Proof. Since the birational map ${\it\varphi}^{-1}:X^{\prime }{\dashrightarrow}X$ does not contract any divisor, it induces an isomorphism from an open subset $W^{\prime }$ of $X^{\prime }$ such that $\text{codim}\,X^{\prime }\setminus W\geqslant 2$ onto an open subset $W$ of $X$. By Proposition 2.1, we have $H^{0}(X^{\prime },({\rm\Omega}_{X^{\prime }}^{1})^{[\otimes m]})\cong H^{0}(W^{\prime },({\rm\Omega}_{W^{\prime }}^{1})^{[\otimes m]})$ for any $m>0$. Moreover, since $W$ is an open subset of $X$, we obtain $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]}){\hookrightarrow}H^{0}(W,({\rm\Omega}_{W}^{1})^{[\otimes m]})\cong H^{0}(W^{\prime },({\rm\Omega}_{W^{\prime }}^{1})^{[\otimes m]})\cong H^{0}(X^{\prime },({\rm\Omega}_{X^{\prime }}^{1})^{[\otimes m]})$ for all $m>0$.◻

Proof. On $W$, we have $H^{0}(W,\mathscr{A}|_{W}^{\otimes r}\otimes \mathscr{C}|_{W}^{\otimes t})=\{0\}$ for all $t>0$ and $r\geqslant 0$ by Proposition 2.1. If we fix $m>0$, by Lemma 2.3, we have a filtration $\mathscr{B}|_{W}^{\otimes m}=\mathscr{R}_{0}\supseteq \cdots \supseteq \mathscr{R}_{m+1}=0$ such that $\mathscr{R}_{i}/\mathscr{R}_{i+1}$ is isomorphic to the direct sum of copies of $\mathscr{A}|_{W}^{\otimes i}\otimes \mathscr{C}|_{W}^{\otimes m-i}$ for all $0\leqslant i\leqslant m$. Since $H^{0}(X,\mathscr{A}|_{W}^{\otimes r}\otimes \mathscr{C}|_{W}^{\otimes t})=\{0\}$ for all $t>0$ and $r\geqslant 0$, we have $H^{0}(W,\mathscr{R}_{i})\cong H^{0}(W,\mathscr{R}_{i+1})$ for $0\leqslant i\leqslant m-1$. Thus $H^{0}(W,\mathscr{B}|_{W}^{\otimes m}/\mathscr{A}|_{W}^{\otimes m})=\{0\}$ and $H^{0}(W,B|_{W}^{\otimes m})\cong H^{0}(W,A|_{W}^{\otimes m})$. By Proposition 2.1, we have $H^{0}(X,\mathscr{B}^{[\otimes m]})\cong H^{0}(X,\mathscr{A}^{[\otimes m]})$.◻

Proof. We have an exact sequence of coherent sheaves $0\rightarrow \mathscr{F}\rightarrow {\rm\Omega}_{X}^{[1]}\rightarrow \mathscr{G}\rightarrow 0$ on $X$, where $\mathscr{F}=(p^{\ast }{\rm\Omega}_{Z}^{1})^{sat}$ and $\mathscr{G}$ is a torsion-free sheaf. In particular, $\mathscr{G}$ is locally free in codimension $1$. Let $V$ be the smooth locus of $Z$ and let $W=p^{-1}(V)$. If $U$ is the largest open subset of $W$ on which $\mathscr{F}$, ${\rm\Omega}_{X}^{1}$ and $\mathscr{G}$ are locally free, then $\text{codim}\,X\setminus U\geqslant 2$ and we have an exact sequence $0\rightarrow \mathscr{F}|_{U}\rightarrow {\rm\Omega}_{U}^{1}\rightarrow \mathscr{G}|_{U}\rightarrow 0$ of locally free sheaves on $U$.

If $F$ is a general fiber of $p|_{U}$, then $\mathscr{F}|_{F}$ is the direct sum of $\mathscr{O}_{F}$ and $\mathscr{G}|_{F}$ is isomorphic to ${\rm\Omega}_{F}^{1}$. Since general fibers of $p$ do not carry any non-zero pluri-form, neither does $F$ by Proposition 2.1. Hence, we have $H^{0}(U,\mathscr{F}|_{U}^{\otimes r}\otimes \mathscr{G}|_{U}^{\otimes t})=\{0\}$ for all $t>0$ and $r\geqslant 0$. By Lemma 2.4, we have an isomorphism from $H^{0}(U,\mathscr{F}|_{U}^{\otimes m})$ to $H^{0}(U,({\rm\Omega}_{U}^{1})^{\otimes m})$ for all $m>0$. By Proposition 2.1, we have $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})\cong H^{0}(X,\mathscr{F}^{[\otimes m]})$ for all $m>0$.◻

Proof. Assume the opposite. Then the discrepancy of $E$ in $X$ is strictly smaller than the one in $X^{\prime }$ (see [Reference Kollár and MoriKM98, Lemma 3.38]). This implies that $X$ does not have canonical singularities along the center of $E$ in $X$, which is a contradiction.◻

Proof. Since there is a natural birational projection $p_{1}:{\rm\Gamma}\rightarrow X$, we have an injection from $H^{0}({\rm\Gamma},({\rm\Omega}_{{\rm\Gamma}}^{1})^{[\otimes m]})$ to $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})$ by Lemma 2.2. Let $p_{2}:{\rm\Gamma}\rightarrow X^{\prime }$ be the natural projection. Let ${\it\sigma}_{X}\in H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})$ be a non-zero element. Since $X$, ${\rm\Gamma}$ and $X^{\prime }$ are birational, ${\it\sigma}_{X}$ induces a rational section ${\it\sigma}_{{\rm\Gamma}}$ of $({\rm\Omega}_{{\rm\Gamma}}^{1})^{[\otimes m]}$ and an element ${\it\sigma}_{X^{\prime }}$ of $H^{0}(X^{\prime },({\rm\Omega}_{X^{\prime }}^{1})^{[\otimes m]})$ (see Lemma 2.2). In order to prove that ${\it\sigma}_{{\rm\Gamma}}$ is a regular section it is sufficient to prove that ${\it\sigma}_{{\rm\Gamma}}$ does not have a pole along any $p_{1}$-exceptional divisor. Let $E$ be an exceptional divisor for $p_{1}$. Let $C\subseteq E$ be a curve that is exceptional for $p_{1}$, then $C$ is not contracted by $p_{2}$ since the graph of $f$ is included in $X\times X^{\prime }$ and the normalization map is finite. Hence, $Y=p_{2}(E)\subseteq X^{\prime }$ is a curve and $f^{-1}$ is not regular around the generic point of $Y$. By Lemma 2.8, $X^{\prime }$ has terminal singularities around the generic point of $Y$. Hence, it is smooth around the generic point of $Y$ since $\text{codim}_{X^{\prime }}\,Y=2$ (see [Reference Kollár and MoriKM98, Corollary 5.18]). Moreover, since $T$, $X$ and $X^{\prime }$ are birational, the form ${\it\sigma}_{{\rm\Gamma}}$ is just the pullback of ${\it\sigma}_{X^{\prime }}$ by $p_{2}$. Hence, ${\it\sigma}_{{\rm\Gamma}}$ does not have a pole along $E$.◻

Construction 2.13. We want to construct a fibration $p$ from a normal threefold $T$ with canonical singularities to a smooth surface $S$ such that $p^{\ast }C=2(p^{\ast }C)_{\mathit{red}}$, where $C$ is a smooth curve in $S$.

Let $T_{0}=\mathbb{P}^{1}\times S$, where $S$ is a smooth projective surface. Let $p_{1}$, $p_{2}$ be the natural projections $T_{0}\rightarrow \mathbb{P}^{1}$ and $T_{0}\rightarrow S$. Let $C\subseteq S$ be a smooth curve, and let $C_{0}=\{z\}\times C$ be a section of $p_{2}$ over $C$ in $T_{0}$, where $z$ is a point of $\mathbb{P}^{1}$. Let $E_{0}=p_{2}^{\ast }C$, which is a smooth divisor in $T_{0}$. We perform a sequence of birational transformations of $T_{0}$.

First, we blow up $C_{0}$, and we obtain a morphism $T_{1}\rightarrow T_{0}$ with exceptional divisor $E_{1}$. Denote still by $E_{0}$ the strict transform of $E_{0}$ in $T_{1}$. If $F$ is a fiber of $T_{1}\rightarrow S$ over a point of $C$, then $F=F_{1}\cup F_{0}$, where $F_{i}\subseteq E_{i}$ are rational curves and $K_{T_{1}}\cdot F_{i}=-1$, $E_{i}\cdot F_{i}=-1$.

Now we blow up $C_{1}$, the intersection of $E_{0}$ and $E_{1}$. We obtain a morphism $T_{2}\rightarrow T_{1}$ with exceptional divisor $E_{2}$. Denote still by $E_{0}$, $E_{1}$ the strict transforms of $E_{0}$ and $E_{1}$ in $T_{2}$. If $F$ is any set-theoretic fiber of $T_{2}\rightarrow S$ over a point of $C$, then $F=F_{0}\cup F_{1}\cup F_{2}$, where $F_{i}\subseteq E_{i}$ are rational curves and $K_{T_{2}}\cdot F_{i}=0$, $E_{i}\cdot F_{i}=-2$ for $i=0,1$, $K_{T_{2}}\cdot F_{2}=-1$, $E_{2}\cdot F_{2}=-1$. Let $q$ be the fibration $T_{2}\rightarrow S$.

We blow down $E_{1}$ and $E_{0}$ in $T_{2}$. Let $H$ be an ample $\mathbb{Q}$-divisor such that $(H+E_{0})\cdot F_{0}=0$. Since $E_{0}\cdot F_{1}=0$, there is a positive rational number $b$ such that $(H+E_{0}+bE_{1})\cdot F_{1}=0$. Moreover, $(H+E_{0}+bE_{1})\cdot F_{0}=0$ since $E_{1}\cdot F_{0}=0$. Let $A$ be a sufficiently ample divisor on $S$. Then the $\mathbb{Q}$-divisor $D=q^{\ast }A+H+E_{0}+bE_{1}$ is nef and big. Moreover, any curve $B$ which has intersection number $0$ with $D$ must be contracted by $T_{2}\rightarrow S$ since $A$ is sufficiently ample. The curve $B$ is also contained in $E_{0}\cup E_{1}$ since $H$ is ample. Since $K_{T_{2}}\cdot F_{0}=K_{T_{2}}\cdot F_{1}=0$, there exists a large integer $k$ such that $kD-K_{T_{2}}$ is nef and big. Then, by the basepoint-free theorem (see [Reference Kollár and MoriKM98, Theorem 3.3]), there is a large enough integer $a$ such that $aD$ is Cartier and $|aD|$ is basepoint-free. The linear system $|aD|$ induces a contraction which contracts $E_{0}$ and $E_{1}$.

By contracting $E_{0}$ and $E_{1}$, we obtain a threefold $T$. Moreover, there is an induced fibration $p:T\rightarrow S$ such that $p^{\ast }C=2(p^{\ast }C)_{\mathit{red}}$. Note that $T_{2}\rightarrow T$ is a resolution of singularities and $K_{T_{2}}=f^{\ast }K_{T}$. Hence, $V$ has canonical singularities.

Proof. Assume that $X$ is uniruled and let ${\it\pi}:X{\dashrightarrow}Z$ be the MRC fibration for $X$ (see, for example, [Reference KollárKol96, Theorem V.5.2]). We argue by contradiction. Assume that $\text{dim}\,X>\text{dim}\,Z>0$. Then there is a Zariski open subset $Z_{0}$ of $Z$ such that ${\it\pi}$ is regular and proper over $Z_{0}$. Let $C$ be a curve contained in some fiber of ${\it\pi}|_{Z_{0}}$. Let $D$ be a non-zero effective Cartier divisor on $Z_{0}$. Let $E_{0}={\it\pi}^{\ast }D$. Then $E_{0}$ is a non-zero effective divisor on $X_{0}={\it\pi}^{-1}Z_{0}$. Let $E$ be the closure of $E_{0}$ in $X$, then $E$ is a non-zero effective $\mathbb{Q}$-Cartier Weil divisor on $X$. However, the intersection number of $E$ and $C$ is zero, which contradicts the fact that $X$ has Picard number $1$.◻

Proof. The problem is local and we may assume that $Z$ is affine. We know that $p_{\ast }\mathscr{O}_{W}(mD)$ is an invertible sheaf. If ${\it\theta}$ is a section of $p_{\ast }\mathscr{O}_{W}(mD)$, then it is a rational function on $Z$ whose pullback on $W$ is a rational function which can only have a pole along $p^{\ast }z$. Hence, ${\it\theta}$ can only have a pole at $z$. Let $d_{}$ be the degree of the pole, then the pullback of ${\it\theta}$ in $V$ is a section of $\mathscr{O}_{W}(mD)$ if and only if $m(p,z)d_{}\leqslant m(m(p,z)-1)$, that is,$d_{}\leqslant [(m(m(p,z)-1))/m(p,z)]$.◻

Proof. The fibration $p:X\rightarrow \mathbb{P}^{1}$ is induced by $q$. Since general fibers of $q$ are rationally connected, general fibers of $p$ are also rationally connected. Hence, $X$ is rationally connected by [Reference Graber, Harris and StarrGHS03, Theorem 1.1]. If $F_{p}$ is a general fiber of $p$, then $F_{p}$ does not carry any non-zero pluri-forms either. Hence, $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})$ is isomorphic to $H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(-2m)\otimes p_{\ast }\mathscr{O}_{X}(mR))$ for $m>0$, where $R$ is the ramification divisor of $p$. However, since $m(q,b)=1$ for all $b\in B$, we have $\sum _{i=1}^{r}[((s_{i}-1)m)/s_{i}]=\sum _{z\in \mathbb{P}^{1}}[((m(p,z)-1)m)/m(p,z)]$ by the definition of $s_{i}$. Finally, we have $p_{\ast }\mathscr{O}_{X}(mR)\cong \mathscr{O}_{\mathbb{P}^{1}}(\sum _{i=1}^{r}[((s_{i}-1)m)/s_{i}])$ and $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})\cong H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(-2m+\sum _{i=1}^{r}[((s_{i}-1)m)/s_{i}]))$ by Lemma 3.1.◻

Proof. Assume the opposite. If $S^{\prime }$ is the result of an MMP for $S$, then $S^{\prime }$ is a Mori fiber space. By [Reference OuOu14, Theorem 3.1], $S^{\prime }$ has Picard number $2$ and we have a Mori fibration $S^{\prime }\rightarrow \mathbb{P}^{1}$. Let $p_{S}$ be the composition of $S\rightarrow S^{\prime }\rightarrow \mathbb{P}^{1}$. By [Reference OuOu14, Theorem 1.5], there is a smooth curve $B$ of positive genus and a finite morphism $B\rightarrow \mathbb{P}^{1}$ such that the natural morphism $S_{B}\rightarrow S$ is étale in codimension $1$, where $S_{B}$ is the normalization of $S\times _{\mathbb{P}^{1}}B$. Hence, $S_{B}$ is a Fano surface. Moreover, it has canonical singularities (see [Reference Kollár and MoriKM98, Proposition 5.20]). Therefore, it is rationally connected by [Reference Hacon and McKernanHM07, Corollaries 1.3 and 1.5]. Hence, $B$ is also rationally connected, which is a contradiction since its genus is positive.◻

Proof. By Theorem 3.4 and Lemma 2.5, we know that every element in the space $H^{0}(X^{\ast },({\rm\Omega}_{X^{\ast }}^{1})^{[\otimes m]})$ comes from the base $\mathbb{P}^{1}$, and so does every element in $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})$ by Lemma 2.2. We have a rational map $X{\dashrightarrow}\mathbb{P}^{1}$ induced by $p$. By Proposition 2.9, we know that $H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})\cong H^{0}(Y,({\rm\Omega}_{Y}^{1})^{[\otimes m]})$. Moreover, we have a natural fibration $g:Y\rightarrow \mathbb{P}^{1}$ which is the composition of $Y\rightarrow X^{\ast }\rightarrow \mathbb{P}^{1}$. If $F_{g}$ is a general fiber of $g$, then there is a birational morphism $F_{g}\rightarrow S$, where $S$ is a general fiber of $p:X^{\ast }\rightarrow \mathbb{P}^{1}$. Since $S$ does not carry any non-zero pluri-forms by Theorem 3.4, neither does $F_{g}$ by Lemma 2.2. From Remark 3.3, we know that $Y$ can be constructed by the method of Theorem 3.2.◻

Proof. Let $D=D_{1}+\cdots +D_{k}$. By [Reference Kawamata, Matsuda and MatsukiKMM87, Lemma 5-1-5], $Z$ is $\mathbb{Q}$-factorial. Note that the problem is local in $Z$, and we may assume that $Z$ is affine.

We construct by induction a finite morphism $c_{k}:Z_{k}\rightarrow Z$ which is ramified over $D$ such that $c_{k}^{\ast }D_{i}=m_{i}(c_{k}^{\ast }D_{i})_{\mathit{red}}$ for all $i$. Let $k_{1}$ be the smallest positive integer such that $k_{1}D_{1}$ is a Cartier divisor. By taking the $k_{1}$th root of the function defining the Cartier divisor $k_{1}D_{1}$, we can construct a finite morphism $c_{1,1}:Z_{1,1}\rightarrow Z$ which is étale in codimension $1$. Moreover, $c_{1,1}^{\ast }D_{1}$ is a reduced Cartier divisor (see [Reference MoriMor88, Proposition-Definition 1.11]). By taking the $m_{1}$th root of the function defining the Cartier divisor $c_{1,1}^{\ast }D_{1}$, we can find a finite morphism $c_{1,2}:Z_{1}\rightarrow Z_{1,1}$ which is ramified exactly over $D_{1}$ with ramification degree $m_{1}$ (see [Reference LazarsfeldLaz04, Section 4.1B]). Let $c_{1}=c_{1,2}\circ c_{1,1}:Z_{1}\rightarrow Z$. Then $c_{1}^{\ast }D_{1}=m_{1}(c_{1}^{\ast }D_{1})_{\mathit{red}}$. Assume that we have constructed a finite morphism $c_{j}:Z_{j}\rightarrow Z$ such that $c_{j}^{\ast }D_{i}=m_{i}(c_{j}^{\ast }D_{i})_{\mathit{red}}$ for any $1\leqslant i\leqslant j$ and $c_{j}^{\ast }D_{i}$ is reduced for any $i>j$, where $1\leqslant j<k$. Let $E=c_{j}^{\ast }D_{j+1}$. Then $E$ is a reduced divisor. By the same method as above, we can construct a finite morphism $d:Z_{j+1}\rightarrow Z_{j}$ which is ramified exactly over $E$ with ramification degree $m_{j+1}$. Let $c_{j+1}:Z_{j+1}\rightarrow Z$ be the composition $c_{j}\circ d$. Then $c_{j+1}^{\ast }D_{i}=m_{i}(c_{j+1}^{\ast }D_{i})_{\mathit{red}}$ for any $1\leqslant i\leqslant j+1$ and $c_{j+1}^{\ast }D_{i}$ is reduced for any $i>j+1$. By induction, we can construct the finite morphism $c_{k}$.

We have $K_{Z_{k}}=c_{k}^{\ast }(K_{Z}+\sum _{i=1}^{k}((m_{i}-1)/m_{i})D_{i})$. Let $X_{k}$ be the normalization of $X\times _{Z}Z_{k}$. Then the natural projection $c_{X}:X_{k}\rightarrow X$ is étale in codimension 1 and $K_{X_{k}}=c_{X}^{\ast }K_{X}$. Hence, $X_{k}$ is klt by [Reference Kollár and MoriKM98, Proposition 5.20]. Moreover, $X_{k}\rightarrow Z_{k}$ is a Fano fibration since $X\rightarrow Z$ is a Mori fibration. Hence, there is a $\mathbb{Q}$-divisor ${\rm\Delta}_{k}$ such that the pair $(Z_{k},{\rm\Delta}_{k})$ is klt by [Reference FujinoFuj99, Corollary 4.7]. Since $K_{Z_{k}}$ is $\mathbb{Q}$-Cartier, $Z_{k}$ is klt (see [Reference Kollár and MoriKM98, Corollary 2.35]). Therefore, the pair $(Z,\sum _{i=1}^{k}((m_{i}-1)/m_{i})D_{i})$ is also klt by [Reference Kollár and MoriKM98, Proposition 5.20].◻

Proof. We may assume that $E$ is smooth around $x$ and $D$ is smooth around $p(x)$. There exist local coordinates $(a_{1},a_{2},\ldots ,a_{n})$ and $(b_{1},b_{2},\ldots ,b_{d})$ of $X$ and $Z$ around $x$ and $p(x)$ such that $E$ is defined by $\{a_{1}=0\}$, $D$ is defined by $\{b_{1}=0\}$ and $p$ is given by $(a_{1},a_{2},\ldots ,a_{n})\mapsto (a_{1}^{k},a_{2},\ldots ,a_{d})$. With these coordinates, the natural morphism $p^{\ast }{\rm\Omega}_{Z}^{1}\rightarrow {\rm\Omega}_{X}^{1}$ is given by

$$\begin{eqnarray}(\text{d}b_{1},\text{d}b_{2},\ldots ,\text{d}b_{d})\mapsto (ka_{1}^{k-1}\text{d}a_{1},\text{d}a_{2},\ldots ,\text{d}a_{d})\end{eqnarray}$$

and the image of the natural morphism $p^{\ast }{\rm\Omega}_{Z}^{d}\rightarrow {\rm\Omega}_{X}^{d}$ is generated by $p^{\ast }(\text{d}b_{1}\wedge \text{d}b_{2}\wedge \cdots \wedge \text{d}b_{d})=(ka_{1}^{k-1}\text{d}a_{1})\wedge \text{d}a_{2}\wedge \cdots \wedge \text{d}a_{d}$. Hence, $(p^{\ast }{\rm\Omega}_{Z}^{d})^{sat}$ is generated by $\text{d}a_{1}\wedge \text{d}a_{2}\wedge \cdots \wedge \text{d}a_{d}$. Since $\{a_{1}=0\}$ defines the divisor $E$, we have $(p^{\ast }{\rm\Omega}_{Z}^{d})^{sat}|_{U}\cong \mathscr{O}_{X}(p^{\ast }K_{Z}+(k-1)E)|_{U}$ for some open neighborhood $U$ of $x$.◻

Proof. By Proposition 2.1, we only have to prove the assertion on an open subset of $X$ whose complement is of codimension at least $2$. Hence, we may assume that both $X$ and $Z$ are smooth and that $\sum _{i=1}^{r}D_{i}$ is smooth. In this case, by Lemma 4.3, the divisor $M$ is linearly equivalent to $p^{\ast }K_{Z}+\sum _{i=1}^{r}\sum _{j=1}^{s_{i}}(n_{i,j}-1)E_{i,j}$, where the $E_{i,1},\ldots ,E_{i,s_{i}}$ are the components of $p^{\ast }D_{i}$ and $n_{i,j}$ is the coefficient of $E_{i,j}$ in $p^{\ast }D_{i}$. Since $m(p,D_{i})$ is the smallest integer among $n_{i,1},\ldots ,n_{i,s_{i}}$, we have $n_{i,j}-1\geqslant ((m(p,D_{i})-1)/m(p,D_{i}))\cdot n_{i,j}$. Thus,

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{r}\mathop{\sum }_{j=1}^{s_{i}}(n_{i,j}-1)E_{i,j}\geqslant \mathop{\sum }_{i=1}^{r}\frac{m(p,D_{i})-1}{m(p,D_{i})}p^{\ast }D_{i}.\end{eqnarray}$$

Proof. Let $D$ be the sum of the divisors in $Z$ over which $p$ does not have reduced fibers. Applying Proposition 2.1 several times, we can suppose without loss of generality that $Z$ is smooth and $D$ is a simple normal crossing divisor. Let $\mathscr{M}=p_{\ast }(((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})^{[\wedge r]})$.

First, we prove that there is a natural injection from ${\rm\Omega}_{Z}^{r}$ to $\mathscr{M}$. We have an injection from $p^{\ast }{\rm\Omega}_{Z}^{1}$ to $(p^{\ast }{\rm\Omega}_{Z}^{1})^{sat}$. Hence, the natural morphism from $(p^{\ast }{\rm\Omega}_{Z}^{1})^{\wedge r}$ to $((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})^{[\wedge r]}$ is generically injective. Since $(p^{\ast }{\rm\Omega}_{Z}^{1})^{\wedge r}$ is without torsion, the natural morphism from $(p^{\ast }{\rm\Omega}_{Z}^{1})^{\wedge r}$ to $((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})^{[\wedge r]}$ is injective. Since $p(X)=Z$, we have an injection from $p_{\ast }((p^{\ast }{\rm\Omega}_{Z}^{1})^{\wedge r})$ to $\mathscr{M}$. By the projection formula, this implies that ${\rm\Omega}_{Z}^{r}$ is a subsheaf of $\mathscr{M}$.

Now we prove that ${\rm\Omega}_{Z}^{r}\cong \mathscr{M}$. Let $W=p^{-1}(Z\setminus D)$. Then the morphism $p|_{W}:W\rightarrow Z\setminus D$ is smooth in codimension $1$. Thus,

$$\begin{eqnarray}((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})^{[\wedge r]}|_{W}\cong p^{\ast }{\rm\Omega}_{Z}^{r}|_{W}\end{eqnarray}$$

(see the proof of Lemma 4.3). Then we obtain $\mathscr{M}|_{Z\setminus D}\cong {\rm\Omega}_{Z\setminus D}^{r}$ by the projection formula. Let $U$ be any open set in $Z$, and let ${\it\beta}$ be any element of $\mathscr{M}(U)$, that is, a section of $\mathscr{M}|_{U}$. Then ${\it\beta}$ is a rational section of ${\rm\Omega}_{Z}^{r}|_{U}$ which can only have a pole along $D$ since $\mathscr{M}|_{Z\setminus D}\cong {\rm\Omega}_{Z\setminus D}^{r}$. However, by the definition of $\mathscr{M}$, ${\it\beta}$ induces a regular section of ${\rm\Omega}_{X}^{r}$ on the smooth locus of $p^{-1}(U)$. This implies that ${\it\beta}$ does not have pole along $D$. Thus, $\mathscr{M}(U)={\rm\Omega}_{Z}^{r}(U)$. Hence, ${\rm\Omega}_{Z}^{r}\cong \mathscr{M}$.◻

Proof. If $D$ is a prime divisor in $Z$, then $p_{1}^{\ast }c_{Z}^{\ast }D=c_{X}^{\ast }p^{\ast }D$. Let $E$ be any irreducible component of $c_{Z}^{\ast }D$. Since $c_{X}$ is étale in codimension $1$, by comparing the coefficients of $E$ in $p_{1}^{\ast }c_{Z}^{\ast }D=c_{X}^{\ast }p^{\ast }D$, the coefficient $m_{E}$ of $E$ in $c_{Z}^{\ast }D$ is not larger than $m(p,D)$. This implies that $(m_{E}-1)\leqslant m_{E}\cdot ((m(p,D)-1)/m(p,D))$. Now, from Lemma 4.3, we obtain $K_{Z_{1}}=c_{Z}^{\ast }K_{Z}+{\rm\Sigma}_{j=1}^{s}(m_{j}-1)E_{j}$, where the $E_{j}$ are all prime divisors along which $c_{Z}^{\ast }$ is ramified, and $m_{j}$ is the degree of ramification of $c_{Z}$ along $E_{j}$. By the discussion above, we have $(m_{j}-1)\leqslant m_{j}\cdot ((m(p,p(E_{j}))-1)/m(p,p(E_{j})))$. Hence, $K_{Z_{1}}\leqslant c_{Z}^{\ast }(K_{Z}+\sum _{i=1}^{r}((m(p,D_{i})-1)/m(p,D_{i}))D_{i})$.◻

Proof. Assume the opposite. First, we assume further that $m_{0}=1$ and $\mathscr{O}_{Z}(D_{0})$ has non-zero global sections. Let $E=p^{\ast }D_{0}$, which is effective. Then $\mathscr{F}=\mathscr{O}_{X}(E)$. Since general fibers of $p$ are isomorphic to $\mathbb{P}^{1}$, which does not carry any non-zero pluri-forms, by Lemma 2.5, the injection $\mathscr{O}_{X}(E){\hookrightarrow}{\rm\Omega}_{X}^{[1]}$ factorizes through $(p^{\ast }{\rm\Omega}_{Z})^{sat}$. Hence, by the projection formula, we have an injection from $\mathscr{O}_{Z}(D_{0})$ to $p_{\ast }((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})$. However, by Lemma 4.6, we have $p_{\ast }((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})={\rm\Omega}_{Z}^{1}$. By Proposition 2.1, we get an injection from $\mathscr{O}_{V}(\bar{D}_{0})$ to ${\rm\Omega}_{V}^{[1]}$. Hence, by the Bogomolov–Sommese theorem (see [Reference GrafGra15, Corollary 1.3]), the Kodaira dimension of $\mathscr{O}_{V}(\bar{D}_{0})$ is not larger than $1$. Hence, $\bar{D}_{0}$ is not ample. We obtain a contradiction.

Now we treat the general case. We show that we can reduce to the previous case. By replacing $m_{0}$ with a large multiple, we may assume that $\bar{D}_{0}$ is very ample. We may also assume that both $\bar{D}_{0}$ and $p^{\ast }D_{0}$ are prime and that the pair $(V,\sum _{i=0}^{r}((m_{i}-1)/m_{i})\bar{D}_{i})$ is klt. Let $c_{X}:X_{1}\rightarrow X$ be the normalization of the ramified cyclic cover with respect to $\mathscr{F}$, $m_{0}$ and $p^{\ast }D_{0}$ (see [Reference Kollár and MoriKM98, Definition 2.52]). Then $c_{X}$ is ramified over $p^{\ast }D_{0}$ with degree $m$ and $(c_{X}^{\ast }\mathscr{F})^{\ast \ast }\cong \mathscr{O}_{X_{1}}(E)$, where $E=(c_{X}^{\ast }p^{\ast }D_{0})_{\mathit{red}}$. Moreover, over the smooth locus of $X$, there is an injection of sheaves from $c_{X}^{\ast }{\rm\Omega}_{X}^{1}$ to ${\rm\Omega}_{X_{1}}^{1}$. Hence, by Proposition 2.1, we have an injection $\mathscr{O}_{X_{1}}(E){\hookrightarrow}{\rm\Omega}_{X_{1}}^{[1]}$.

Let $X_{1}\overset{p_{1}}{\longrightarrow }Z_{1}\overset{c_{Z}}{\longrightarrow }Z$ be the Stein factorization. Let $c_{V}:V_{1}\rightarrow V$ be the normalization of $V$ in the function field of $X_{1}$. Then we obtain an open embedding $j_{1}:Z_{1}\rightarrow V_{1}$ such that $\text{codim}\,V_{1}\setminus Z_{1}\geqslant 2$. If $F_{p}$ is a general fiber of $p$ and if $F_{p_{1}}$ is a general fiber of $p_{1}$ which is mapped to $F_{p}$, then $F_{p_{1}}\rightarrow F_{p}$ is étale. Since $p$ is a Mori fibration, $F_{p}$ is a smooth rational curve which is simply connected. Hence, $F_{p_{1}}\rightarrow F_{p}$ is an isomorphism and $F_{p_{1}}\cong \mathbb{P}^{1}$. Hence, $c_{Z}$ is of degree $m_{0}$. Let $H=(c_{Z}^{\ast }D_{0})_{\mathit{red}}$. Since $c_{Z}\circ p_{1}:X_{1}\rightarrow Z$ has connected fibers over $D_{0}$, we have $c_{Z}^{\ast }D_{0}=m_{0}H$. Hence, $E_{1}=p_{1}^{\ast }H$. Let $\bar{H}$ be the closure of $H$ in $V_{1}$. Then it is ample since $c_{V}$ is finite.

Note that $c_{X}|_{X_{1}\setminus E}$ is étale in codimension $1$ and $c_{Z}^{\ast }D_{0}=m_{0}H$. By Lemma 4.7, we conclude that there is an effective $\mathbb{Q}$-divisor ${\rm\Delta}_{1}^{\prime }$ in $Z_{1}$ such that $K_{Z_{1}}+{\rm\Delta}_{1}^{\prime }=p^{\ast }(K_{Z}+\sum _{i=0}^{r}((m_{i}-1)/m_{i})D_{i})$. Hence, if ${\rm\Delta}_{1}$ is the closure of ${\rm\Delta}_{1}^{\prime }$ in $V_{1}$, then $K_{V_{1}}+{\rm\Delta}_{1}=p^{\ast }(K_{V}+\sum _{i=0}^{r}((m_{i}-1)/m_{i})\bar{D}_{i})$. This implies that $(V_{1},{\rm\Delta}_{1})$ is klt by [Reference Kollár and MoriKM98, Proposition 5.20 and Corollary 2.35].

Hence, $p_{1}:X_{1}\rightarrow Z_{1}$ satisfies the conditions in the lemma. There is an injection from $\mathscr{O}_{X_{1}}(E)$ to ${\rm\Omega}_{X_{1}}^{[1]}$, and $\mathscr{O}_{X_{1}}(E)\cong \mathscr{O}_{X_{1}}(p_{1}^{\ast }H)$, such that $\bar{H}$ is ample in $V_{1}$. We are in the same situation as in the first case. This leads to a contradiction.◻

Proof. Assume the opposite. Let $\mathscr{F}=(p^{\ast }{\rm\Omega}_{Z}^{1})^{sat}$ be the saturation of the image of $p^{\ast }{\rm\Omega}_{Z}^{1}$ in ${\rm\Omega}_{X}^{[1]}$. Since general fibers of $p$ do not carry any non-zero pluri-forms, we have $H^{0}(X,{\rm\Omega}_{X}^{[r]})\cong H^{0}(X,\mathscr{F}^{[\wedge r]})$ by Lemma 2.5. Hence, there is an injection from $\mathscr{O}_{X}$ to $\mathscr{F}^{[\wedge r]}$. By taking the direct image, we have an injection from $\mathscr{O}_{Z}$ to $p_{\ast }(\mathscr{F}^{[\wedge r]})$. By Lemma 4.6, $p_{\ast }(\mathscr{F}^{[\wedge r]})\cong {\rm\Omega}_{Z}^{r}$. This implies that $H^{0}(V,{\rm\Omega}_{V}^{[r]})\neq \{0\}$ by Proposition 2.1, which is a contradiction.◻

Proof of Theorem 4.2.

We argue by contradiction. Let $f:Z\rightarrow Z^{\prime }$ be the result of a $(K_{Z}+{\rm\Delta})$-MMP. Assume that $Z^{\prime }$ has Picard number $1$. Set ${\rm\Delta}^{\prime }=f_{\ast }{\rm\Delta}$. Then $K_{Z^{\prime }}+{\rm\Delta}^{\prime }$ is not pseudo-effective either. Thus,$-(K_{Z^{\prime }}+{\rm\Delta}^{\prime })$ is ample. We know that there is a smooth open subset $Z_{0}^{\prime }\subseteq Z^{\prime }$ with $\text{codim}\,Z^{\prime }\setminus Z_{0}^{\prime }\geqslant 2$ such that $f^{-1}$ is an isomorphism from $Z_{0}^{\prime }$ onto its image. Let $Z_{0}$ be $f^{-1}(Z_{0}^{\prime })$, which is an open subset in $Z$. Let $X_{0}=p^{-1}(Z_{0})$.

Since $\text{codim}\,Z^{\prime }\setminus Z_{0}^{\prime }=2$, there is a projective curve $C_{0}^{\prime }$ in $Z_{0}^{\prime }$ such that it is an ample divisor in $Z^{\prime }$. Let $C_{0}$ be the strict transform of $C_{0}^{\prime }$ in $Z_{0}$. Let ${\it\alpha}$ be the class of the curve $p^{\ast }C_{0}\cap H$, where $H$ is a very ample divisor in $X$. Then the class ${\it\alpha}$ is movable and $p_{\ast }{\it\alpha}$ is proportional to the class of $C_{0}$.

Since general fibers of $p$ do not carry any non-zero pluri-forms, by Lemma 2.5, we obtain that

$$\begin{eqnarray}H^{0}(X,({\rm\Omega}_{X}^{1})^{[\otimes m]})\cong H^{0}(X,((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})^{[\otimes m]})\end{eqnarray}$$

for any $m>0$, where $(p^{\ast }{\rm\Omega}_{Z}^{1})^{sat}$ is the saturation of the image of $(p^{\ast }{\rm\Omega}_{Z}^{1})$ in ${\rm\Omega}_{X}^{[1]}$. Hence, ${\it\mu}_{{\it\alpha}}^{max}((p^{\ast }{\rm\Omega}_{Z}^{1})^{sat})\geqslant 0$ and there is a coherent sheaf $\mathscr{H}$ saturated in $(p^{\ast }{\rm\Omega}_{Z}^{1})^{sat}$ such that ${\it\mu}_{{\it\alpha}}(\mathscr{H})\geqslant 0$.