot \mathscr {O}_{\widetilde {X}}\subseteq \mathscr {O}_{\widetilde {X}}$ . Recall that $\omega _{\widetilde {X}}$ is reflexive since $\widetilde {X}$ is normal. It follows that $\widetilde {\mathscr {C}}$ is reflexive as well. Hence, there exists an effective Weil divisor on $\widetilde {X}$ such that $\widetilde {\mathscr {C}}\cong \mathscr {O}_{\widetilde {X}}(-E)$ , proving (1).

If $E=0$ , then $\mathscr {C}\cong \mathscr {O}_{X}$ . This immediately implies that $\nu $ is an isomorphism, finishing the proof of the lemma.

Proof. Suppose that $m<n$ . If $m=n$ , the proof of the proposition is very similar and therefore omitted.

We may assume without loss of generality that $c_1(\mathscr {M})\not \equiv 0$ . In other words, assume that I is not empty. By Proposition 3.2, we have $c_1(\mathscr {M})^2 \equiv 0$ . It follows that $\mathscr {M}|_{B_i}\equiv 0$ for any $i \in I$ since $\mathscr {M}$ is nef by assumption. By Proposition 3.2 again,

$$ \begin{align*}L:=\langle c_1(\mathscr{L}_1|_Y),\ldots,c_1(\mathscr{L}_{n-1}|_Y), c_1(\mathscr{N}|_Y)\rangle = \langle c_1(\mathscr{M}|_Y)\rangle \subseteq H^1(Y,\Omega^1_Y).\end{align*} $$

Thus, $\mathscr {L}_j|_{B_i}\equiv 0$ and $\mathscr {N}|_{B_i}\equiv 0$ . As a consequence, the vector bundle $\Omega _X^1|_{B_i}$ is numerically flat. Consider the exact sequence

$$ \begin{align*}0 \to \mathscr{N}_{Y/X}^*|_{B_i} \to \Omega_X^1|_{B_i} \to \Omega_Y^1|_{B_i}\to 0.\end{align*} $$

Since $\mathscr {N}_{Y/X}^*|_{B_i}$ is nef by assumption, we infer that $\mathscr {N}_{Y/X}^*|_{B_i}$ and $\Omega _Y^1|_{B_i}$ are numerically flat as well.

Next, consider the exact sequence of sheaves

$$ \begin{align*}0 \to \mathscr{O}_{B_i}(-B_i) \to \Omega_Y^1|_{B_i} \to \Omega_{B_i}^1 \to 0.\end{align*} $$

Since $-B_i|_{B_i}\equiv \frac {1}{b_i}\sum _{j\neq i}b_j B_j|_{B_i}$ , we conclude that $-B_i|_{B_i}\equiv 0$ and that $\mathscr {O}_{B_i}(-B_i)$ is a subbundle of $\Omega _Y^1|_{B_i}$ (see [Reference Demailly, Peternell and Schneider22, Proposition 1.16]). As a consequence, the sheaf $\Omega _{B_i}^1$ is locally free and numerically flat. Moreover, $B_i \cap B_j =\emptyset $ if $i\neq j$ .

Let $\nu _i\colon \widetilde {B}_i \to B_i$ be the normalization morphism. By the adjunction formula,

$$ \begin{align*}\omega_{B_i} \cong \mathscr{O}_Y(K_Y+B_i)|_{B_i}\equiv 0.\end{align*} $$

In particular, $\omega _{B_i}$ is locally free. By Lemma 3.3 above, there exists an effective Weil divisor $E_i$ on $\widetilde {B}_i$ such that $\omega _{\widetilde {B}_i}\cong \nu _i^*\omega _{B_i}(-E_i)$ .

Suppose that $E_i \neq 0$ . Then $K_{\widetilde {B}_i}$ is not pseudo-effective, and hence $B_i$ is uniruled by [Reference Boucksom, Demailly, Păun and Peternell5, Corollary 0.3] applied to a resolution of $\widetilde {B}_i$ . Let $C \subseteq B_i$ be a rational curve. Then $\Omega ^1_Y|_C$ is the trivial vector bundle since $\Omega ^1_Y|_C$ is numerically flat, yielding a contradiction. This shows that $E_i=0$ , and hence $B_i$ is normal by Lemma 3.3. Then [Reference Boucksom, Demailly, Păun and Peternell5, Corollary 0.3] applies to show that $B_i$ has canonical singularities since $K_{B_i}\equiv 0$ . By the solution of the Zariski–Lipman conjecture for canonical spaces (see [Reference Greb, Kebekus, Kovács and Peternell29, Theorem 6.1] or [Reference Druel23, Theorem 1.1]), we conclude that $B_i$ is smooth. In addition, we have $c_1(B_i)\equiv 0$ and $c_2(B_i)\equiv 0$ by [Reference Demailly, Peternell and Schneider22, Corollary 1.19]. As a classical consequence of Yau’s theorem on the existence of a Kähler–Einstein metric, $B_i$ is then covered by a complex torus (see [Reference Kobayashi47, Chapter IV Corollary 4.15]). This finishes the proof of the proposition.

Proof. Suppose that $m<n$ . If $m=n$ , the proof of the proposition is very similar and therefore omitted.

The Chern polynomial $c_t(T_X|_Y)$ of $T_X|_Y$ satisfies

$$ \begin{align*}c_t(T_X|_Y)=c_t(\mathscr{N}|_Y)\cdot\prod_{1\leqslant i\leqslant n-1}c_t(\mathscr{L}_i|_Y),\end{align*} $$

where $\mathscr {N}$ denotes the normal bundle of $\mathscr {F}$ . Therefore, by Proposition 3.2, we have $c_1(T_X|_Y)^2=0$ and $c_k(T_X|_Y)=0$ for any $k \in \{2,\ldots ,n\}$ . On the other hand,

$$ \begin{align*}c_t(T_X|_Y)=c_t(\mathscr{N}_{Y/X})\cdot c_t(T_Y)=c_t(T_Y)\end{align*} $$

since $c_t(\mathscr {N}_{Y/X})=1$ by assumption. This easily implies $c_1(Y)^2=0$ and $c_k(Y)=0$ for any $k \in \{2,\ldots ,m\}$ . Then $\chi (Y,\mathscr {O}_Y)=0$ by the Hirzebruch–Riemann–Roch theorem. This finishes the proof of the proposition.

Proof. By [Reference Höring39, Proof of Lemma 2.2] and [Reference Nakayama51, Lemma 7.3], there exists an open subset $Y^\circ \subseteq Y$ with complement of codimension at least $2$ such that the restriction $f^\circ $ of f to $X^\circ :=f^{-1}(Y^\circ )$ is locally homotopically Q-smooth (we refer the reader to [Reference Nakayama51, Definition 7.2] for this notion). Then [Reference Nakayama51, Theorem 7.8] applies to show that $f^\circ $ is birational to a Q-smooth Q-torus fibration $g^\circ \colon Z^\circ \to Y^\circ $ over $Y^\circ $ . In other words, there exists a finite morphism $Y_1^\circ \to Y^\circ $ such that the normalization $Z_1^\circ $ of $Y_1^\circ \times _{Y^\circ } Z^\circ $ is étale over $Z^\circ $ and such that the morphism $Z_1^\circ \to Y_1^\circ $ is smooth with fibers being finite étale quotients of abelian varieties. This immediately implies that any fiber of $g^\circ $ (with its reduced structure) is a finite étale quotient of an abelian variety. It then follows that $X^\circ /Y^\circ \cong Z^\circ /Y^\circ $ . Let $X_1^\circ \to X^\circ $ be the induced finite étale cover, and denote by $f_1^\circ \colon X_1^\circ \to Y_1^\circ $ the corresponding smooth fibration. By [Reference Nakayama51, Theorem 3.14], there exists a finite étale cover $X_2^\circ \to X_1^\circ $ such that any fiber of the composition $f_2^\circ \colon X_2^\circ \to X_1^\circ \to Y_1^\circ =:Y_2^\circ $ is an abelian variety. Let $F_2$ be a general fiber of $f_2^\circ $ . By [Reference Claudon16, Proof of Lemma 2.3], the homomorphism $\pi _1(F_2) \to \pi _1(X_2^\circ )$ is injective. Moreover, there is a commutative diagram

where F denotes the image of $F_2$ in $X^\circ $ . By [Reference Nakayama51, Corollary 3.7], $\pi _1(F)$ is torsion-free. This easily implies that the homomorphism $\pi _1(F) \to \pi (X^{\circ})$ is also injective since $\pi _1(F_2)$ has finite index in $\pi (F)$ . Finally, the inclusion induces an isomorphism $\pi _1(X^\circ )\cong \pi _1(X)$ of fundamental groups since $X\setminus X^\circ $ is a closed subset of codimension at least $2$ . This finishes the proof of the lemma.

Proof. By [Reference Nakayama51, Theorem 3.14], there exists a finite étale cover $X_1 \to X$ such that the fiber $f_1^{-1}(y_0)$ of the composition $f_1 \colon X_1 \to X \to Y=:Y_1$ is an abelian variety. Notice that $f_1$ is a smooth morphism. By Ehresmann’s theorem, any fiber $F_1$ of $f_1$ is homeomorphic to an abelian variety. Then [Reference Catanese13, Theorem 4.8] applies to show that $F_1$ is an abelian variety. Finally, shrinking $Y_1$ if necessary, we can find a suitable finite étale cover $Y_2\to Y_1$ such that the projection morphism $Y_2 \times _{Y_1} X_1 \to Y_2$ has a section. Then $Y_2 \times _{Y_1} X_1$ is an abelian scheme over $Y_2$ . Replacing $Y_2$ by a further finite étale cover, we may assume that $Y_2 \times _{Y_1} X_1/Y_2$ admits a level three structure, finishing the proof of the lemma.

Proof. Let $\Sigma \subseteq X$ be the neutral section of f, and let N be the normalization of $\Lambda :=\beta (\Sigma )$ . Let also $\mathscr {A}(3)$ be the fine moduli space of polarized abelian varieties with a level three structure.

Recall that there exists a composition $Z \to M$ of a finite number of blow-ups with smooth centers such that the induced rational map $Z \dashrightarrow X$ is a morphism. This easily implies that every positive-dimensional fiber of $\beta $ is rationally chain connected.

Let $C\subseteq X$ be a rational curve such that $\dim \beta (C)=0$ , and set $B:=f(C)$ . By [Reference Kollár48, Lemma 5.9.3], $\big (f^{-1}(B)/B,\Sigma|_{f^{-1}(B)}\big )\cong (B \times A/B, B\times \{0_A\})$ , where $(A,0_A)$ is an abelian variety. Notice that $C=B \times \{a_0\}$ for some point $a_0 \in A$ . In particular, either C is disjoint from $\Sigma $ or C is contained in $\Sigma $ . This immediately implies that $\Sigma =\beta ^{-1}(\Lambda )$ since every positive-dimensional fiber of $\beta $ is rationally chain connected. By the rigidity lemma (see [Reference Debarre19, Lemma 1.15]), the morphism $Y \to \mathscr {A}(3)$ corresponding to f factorizes through the map $\eta \colon Y\cong \Sigma \to N$ induced by $\beta|_\Sigma $ . In other words, the abelian scheme f is the pull-back of an abelian scheme $M_1$ over N via $\eta $ . Let $\beta _1\colon X \to M_1$ be the induced morphism.

To prove the statement, it suffices to show that there exists an isomorphism $\tau \colon M \to M_1$ such that $\beta _1=\tau \circ \beta $ or equivalently that $\beta $ (resp. $\beta _1$ ) factorizes through $\beta _1$ (resp. $\beta $ ).

Let F be a positive-dimensional fiber of $\beta $ . Recall that F is rationally chain connected. Set $G:=f(F)$ . By [Reference Kollár48, Lemma 5.9.3] again, $\big (f^{-1}(G)/G,\Sigma|_{f^{-1}(G)}\big )\cong (G \times A_2/G, G\times \{0_{A_2}\})$ , where $(A_2,0_{A_2})$ is an abelian variety. Moreover, $F=G \times \{a_2\}$ for some point $a_2 \in A_2$ . Since G is a fiber of $\eta $ , F is contracted by $\beta _1$ . By the rigidity lemma (see [Reference Debarre19, Lemma 1.15]), $\beta _1$ factorizes through $\beta $ . Let now $F_1$ be a fiber of $\beta _1$ . Set $G_1:=f(F_1)$ . Then $G_1$ is a fiber of $\eta $ . Moreover, $\big (f^{-1}(G_1)/G_1,\Sigma|_{f^{-1}(G_1)}\big ) \cong (G_1 \times A_1/G_1, G_1 \times \{0_{A_1}\})$ for some abelian variety $(A_1,0_{A_1})$ , and $F_1\cong G_1 \times \{a_1\}$ for some point $a_1 \in A_1$ . Since $G_1 \times \{0_{A_1}\}\cong \Sigma \cap f^{-1}(G_1)$ is contracted by $\beta $ , $F_1$ is contracted by $\beta $ as well. By the rigidity lemma (see [Reference Debarre19, Lemma 1.15]), $\beta $ factorizes through $\beta _1$ . This finishes the proof of the lemma.

Proof. Let m be a positive integer such that the set of points $y\in Y$ with $\mathscr {L}^{\otimes m}|_{X_y} \cong \mathscr {O}_{X_y}$ is dense. Because the functions $y \mapsto h^0\big (X_y,\mathscr {L}^{\otimes \pm m}|_{X_y}\big )$ are upper semicontinuous in the Zariski topology on Y (see [Reference Hartshorne36, Theorem 12.8]), we have $h^0\big (X_y,\mathscr {L}^{\otimes \pm m}|_{X_y}\big )=1$ for every $y\in Y$ . This implies that $\mathscr {L}^{\otimes m}|_{X_y} \cong \mathscr {O}_{X_y}$ since $X_y$ is integral by assumption. By [Reference Hartshorne36, Corollary 12.9] together with the base change theorem [Reference Hartshorne36, Theorem 12.11], $\mathscr {M}:=f_*\mathscr {L}^{\otimes m}$ is a line bundle and the formation of $f_*\mathscr {L}^{\otimes m}$ commutes with arbitrary base change. This easily implies that $\mathscr {L}^{\otimes m} \cong f^*\mathscr {M}$ .

Proof. Let $B^\circ \subseteq B$ be a dense open set such that $f^\circ :=f|_{X^\circ }$ is a smooth morphism, where $X^\circ :=f^{-1}(B^\circ )$ . Let m be a positive integer. Because the functions $b \mapsto h^0\big (X_b,\mathscr {L}^{\otimes \pm m}|_{X_b}\big )$ are upper semicontinuous in the Zariski topology on B (see [Reference Hartshorne36, Theorem 12.8]), the set of points b on $B^\circ $ such that $\mathscr {L}^{\otimes m}|_{X_b}\cong \mathscr {O}_{X_b}$ is closed. As a consequence, there exists a positive integer $m_1$ such that $\mathscr {L}^{\otimes m_1}|_{X_b}\cong \mathscr {O}_{X_b}$ for all $b \in B^\circ $ . From [Reference Hartshorne36, Corollary 12.9], we see that $f^\circ _*(\mathscr {L}^{\otimes m_1}|_{X^\circ })$ is a line bundle. Thus, there exists a divisor D on X with $f(\text {Supp}\,D)\subsetneq B$ such that $\mathscr {L}^{\otimes m_1}\cong \mathscr {O}_X(D)$ .

Let $S \subset X$ be a two-dimensional complete intersection of general elements of a very ample linear system on X. This is a smooth surface in X and the restriction $f|_S$ of f to S is a surjective morphism with connected fibers onto B. By Zariski’s lemma (see [Reference Barth, Peters and Van de Ven10, Lemma III.8.2]), we have $D|_S^2 \leqslant 0$ . Since D is nef by assumption, we must have $D|_S^2 = 0$ . By Zariski’s lemma again, there exist a positive integer $m_2$ and a divisor G on B such that $m_2 D|_S=f|_S^*G$ . The claim now follows from the Grothendieck–Lefchetz hyperplane theorem.

Proof. Notice that a general fiber of f is irreducible by [Reference Debarre19, 5.5]. By a theorem of Chevalley (see [Reference Grothendieck34, Corollaire 14.4.4], the morphism f is universally open.

Let $y \in Y$ be a point, and let $B \to Y$ be a morphism from a smooth complete curve B whose image contains y and a general point in Y. Observe that product $B \times _Y X$ is irreducible since the projection $B \times _Y X \to B$ is open with irreducible general fibers. Let Z be a resolution of $(B \times _Y X)_{\text {red}}$ . Our claim follows from Proposition 3.11 applied to the pull-back of $\mathscr {L}$ to Z.

Proof. Let us first show that $\dim Y =1$ . We argue by contradiction and assume that $\dim Y \geqslant 2$ . Let $S \subseteq X$ be a two-dimensional complete intersection of general elements of $|H|$ , and set $C:= S \cap D$ . By Bertini’s theorem (see [Reference Jouanolou41, Théorème I.6.10]), we may assume without loss of generality that S and C are reduced and irreducible. Then $D^2\cdot S = (D|_S)^2 <0$ since C is contracted by the generically finite morphism $f|_S$ , yielding a contradiction since $D^2\cdot S = D^2\cdot H^{n-2} \geqslant 0$ by assumption. This shows $\dim Y =1$ .

Zariski’s lemma then applies to show that some nonzero multiple of $D|_S=C$ is a fiber of $f|_S$ . This easily implies that some nonzero multiple of D is a fiber of f, and hence $\kappa (D) = 1$ . This finishes the proof of the lemma.

Proof. Let D be an effective $\mathbb {Q}$ -divisor such that $K_X \sim _{\mathbb {Q}} D$ . Notice that $D \cdot B_i = K_X\cdot B_i \equiv 0$ since $B_i^2\equiv 0$ and $K_{B_i}\equiv 0$ by assumption. Thus, there is a decomposition $D=D_1+D_2$ , where $D_1$ and $D_2$ are effective $\mathbb {Q}$ -divisor with $\text {Supp}\, D_1 \cap \text {Supp}\, D_2 = \emptyset $ and $\text {Supp}\, D_2 \subseteq \text {Supp}\, B$ .

Let us show that $D_1$ is nef. We argue by contradiction and assume that there exists a curve $C \subset X$ such that $D_1 \cdot C < 0$ . Then $C \subseteq \text {Supp}\, D_1$ and hence $C \cap \text {Supp}\, D_2 =\emptyset $ . Thus, $K_X \cdot D =D \cdot C = D_1 \cdot C <0$ , yielding a contradiction. Notice also that $B_i$ is nef since $B_i^2 \equiv 0$ by assumption.

Pick $i_0 \in I$ . Then $K_X+B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I\setminus \{i_0\}}B_i\big )$ is nef as well for every $\varepsilon>0$ . Let $0<\varepsilon \ll 1$ be a rational number such that the pair $\big (X,B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I\setminus \{i_0\}}B_i\big )\big )$ is plt. Then [Reference Demailly, Hacon and Păun21, Corollary 1.8] applies to show that there exists an effective $\mathbb {Q}$ -divisor $G_1$ such that $K_X+B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I \setminus \{i_0\}}B_i\big )\sim _{\mathbb {Q}} G_1$ and $B_{i_0} \cap G_1 =\emptyset $ since $\big (K_X+B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I \setminus \{i_0\}}B_i\big )\big )|_{B_{i_0}}\sim _{\mathbb {Q}} K_{B_{i_0}}$ is torsion. Set $G_2:=D+B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I \setminus \{i_0\}}B_i\big )$ . Then $G_2 \sim _{\mathbb {Q}} G_1$ and $\text {mult}_{B_{i_0}}\, G_2 \geqslant 1$ .

Let m be a positive integer such that $mG_1$ and $m G_2$ are $\mathbb {Z}$ -divisors, and let $X \dashrightarrow \mathbb {P}^1$ be the corresponding rational map. Let $p\colon Y \to X$ and $q\colon Y \to \mathbb {P}^1$ be a common resolution of singularities. We may assume without loss of generality that p restricts to an isomorphism over some Zariski open neighborhood of $B_{i_0}$ since $B_{i_0} \cap G_1 =\emptyset $ by construction. Let $g \colon Y \to C$ be the Stein factorization of q. By Zariski’s lemma, some positive multiple of the strict transform of $B_{i_0}$ in Y is a fiber of g since $B_{i_0}^2\equiv 0$ . This immediately implies that the rational map $f \colon X \dashrightarrow C$ is a morphism. By Zariski’s lemma again, for every $i\in I$ , some positive multiple of $B_i$ is a fiber of f. Let now F be a general fiber of f. Then, by construction, $\big (K_X+B_{i_0}+\varepsilon \big (D_1+\sum _{i \in I \setminus \{i_0\}}B_i\big )\big )|_F = K_X|_F$ is torsion. But $K_X|_F \sim _{\mathbb {Z}} K_F$ by the adjunction formula. The statement easily follows from Proposition 3.11.

Proof. Let $\mathscr {N}$ denote the normal bundle of $\mathscr {F}$ , and let $f\colon S \to \mathbb {P}^1$ denote the natural morphism. Recall that $\text {N}^1(S)\cong \mathbb {Z}[C_0]\oplus \mathbb {Z}[F]$ , where F is any fiber of f and $C_0$ is a section of f with $C_0^2=-e\leqslant 0$ .

Set $\mathscr {N}_S:=\mathscr {N}|_S$ . By Lemma 3.1, we have $c_1(\mathscr {N}_S)^2 = 0$ . Write $c_1(\mathscr {N}_S)\equiv a C_0 + b F$ . Then

$$ \begin{align*}0=c_1(\mathscr{N}_S)^2=(a C_0 + b F)^2=a(-ae+2b),\end{align*} $$

$$ \begin{align*}c_1(\mathscr{N}_S)\cdot F=a\end{align*} $$

and

$$ \begin{align*}c_1(\mathscr{N}_S)\cdot C_0=-ae+b.\end{align*} $$

We may obviously assume that F is not tangent to $\mathscr {F}$ . Then the composition $T_{\mathbb {P}^1}\cong T_F \to T_X|_F \to \mathscr {N}|_F=\mathscr {N}_S|_F$ is nonzero, and hence $a=c_1(\mathscr {N}_S)\cdot F\geqslant 2>0$ . It follows that $2b=ae$ . In particular, we have $b \geqslant 0$ .

If $C_0$ is contracted by $\gamma $ , then $c_1(\mathscr {N}_S)\cdot C_0=0$ . This immediately implies $b=e=0$ . Then $S \cong \mathbb {P}^1 \times \mathbb {P}^1$ and f identifies with the projection onto the first factor. Moreover, the image of a general fiber of the projection onto the second factor is tangent to $\mathscr {F}$ .

Suppose from now on that $C_0$ is not contracted by $\gamma $ and that $\gamma (C_0)$ is not tangent to $\mathscr {F}$ . Then, as above, we must have $c_1(\mathscr {N}_S) \cdot C_0 \geqslant 2$ . It follows that $b \geqslant ae+2$ , and hence $b+2 \leqslant 0$ , yielding a contradiction. This finishes the proof of the lemma.

Proof. Let $\mathscr {N}$ denote the normal bundle of $\mathscr {F}$ , and let $C \subseteq F$ be a rational curve. Suppose that $\mathscr {N}\cdot C=0$ . Then C is tangent to $\mathscr {F}$ and moreover $\mathscr {N}|_F\equiv 0$ since $\varphi $ is elementary by assumption. This in turn implies that any irreducible component of F is tangent to $\mathscr {F}$ since F is rationally chain connected by [Reference Hacon and McKernan38, Corollary 1.4], yielding a contradiction. This shows that $\mathscr {N}\cdot C \neq 0$ . On the other hand, $c_1(\mathscr {N})^2\equiv 0$ by Lemma 3.1. This immediately implies that any positive-dimensional fiber of $\varphi $ is one-dimensional. The claim now follows from [Reference Wiśniewski62, Theorem 1.2].

Proof. Let $\mathscr {N}$ denote the normal bundle of $\mathscr {F}$ , and let $\mathbb {P}^1 \to C \subset X$ be the normalization morphism. Since $\mathscr {F}$ is projectively flat, there exists $a \in \mathbb {Z}$ such that $\mathscr {F}|_{\mathbb {P}^1}\cong \mathscr {O}_{\mathbb {P}^1}(a)^{\oplus n-1}$ . Notice that $a \geqslant 0$ since $K_{\mathscr {F}} \cdot C \leqslant 0$ by assumption.

If C is tangent to $\mathscr {F}$ , then $\mathscr {N}|_{\mathbb {P}^1}$ is a flat line bundle by Lemma 3.1, and hence $\mathscr {N}|_{\mathbb {P}^1}\cong \mathscr {O}_{\mathbb {P}^1}$ . If C is not tangent to $\mathscr {F}$ , then the composition $T_{\mathbb {P}^1} \to T_X|_{\mathbb {P}^1} \to \mathscr {N}|_{\mathbb {P}^1}$ is nonzero, and hence $\mathscr {N} \cdot C \geqslant 2$ . In either case, the exact sequence

$$ \begin{align*}0 \to \mathscr{F}|_{\mathbb{P}^1} \to T_X|_{\mathbb{P}^1} \to \mathscr{N}|_{\mathbb{P}^1} \to 0\end{align*} $$

shows that $T_X|_{\mathbb {P}^1}$ is nef. But then X is uniruled by [Reference Kollár49, Theorem IV.1.9].

Proof. This follows easily from [Reference Touzet56, Théorème 1.2] together with [Reference Druel24, Lemma 5.9].

Proof. By assumption, $\mathscr {F}$ is induced by a morphism with irreducible fibers $f\colon X \to C$ onto a smooth complete curve C. By [Reference Jahnke and Radloff42, Theorem 0.1], any leaf of $\mathscr {F}$ is a finite étale quotient of an abelian variety. By Lemma 3.6 and [Reference Kollár48, Theorem 6.3] together, after replacing X by a finite étale cover, the fibration f is birational to an abelian scheme over C. This easily implies that f is an abelian scheme. Replacing X by a further finite étale cover, we may also assume that f is polarized with a level three structure. Let $\mathscr {A}(3)$ be the corresponding fine moduli space of polarized abelian varieties with a level three structure, and let $C \to \mathscr {A}(3)$ be the moduli map. By [Reference Kollár49, Lemma 5.9.3], either the moduli map is the constant morphism or C has genus at least $2$ . The proposition now follows easily.

Proof. Replacing X by a finite étale cover, if necessary, we may assume without loss of generality that X is an abelian variety. Then $\mathscr {N}$ is generated by its global section. On the other hand, we have $\mathscr {N}^2\equiv 0$ by Lemma 3.1. If $\mathscr {N}\cong \mathscr {O}_X$ , then $K_{\mathscr {F}}\equiv 0$ by the adjunction formula. Suppose that $\mathscr {N} \not \equiv 0$ . Then, after a further finite étale cover, we have $X \cong A \times E$ , where A is an abelian variety of dimension $n-1$ and E is an elliptic curve by Poincaré’s complete reducibility theorem. Moreover, $\mathscr {N}\cong f^*\mathscr {M}$ for some line bundle $\mathscr {M}$ on E, where $f\colon X \to E$ denotes the morphism induced projection onto E. Then Lemma 3.23 below applies to show that $K_{\mathscr {F}} \equiv 0$ since $n \geqslant 3$ by assumption. This implies $\mathscr {N}\equiv 0$ , yielding a contradiction. This finishes the proof of the lemma.

Proof. Let $\omega \in H^0(X,\Omega _X^1\otimes \mathscr {N})$ be a twisted $1$ -form defining $\mathscr {F}$ . Let $F \cong A$ be a fiber of f. Then $T_X|_F\cong \mathscr {O}_F^{\oplus n}$ and $\mathscr {N}|_F \cong \mathscr {O}_F$ . This immediately implies that $\mathscr {F}|_F\cong \mathscr {O}_F^{\oplus n-1}$ and that $\mathscr {F}$ is regular since the zero set of $\omega $ has codimension at least $2$ in X. Moreover, there exists a vector bundle $\mathscr {E}$ on C such that $\mathscr {F} \cong f^*\mathscr {E}$ .

We may obviously assume that $\mathscr {F}\neq T_{X/C}$ . If F is a general fiber of f, then $\mathscr {F}\cap T_{X/C}|_F \subset T_F$ is a linear codimension $1$ foliation on $F \cong A$ . Hence, there exists a subbundle $\mathscr {K} \subset \mathscr {E}$ of rank $n-2$ such that $\mathscr {F}\cap T_{X/C} = f^*\mathscr {K} \subset \mathscr {F}$ . Set $g:=\text {pr}_A$ . Write $\omega =\alpha + \beta \in H^0(X,\Omega _X^1\otimes \mathscr {N}) \cong H^0(X,f^*\omega _C\otimes \mathscr {N}) \oplus H^0(X,g^*\Omega _A^1\otimes \mathscr {N})$ with $\alpha \in H^0(X,f^*\omega _C\otimes \mathscr {N})$ and $\beta \in H^0(X,g^*\Omega _A^1\otimes \mathscr {N})\cong H^0(A,\Omega _A^1) \otimes H^0(C,\mathscr {L})$ . By construction, there is an effective divisor $D_1$ on C and an exact sequence

$$ \begin{align*}0 \to \mathscr{K} \to H^0(A,\Omega_A^1)^*\otimes \mathscr{O}_C \to \mathscr{L}(-D_1) \to 0,\end{align*} $$

where the map $H^0(A,\Omega _A^1)^*\otimes \mathscr {O}_C \to \mathscr {L}(-D_1)$ is induced by $\beta $ . Since $\mathscr {F}\neq T_{X/C}$ , the natural map $\mathscr {O}_X^{\oplus n-1}\cong T_{X/C} \to \mathscr {N}$ is nonzero, and hence $h^0(X,\mathscr {L}) \geqslant 1$ . Let $D_2$ be any effective divisor on C such that $\mathscr {L} \cong \mathscr {O}_C(D_2)$ . We view $\omega $ , $\alpha $ and $\beta $ as rational $1$ -forms on X with poles along the divisor $f^*D_2$ . We have $\text {d}\alpha =0$ and $\text {d}\beta \wedge \alpha =0$ . But then $\text {d}\beta \wedge \beta =\text {d}\omega \wedge \omega =0$ . Let $\gamma _1,\ldots ,\gamma _{n-1}$ be a basis of $H^0(A,\Omega _A^1)$ , and let $U \subset C\setminus \text {Supp}\, D_2$ be an open affine subset. Then

$$ \begin{align*}\beta|_{U \times A}=\sum_{1 \leqslant i \leqslant n-1} t_i \gamma_i\end{align*} $$

for some regular functions $t_i$ on U. Moreover, $\text {d}\beta \wedge \beta =0$ if and only if $\frac {\text {d}t_i}{t_i}=\frac {\text {d}t_j}{t_j}$ for all indices i and j. Suppose that $t_j\neq 0$ . Then $t_i=c_{ij}t_j$ for some $c_{ij}\in \mathbb {C}$ . This implies that there exists $s \in H^0(Y,\mathscr {L})$ such that $\beta = \gamma \otimes s$ , where $\gamma := \sum _{1 \leqslant i \leqslant n-1} c_{ij} \gamma _i \in H^0(A,\Omega _A^1)$ . Then $\text {div} \, s = D_1$ and $\mathscr {K}\cong \mathscr {O}_C^{\oplus n-2}$ .

Set $\mathscr {Q}:=\mathscr {E}/\mathscr {K}$ . Then $\mathscr {Q} \cong T_C\otimes \mathscr {L}^*$ . If $h^0(C,\mathscr {L}^*)=0$ , then $h^1(C,\mathscr {Q}^*)=h^1(C,\omega _C\otimes \mathscr {L})=h^0(C,\mathscr {L}^*)=0$ by Serre duality, and hence the exact sequence

$$ \begin{align*}0 \to \mathscr{K} \to \mathscr{E} \to \mathscr{Q} \to 0\end{align*} $$

splits. Then $\mathscr {F}\cong \mathscr {O}_X^{\oplus n-2}\oplus f^*(T_C \otimes \mathscr {L}^*)$ .

Suppose finally that $h^0(C,\mathscr {L}^*) \geqslant 1$ . Then $\mathscr {L}\cong \mathscr {O}_C$ since we also have $h^0(C,\mathscr {L}) \geqslant 1$ . Then $\mathscr {Q}\cong T_C$ , finishing the proof of the lemma.