Superadditivity of anticanonical Iitaka dimension]Superadditivity of anticanonical Iitaka dimension for contractions with $F$-split fibres

Marta Benozzo
Imperial College London, London, SW7 2AZ, UK
University College London, London, WC1E 6BT, UK

Iacopo Brivio

National Center for Theoretical Sciences, Taipei, 106, Taiwan

Chi-Kang Chang
Astronomy Mathematics Building 5F, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan

In this paper, we study a version of the Iitaka conjectures for anticanonical divisors over perfect fields of positive characteristic. That is, we prove the inequality $\kappa(X,-K_X)\leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$, for a contraction $f\colon X\rightarrow Y$ with general fibre $X_y$ having good arithmetic properties.

§ INTRODUCTION

A projective algebraic variety $X$ is classified according to the positivity properties of its canonical divisor $K_X$. The most basic measure of such positivity is its Iitaka dimension $\kappa(X,K_X)$, an invariant which measures the rate of growth of the spaces $H^0(X,mK_X)$ as a function of $m$. For varieties over the complex numbers, Iitaka proposed the following Conjecture

Let $f\colon X\to Y$ be a contraction of smooth projective complex varieties, of dimensions $n$ and $m$ respectively, and let $y\in Y$ be a general point. Then

\[
\kappa(X,K_X)\geq \kappa(X_y,K_{X_y})+\kappa(Y,K_Y).
\]

Although still open in general, over fields of characteristic $0$ this conjecture is proven for many important classes of contractions (<cit.>).
In particular, $C_{n,m}$ holds when $\dim(Y) \leq 2$, and when $X_y$ admits a good minimal model.

More recently it was shown in <cit.> that a similar inequality holds for the anticanonical divisors.

Let $f\colon X\to Y$ be a contraction of smooth complex projective varieties, of dimensions $n$ and $m$ respectively, such that the stable base locus $\bB(-K_X)$ does not dominate $Y$, and let $y\in Y$ be a general point. Then

\[
\kappa(X,-K_X)\leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y).
\]

The condition on the stable base locus is necessary, as shown by <cit.>.

Broadly speaking, for both <ref> and <ref> the main tools are semipositivity results for the sheaves $f_*\omega_{X/Y}^m$. In particular, <cit.> makes use of the canonical bundle formula for klt-trivial fibrations (<cit.>), the proof of which relies on Hodge-theoretic methods. Given that these techinques are not available in characteristic $p>0$ (<cit.>), it is natural to ask whether the above statements still hold in this case. As it turns out, both $C_{n,m}$ and $C_{n,m}^-$ can fail in general (<cit.>).
However, <ref> has been proven for generically smooth contractions of relative dimension one, or when $\dim(X)\leq 3$ and $p>5$ (<cit.>, <cit.>, <cit.>, <cit.> and <cit.>).
Due to the lack of generic smoothness results, over fields of positive characteristic, the general fibre of a contraction may be singular and even non-reduced.
In this case, Patakfalvi showed <ref> when $Y$ is of general type and $X_y$ has non-nilpotent Hasse-Witt matrix (<cit.>). This suggests that we may expect $C_{n,m}$ to hold for contractions whose fibres have “arithmetically nice” singularities. Similarly, it was proven in <cit.> that, when we have enough control on the singularities of the general fibre, $C_{n,1}^-$ and $C_{3,m}^-$ hold (the latter provided that $p\geq 5$), as well as $C_{n,n-1}^-$ if $\kappa(Y,-K_Y)=0$.

In this paper we extend <ref> to certain contractions in positive characteristic in any dimension.

Let $f \colon X \to Y$ be a contraction of smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$, such that a general fibre $X_{y}$ is $K$-globally $F$-regular[See <ref>]. Assume there is $m\in\bN\setminus p\bN$ such that $-mK_X$ is Cartier and $|V|\subseteq |-mK_X|$ such that $|V|_{X_y}$ induces a morphism with Stein degree not divisible by $p$. Then

\[
\kappa(X,-K_X)\leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y).
\]

§.§ Proof outline

The key statement is <cit.>, a result stating that negativity of $-K_X$ descends along contractions in a controlled way. More precisely: if $|-K_X-f^*E|_{\bQ}\neq \emptyset$ then $|-K_Y-\epsilon E|_{\bQ}\neq\emptyset$ for small $\epsilon>0$.
This can be used to prove an injectivity theorem (<cit.>) which implies <ref> when $\kappa(Y,-K_Y)=0$.
The proof of <cit.> is a consequence of the following facts.

(1) As $\bB(-K_X)$ does not dominate $Y$ we can take $\Delta\in |-K_X|_{\bQ}$ such that $(X_y,\Delta_{X_y})$ has “nice” (i.e. klt) singularities;

(2) $(X,\Delta)\xrightarrow{f}Y$ is now a log Calabi-Yau contraction, hence we have a canonical bundle formula $K_X+\Delta\sim_{\bQ}f^*(K_Y+B_Y+M_Y)$.

(3) If $\Gamma\in |-K_X-f^*E|_{\bQ}$, then a small perturbation $(X,\epsilon\Gamma)$ will still have “nice” singularities over the generic point of $Y$ and $\bB(-K_X-\epsilon\Gamma)$ does not dominate $Y$, hence we can apply (1) and (2) to the pair $(X,\epsilon\Gamma)$.

There are several obstacles to generalising this approach to characteristic $p>0$.
First of all, the canonical bundle formula is known to be false in general (<cit.>).
On the positive side, a weak form of it is known to hold for contractions whose generic fibre is globally $F$-split (<cit.>), so we chose to restrict ourselves to this setup.
Even so, there are still issues with the first and last point. If $X_y$ is globally $F$-split, it is known that we can find a $\Delta_{X_y}\in |-K_{X_y}|_{\bQ}$ such that $(X_y,\Delta_{X_y})$ is a globally $F$-split Calabi-Yau variety.
However, one must show that such divisor lifts to an element of $|-K_X|_{\bQ}$, and we are able to show that this is the case when, for some $m\geq 1$ not divisible by $p$, the rational map $\phi_{|-mK_X|}$ restricts to a morphism on $X_y$ with Stein degree not divisible by $p$.
Lastly, globally $F$-split singularities behave poorly under perturbations, so this makes point (3) problematic. Our solution is to introduce $K$-globally $F$-regular varieties, a notion that interpolates between globally $F$-split and globally $F$-regular varieties.
Roughly speaking, $X_y$ is $K$-globally $F$-regular if it is globally $F$-split and the Iitaka fibration of $-K_{X_y}$ maps $X_y$ to a variety which is globally $F$-regular. The advantage is that this class of varieties is stable under small perturbations by members of the anticanonical $\bQ$-linear system.
Under these assumptions we are able to prove the injectivity result <ref>.

In <cit.>, the author concludes by reducing to the $\kappa(Y,-K_Y)=0$ case considering $g\colon Y\to Z$, the Iitaka fibration of $-K_Y$.
Over fields of positive characteristic, this contraction may have highly singular fibers. However the singularities of the general fiber manifest themselves on the total space after a base-change by a sufficiently high power of the Frobenius. This allows us to circumvent the issue by considering

X_e[r][d,"f_e"][dd,"h_e",swap,bend right=30]     X[d,"f",swap][dd,"h",bend left=30]

Y_e[r][d,"g_e"]     Y[d,"g",swap]

Z^e[r,"^e"]     Z,

where $X_e$ (resp. $Y_e$) is the normalization of the reduction of $Y\times_Z Z^e$ (resp. of $X\times_Z Z^e$). The resulting contractions $f_e,g_e$ and $h_e$ will be (universally) homeomorphic to the original ones, but their fibers will now be normal, thus the arguments of <cit.> apply in this case. We can then conclude by comparing the canonical divisors of $X,Y$ and $Z$ with those of $X_e,Y_e$ and $Z_e$ using the correspondence between purely inseparable morphisms and foliations (<cit.>).

§.§ Acknowledgements

We would like to thank our advisors Paolo Cascini and Jungkai Chen for their guidance and helpful suggestions throughout the project.
We thank Karl Schwede, Yoshinori Gongyo, and Hiromu Tanaka for answering our questions and pointing out useful papers.
The first author was supported by the Engineering and Physical Sciences Research Council [EP/S021590/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. The second author is supported by the National Center for Theoretical Sciences and a grant from the Ministry of Science and Technology, grant number MOST-110-2123-M-002-005. The third author is supported by the PhD student's scholarship of National Taiwan University, and by the scholarship of National Science and Technology Council for PhD students to study abroad, which he used to visit the University of Tokyo.
We would also like to thank the National Centre for Theoretical Sciences Mathematics division for their support that allowed the first two authors to meet in person in Taipei.

§ CONVENTIONS AND NOTATION

* All of our schemes $X$ will be irreducible, separated, essentially of finite type over a field $k$ of characteristic $p>0$, and $F$-finite. Note that for any such scheme the dualising complex is well-defined (<cit.>).

* A k-variety $X$ is a separated integral $k$-scheme of $k$-finite type. We denote by $\sing(X)$ the locus of singular points of $X$.

* When $X$ is an integral $k$-scheme we denote by $k(X)$ its field of rational functions.

* A $\bK$-divisor $D$ on a scheme $X$ is a formal finite linear combination $D=\sum_ia_iD_i$, where $D_i$ are irreducible closed codimension-one subsets of $X$ and $a_i\in\bK$. We will take $\bK\in \lbrace \bZ,\bZ_{(p)},\bQ\rbrace$. If $\bK=\bZ$ we refer to $D$ as an integral divisor or simply a divisor. We define the positive part (resp. negative part) of $D$ to be $D^+\coloneqq\sum_{a_i>0}a_iD_i$ (resp. $D^-\coloneqq\sum_{a_i<0}(-a_i)D_i$).

* Given a divisor $D$ on a normal variety $X$, we denote by $|D|$ the complete linear system it defines. If $V\subseteq H^0(X,D)$ is a subspace, we denote by $|V|\subseteq |D|$ the corresponding linear subsystem. If $Y\subseteq X$ is a closed subvariety, we denote by $|V|_{Y}$ the linear subsystem given by the restriction of $V$ to $Y$. The corresponding rational maps will be denoted by $\phi_{|D|},\phi_{|V|}$ and $\phi_{|V|_Y}$ respectively.

* A $\bQ$-divisor $D$ is $\bQ$-Cartier if $mD$ is Cartier for some $m$. If there exists such $m$ with $p \nmid m$, then we say $D$ is a $\bZ_{(p)}$-Cartier $\bZ_{(p)}$-divisor.

* If $D_1,D_2$ are $\bQ$-divisors on a scheme $X$ such that $mD_i$ is integral for $i=1,2$ and $mD_1\sim mD_2$ for some positive integer $m$, then we say $D_1$ and $D_2$ are $\bQ$-linearly equivalent $\bQ$-divisors, denoted by $D_1\sim_{\bQ}D_2$. If $p\nmid m$ then we say $D_1$ and $D_2$ are $\bZ_{(p)}$-linearly equivalent $\bZ_{(p)}$-divisors, denoted $D_1\sim_{\bZ_{(p)}}D_2$.

* Let $f\colon X\to Y$ be a morphism of schemes and let $D$ be a divisor on $X$: we write $D\sim_{Y}0$ if $D\sim f^*M$ where $M$ is a Cartier divisor on $Y$. If $D$ is a $\bQ$-divisor (resp. a $\bZ_{(p)}$-divisor) we write $D\sim_{\bQ,Y}0$ (resp. $D\sim_{\bZ_{(p)},Y}0$) if for some $m\geq 1$ (resp. for some $m\geq 1$ such that $p\nmid m$) we have that $mD$ is integral and $mD\sim_Y0$. In particular, we have that $D$ is Cartier (resp. $\bQ$ or $\bZ_{(p)}$-Cartier).

* Let $D$ be a $\bQ$-divisor on a scheme $X$: we say $D$ is effective ($D\geq 0$) if all of its coefficients are non-negative. We say $D$ is $\bQ$-effective (resp. $\bZ_{(p)}$-effective) if, for some $m\geq 1$ (resp. for some $m\geq 1$ such that $p\nmid m$) $mD$ is integral and $H^0(X,mD)\neq 0$. Given $\bQ$-divisors $D_1,D_2$ we write $D_1\geq D_2$ if $D_1-D_2$ is effective, and we write $D_1\geq_{\bQ}D_2$ (resp. $D_1\geq_{\bZ_{(p)}}D_2$) if $D_1-D_2$ is $\bQ$-effective (resp. $\bZ_{(p)}$-effective).

* A sub-couple $(X,B)$ consists of an integral normal scheme $X$ and a $\bQ$-divisor $B$. If $B\geq 0$ we say $(X,B)$ is a couple. A sub-pair is a sub-couple $(X,B)$ such that $K_X+B$ is $\bQ$-Cartier. If $B\geq 0$ we say $(X,B)$ is a pair.

* Let $f\colon X\to Y$ be a morphism of $k$-schemes, where $k$ is algebraically closed. A general fibre of f is $X_y\coloneqq f^{-1}(y)$ where $y$ is a $k$-point belonging to a dense open subset of $Y$. We say $X_y$ is a very general fibre if $y$ is a $k$-point belonging to a countable intersection of dense open subsets of $Y$.

* A contraction is a projective morphism of schemes $f\colon X\to Y$ such that $f_*\cO_X=\cO_Y$.

* Let $f\colon X \rightarrow Y$ be a contraction with general fibres that are normal, $D$ a divisor on $X$ and $X_y$ a general fibre.
Then $D$ is $\bQ$-Cartier along any codimension $1$ point of $X_y$, hence we can define its restriction to $X_y$, $D_{X_y}:=D|_{X_y}$.

* Given a contraction $f \colon X \rightarrow Y$ and $D$ a divisor on $X$, if $\eta$ is the generic point of $Y$, we denote by $D_{\eta} \coloneqq D|_{X_{\eta}}$. As above, this restriction is well-defined.
When the geometric generic fibre is normal, we use an analogous notation $D_{\overline{\eta}} \coloneqq D|_{X_{\overline{\eta}}}$.

* Let $f\colon X\to Y$ be a surjective projective morphism of schemes, and let $f\colon X\xrightarrow{g} Z\xrightarrow{h} Y$ be its Stein factorisation. The degree of $h$ is called the Stein degree of $f$ and it is denoted by $\stdeg(f)$. Note that, when $Y$ is normal and $p\nmid\stdeg(f)$, then $f$ is a split morphism, that is the natural map $\cO_Y\to f_*\cO_X$ is a split inclusion, via the trace map $\Tr_{X/Y}\colon f_*\cO_X\to\cO_Y$.

* Let $(X,B)$ be a sub-couple over $\bC$. A model of $(X,B)$ is a normal, integral, separated, projective $A$-scheme of finite type $\cX\to\Spec(A)$, where $A$ is a finitely generated $\bZ$-algebra, together with a $\bQ$-divisor $\cB$ such that $(X,B)=(\cX,\cB)\times_{\Spec(A)}\Spec(\bC)$.

We refer the reader to <cit.> for the definitions of the various classes of singularities appearing in the Minimal Model Program.

In the remainder, we will mostly deal with normal varieties, thus we will tacitly use their $S2$ property.
More precisely, when we use reflexive sheaves, we work locally over the regular locus, then extend the result to all of $X$.
This is needed, for example, when applying Grothendieck duality to the Frobenius morphism.

In the sequel, sometimes we will pull-back divisors under equidimensional morphisms $f\colon X \to Y$ of normal varieties, without requiring any Cartier assumption.
This operation is well-defined since, given a prime divisor $D$ on $Y$, its preimage under $f$ is a divisor on $X$.

§ PRELIMINARIES

§.§ Iitaka dimension

In this section, after recalling its definition, we collect some results on the behaviour of the Iitaka dimension.
We refer to <cit.> for more details.

Let $X$ be a normal projective variety and $L$ a $\bQ$-divisor on it.
For every $m>0$ such that $mL$ is integral and the linear system $|mL|$ is not empty, $|mL|$ defines a rational map $\phi_{|mL|} \colon X \dashrightarrow \bP^{N_m}$.
For $m \gg 0$ and sufficiently divisible, $\dim(\phi_{|mL|}(X))$ stabilises.
The Iitaka dimension of $L$ is defined as:

\[ \kappa(X, L) \coloneqq
\begin{cases}
-\infty \quad \text{if} \; |mL|= \emptyset \; \text{for all} \; m \geq 0;\\
\max_{m\geq 1}\dim(\phi_{|mL|}(X)) \quad \text{otherwise}.
\end{cases}
\]

Sometimes it is useful to work with different characterisations of the Iitaka dimension. We recall one here.
Let $X$ be a normal projective variety over a field $k$ and $L$ a Cartier divisor on it.
Define the section ring of $L$ as $R(X,L) \coloneqq \bigoplus_{m=0}^{\infty} H^0(X, mL)$.
If $R(X,L) \neq 0$, it is an integral domain. Denote by $Q(X, L)$ its fraction field.
If $\kappa(X,L) \geq 0$, then $\kappa(X,L)= \mathrm{tr.deg}_k Q(X,L)$.
For more details, we refer to <cit.>.

Let $\varphi \colon X' \rightarrow X$ be a surjective morphism between normal projective varieties and $L$ a Cartier divisor on $X$. Then $\kappa(X', \varphi^*L) = \kappa(X, L)$.

Note that, as $\varphi$ is surjective, $\kappa(X,L) \leq \kappa(X', \varphi^*L)$. If $\varphi$ is a contraction, the result follows from the projection formula.
By considering the Stein factorisation of $\varphi$, we can thus reduce to the case where $\varphi$ is finite.

Now suppose $\varphi=\Fr^e$ for some $e >0$, then the statement is trivial since $\Fr^{e*}L=p^eL$.
If $\varphi$ is purely inseparable, there exists $\psi \colon X \rightarrow X'$ such that $\varphi \circ \psi = \Fr^e$, for some $e\geq 0$.

\[
\kappa(X,L) \leq \kappa(X', \varphi^*L) \leq \kappa(X, \psi^*\phi^*L) = \kappa(X,L).
\]

If, instead, $\varphi$ is a Galois cover, the result is proven in <cit.>. Note that they prove it over fields of characteristic $0$, but, assuming $\varphi$ is Galois, the same proof works also over fields of positive characteristic.
If $\varphi$ is separable, there exists $\psi\colon X'' \rightarrow X'$ such that $\varphi \circ \psi$ is Galois.

\[
\kappa(X,L) \leq \kappa(X', \varphi^*L) \leq \kappa(X'', \psi^*\phi^*L) = \kappa(X,L).
\]

In general, we can factor $\varphi$ in its separable and purely inseparable part and conclude by the above discussion.

Let $f \colon X \rightarrow Y$ be a projective morphism between varieties.
Assume that a very general fibre $X_y$ is reduced and normal.
Let $\eta$ be the generic point of $Y$ and $\overline{\eta}$ its geometric generic point.
Let $L$ be a Cartier divisor on $X$.

\[
\kappa(X_{\overline{\eta}}, L_{\overline{\eta}}) =\kappa(X_\eta, L_\eta) = \kappa({X_y}, L_{X_y}).
\]

Moreover, if $\kappa(X_{\eta}, L_{\eta}) \geq 0$, the above equalities hold also for the general fibre $X_y$.

The first equality is a consequence of the flat base change theorem.
As for the second, note that we can assume $f$ is flat without loss of generality, hence we conclude by <cit.>.

Now, suppose $\kappa(X_{\eta}, L_{\eta}) \geq 0$.
Let $H$ be an ample enough Cartier divisor on $Y$ such that $L+f^*H$ is $\bQ$-effective.
By the easy additivity Theorem, <cit.> and <cit.>, for a general fibre $X_y$ we have:

\[
\kappa(X, L+ f^*H) = \kappa(X_y, L_{X_y}) + \dim(Y) \quad \text{and} \quad \kappa(X, L+f^*H) = \kappa(X_\eta, L_{\eta}) + \dim(Y).
\]

Thus, $\kappa(X_y, L_{X_y})=\kappa(X_\eta, L_{\eta})$.

In the above <ref>, if $\kappa(X_\eta, L_\eta) = -\infty$, it may be false that the general fibre $X_y$ satisfies $\kappa({X_y}, L_{X_y})= - \infty$.
Let $\hat{Y}$ be an abelian variety and $Y$ its dual. The Poincaré bundle $L$ on $X\coloneqq \hat{Y} \times Y\to Y$ gives a counterexample.

§.§ Frobenius

In this section, we define the different Frobenius morphisms we will use and outline their relations.

Let $X$ be an $F$-finite scheme over a field of characteristic $p>0$.
The Frobenius morphism on $X$, $\Fr^e_X \colon X^e \rightarrow X$, for $e \in \bN$, is defined to be the identity on points and the $(p^{e})^{\text{th}}$-power on the structure sheaf. When the underlying scheme is clear from the context we will just write $\Fr^e$. Note that $X^e$ and $X$ are the same scheme, the index is just used to differentiate between the target and the source of $\Fr^e$.

Given a morphism of $k$-schemes $\pi\colon X\to V$, we have the following commutative diagram:

\begin{equation}\label{e-relfrob}
\begin{tikzcd}
X^e \arrow[ddrr, "\pi^{e}", bend right=20, swap] \arrow[drr, "\Fr^{e}_{X^e/V^e}"] \arrow[drrrr, bend left=20, "\Fr^{e}_{X}"] & &  \\
& &X_{V^e} \arrow[d, "\pi_{V^e}", swap] \arrow[rr, "(\Fr^{e}_{V})_{X}",swap] & & X \arrow[d, "\pi"]\\
& & V^e \arrow[rr, "\Fr^{e}_{V}"] & & V
\end{tikzcd}
\end{equation}

where $X_{V^e}$ is the fibre product $X \times_{V^e} V$ and $(\Fr^e_V)_X$ is the induced map.
The morphism $\Fr^e_{X^e/V^e}$ denotes the $e^{th}$ relative Frobenius of X over V. When $V=\Spec(k)$ the relative Frobenius is also called the $k$-linear Frobenius.

Note that $\Fr^e \colon X^e \rightarrow X$ is not $k$-linear.
However, if $k$ is perfect, it differs from the $k$-linear Frobenius only by an automorphism of $\Spec(k)$.
On the other hand, if $\Spec(k^e)\to V^e$ is a $k^e$-point, then the base-change

\[
\Fr^e_{X^e/V^e}\otimes_{V^e}k^e\colon X_{k^e}^e\to X_{k^e}
\]

coincides with the $k$-linear Frobenius of $X_k \coloneqq X \times_V \Spec(k)$.

§.§ Frobenius base change

In positive characteristic, it is hard to control the singularities of the general fibre of a contraction due to the failure of generic smoothness theorems.
However, after a base change with a high power of the Frobenius morphism, all the singularities of the general fibre appear on the total space.

Let $f: X \rightarrow Y$ be a contraction between normal projective varieties over a perfect field of characteristic $p>0$ and let $\overline{\eta}$ be the geometric generic point of $Y$.
Consider the base change with a power of the Frobenius morphism:

X_Y^e [r] [swap,d,"f_e"]     X [d,"f"]

Y^e [r,"^e"]     Y.

Then, for $e \gg 0$, $((X_{Y^e,\red})^{\nu})_{\overline{\eta}^e}=(X_{\overline{\eta},\red})^\nu$.

In positive characteristic, there is a correspondence between height one purely inseparable morphisms and foliations.
Thanks to this, we are able to study the behaviour of the canonical divisors under purely inseparable base changes.

A purely inseparable morphism of schemes $a \colon X' \rightarrow X$ is called of height one if there exists $\alpha \colon X \rightarrow X'$ such that $a\circ \alpha = \Fr$.

Let $X$ be a normal variety over a perfect field of characteristic $p>0$.
A foliation on $X$ is a subsheaf of the tangent sheaf, $\mathcal{F} \subseteq T_X$, which is saturated and closed under $p$-powers.

One normally requires that $\cF$ is involutive, that is closed under Lie brackets. However, this follows from closure under $p$-powers by <cit.>.

Let $X'$ be a normal variety over a perfect field of characteristic $p>0$.
There is a $1$-to-$1$ correspondence

{ Height one morphisms
$X'\to X$ with $X$ normal } [rr]         { Foliations $\cF\subseteq T_{X'}$[ll]

given by:

($\leftarrow$) $X:= \Spec_{X'} \cO_{X'}^{\cF}$, where $\cO_{X'}^{\cF}\subseteq \cO_{X'}$ is the subsheaf of $\cO_{X'}$ that is taken to zero by all the sections of $\cF$;

($\rightarrow$) $\cF\coloneqq\lbrace \partial\in T_{X'} \textup{ s.t. } \partial\cO_{X}=0\rbrace$.

Moreover, morphisms of degree $p^r$ correspond to foliations of rank $r$.

Let $X' \rightarrow X$ be a purely inseparable morphism of height one between normal varieties over a perfect field of characteristic $p>0$ and let $\mathcal{F}$ be the corresponding foliation.

\[
\omega_{X'/X} \simeq (\det \mathcal{F})^{[p-1]}.
\]

As a consequence of the flattening lemma <cit.> we have the following.

Let $f\colon X \rightarrow Y$ be a projective dominant morphism of normal varieties.
Then, there is an open subset $U \subseteq Y$ with $\codim(Y \setminus U) \geq 2$ such that $X_U\coloneqq f^{-1}(U)$ is flat over $U$.

Let $X$ be a normal variety over a perfect field $k$ of characteristic $p>0$ and let $f \colon X \rightarrow Y$ be a morphism to a normal variety over $k$.
Let $a \colon Y' \rightarrow Y$ be a finite purely inseparable $k$-morphism of height one from a normal variety, let $X'$ be the normalisation of the reduction of $X \times_Y Y'$ and $f' \colon X' \rightarrow Y'$ the induced morphism.
Set $\mathcal{A}$ to be the foliation induced by $a$.

(i) $K_{X'/X} \sim (p-1)D$ for some Weil divisor $D$ on $X'$;

(ii) there is a non-empty open subset $U \subseteq Y'$ and an effective divisor $C$ on $f'^{-1}(U)$ such that $C \sim -D|_{f'^{-1}(U)}$.

Moreover, assume $X_{\overline{\eta}}$ is reduced, where $\overline{\eta}$ is the geometric generic point of $Y$, and $f$ is equidimensional. Then:

(iii) $f'^*( \det \mathcal{A}) -D \sim C$ for some effective divisor $C$ on $X'$.

Points (i) and (ii) are just points (a) and (b) of <cit.>. As for point (iii), first, assume $X \times_Y Y'$ is reduced.
Note that $f'$ is equidimensional since so is $f$. In particular, as $f$ and $f'$ are both equidimensional, we can freely replace $Y$ by $Y_0\coloneqq Y\setminus(\sing(Y)\cup a(\sing(Y')))$, $Y'$ by $Y_0'\coloneqq a^{-1}(Y_0)$, $X$ by $f^{-1}(Y_0)$ and $X'$ by $f'^{-1}(Y_0')$. Then point (iii) follows from point (d) of <cit.>.
If $X \times_Y Y'$ is not reduced, by <ref>, there exists $U \subseteq Y$ with $\codim(Y \setminus U) \geq 2$ such that $f|_{X_U} \colon X_U \rightarrow U$ is flat, where $X_U\coloneqq f^{-1}(U)$.
Let $U'\coloneqq a^{-1}(U)$.
By <cit.>, the fibre product $X_U \times_U U'$ is reduced since $f|_{X_U}$ is flat and $X_{\overline{\eta}}$ is reduced.
Let $X'_{U'} \subseteq X'$ be the normalisation of $X_U \times_U U'$.
By the above discussion, we conclude $(f'^*( \det \mathcal{A}) -D)|_{X'_{U'}} \sim C|_{X'_{U'}}$ for some divisor $C$ on $X'$ such that $C|_{X'_{U'}} \geq 0$.
Since $f'$ is equidimensional, $\codim(X' \setminus X'_{U'}) \geq 2$, therefore, by normality of $X'$, we can extend the above equation on all of $X'$.

In the sequel, we will need to consider base changes with purely inseparable maps that are not necessarily of height one.
The previous results extend to this situation by induction on the height.

Let $f \colon X \rightarrow Y$ be an equidimensional contraction between normal projective varieties and $g\colon Y \rightarrow Z$ a morphism between normal varieties.
Let $Y_e$ be the normalisation of the reduction of $Y \times_Z Z^e$, $X_e$ the normalisation of the reduction of $X \times_Y Y_e$.
Assume that $X_{\overline{\eta}}$ is reduced, where $\overline{\eta}$ is the geometric generic point of $Y$.
Let $f_e \colon X_e \rightarrow Y_e$ and $g_e\colon Y_e\to Z^e$ be the induced morphisms.

(i) $K_{X_e/X} - f_e^*K_{Y_e/Y} \sim -C$ for some effective Weil divisor $C$ on $X_e$;

(ii) $K_{Y_e/Y} \sim D$ for some Weil divisor $D$ on $Y_e$ and there is a non-empty open subset $U \subseteq Z^e$ such that $-D|_{g_e^{-1}(U)}$ is effective.

We proceed by induction.
When $e=1$, let $\mathcal{A}$ be the foliation on $Y_1$ corresponding to $Y_1\to Y$.
By <ref>, $K_{Y_1/Y} \simeq (\det \mathcal{A})^{\left[ p-1 \right]}$ and, by <ref>,

\[
K_{X_1/X}-(p-1)f_1^*(\det \mathcal{A}) \sim -C
\]

for some effective divisor $C$ on $X_1$, giving point (i).
Point (ii) corresponds to points (i) and (ii) of the above <ref>.
If $e>1$, consider the diagram:

X_e [swap,d,"f_e"] [r,"π_2"]     X_e-1 [r, "π_1"] [d,"f_e-1"]     X [d, "f"]

Y_e [r,"p_2"] [swap, d, "g_e"]     Y_e-1 [r, "p_1"] [d, "g_e-1"]     Y [d, "g"]

Z^e [r, ""]     Z^e-1 [r, "^e-1"]     Z,

where $\pi_1$, $\pi_2$, $p_1$ and $p_2$ are the induced maps.
By the inductive assumptions,

* $K_{X_{e-1}/X} -f_{e-1}^* K_{Y_{e-1}/Y} \sim -C_1$ and $C_1\geq 0$;

* $K_{X_e/X_{e-1}} - f_e^*K_{Y_e/Y_{e-1}} \sim -C_2$ and $C_2\geq 0$;

* $K_{Y_{e-1}/Y} \sim D_1$ and there exists an open $U_1\subseteq Z^{e-1}$ such that $-D_1|_{g_{e-1}^{-1}(U_1)}\geq 0$;

* $K_{Y_e/Y_{e-1}} \sim D_2$ and there exists an open $U_2\subseteq Z^{e}$ such that $-D_2|_{g_{e}^{-1}(U_2)}\geq 0$.

Setting $C\coloneqq\pi_2^*C_1 + C_2$, $D\coloneqq p_2^*D_1 + D_2$, and $U\coloneqq U_1\cap U_2$ we get the claim.

§.§ $F$-singularities

Throughout this subsection we will denote by $(X,B)$ a sub-couple such that $K_X+B$ is a $\bZ_{(p)}$-divisor. If $(1-p^e)(K_X+B)$ is integral for some $e\geq 1$, we will denote by $\cL_{X,B}^{(e)}$ or $\cL^{(e)}_{B}$ the divisorial sheaf $\cO_X((1-p^e)(K_X+B))$. In particular, $\cL^{(e)}=\cL^{(e)}_X=\cO_X((1-p^e)K_X)$.

Suppose $B \geq 0$, let $e\geq 1$ such that $(p^e-1)(K_X+B)$ is integral, and let $L$ be a divisor on $X$. We have a natural map of $\cO_X$-modules induced by Grothendieck duality

\[
T^e_{B}\colon \Fr^e_* \cL^{(e)}_B\subseteq \Fr_*^e\cL^{(e)} \to \cO_X,
\]

which in turn induces

\[
T^e_{B}(L)\colon \Fr^e_* \cL^{(e)}_B\otimes_{\cO_X}\cO_X(L) \to \cO_X(L).
\]

The space of Frobenius stable sections of $\cO_X(L)$ is defined as

\[
S^0(X,B;L)\coloneqq\bigcap_{e>0:\, (1-p^e)(K_X+B) \textup{ is Weil }}\Image(H^0(X,T_B^e(L)))\subseteq H^0(X,\cO_X(L)).
\]

Note that $S^0(X,B;L)=\Image(H^0(X,T_B^e(L)))$ for some $e\gg 0$.

We have correspondences

{ $\bQ$-divisors $\Delta\geq 0$ such that
$(1-p^e)(K_X + \Delta)$ is integral } [rr][d,hook]         { divisorial sheaves $\cL$ and
$\cO_X$-linear maps
$\phi : \Fr^e_* \cL \xrightarrow{\neq 0} \cO_X$ }/∼[ll][d,hook]

{ $\bQ$-divisors $\Delta$ such that
$(1-p^e)(K_X + \Delta)$ is integral } [rr]         { divisorial sheaves $\cL$ and
$\cO_X$-linear maps
$\phi : \Fr^e_* \cL \xrightarrow{\neq 0} k(X)$ }/∼, [ll]

where the horizontal arrows are bijections, and the equivalence relations on the right identify two maps which agree up to multiplication by a unit of $H^0(X, \cO_X)$.

We outline the main ideas for the sake of completeness. When $\Delta\geq 0$ we set $\cL\coloneqq \cO_X((1-p^e)(K_X+\Delta))$ and $\phi\coloneqq T_\Delta^e$. Conversely, given $\phi$, by Grothendieck duality we have

\begin{equation*}
\begin{split}
\phi\in \Hom_{\cO_X} (\Fr^e_*\cL,\cO_X)&\simeq \Hom_{\cO_X} (\Fr^e_*(\cL(p^eK_X)),\cO_X(K_X))\\
&\simeq \Fr_*^e \Hom_{\cO_X} (\cL(p^eK_X),\cO_X(K_X))\\
&\simeq \Fr_*^e H^0(X,\cL^{-1}((1-p^e)K_X)).
\end{split}
\end{equation*}

Hence we can identify $\phi$ with an element $D_{\phi}\in H^0(X,\cL^{-1}((1-p^e)K_X))$, and we can set $\Delta\coloneqq D_{\phi}/(p^e-1)$. Note that if we change $\phi$ by multiplication by a unit, we obtain the same $\Delta$. It is easy to check that the two constructions are one the inverse of the other.

When $\Delta=\Delta^+-\Delta^-$ is not effective one can see, by arguing locally, that there exists a non-zero $\cO_X$-linear map $\phi$ fitting in the diagram

^e_*_X((1-p^e)(K_X+Δ^+))[d,hook][r,"T_Δ^+^e"]     _X[d,hook]

^e_*_X((1-p^e)(K_X+Δ))[r,"ϕ"]     _X(E),

for some effective Weil divisor $E\geq (p^e-1)\Delta^-$. Indeed, working locally and letting $s\in\cO_X((1-p^e)(K_X+\Delta^+))$ and $s/f\in\cO_X((1-p^e)(K_X+\Delta))$, for $f$ a regular function, we have

\[
\phi\left(\Fr^e_*\left(\dfrac{s}{f}\right)\right)=\dfrac{T_{\Delta^+}^e(\Fr_*^e(f^{p^e-1}s))}{f}.
\]

Conversely, given $\phi$ as in the hypothesis, a similar local computation shows that $\Image(\phi)\subseteq\cO_X(E)$, for some effective Weil divisor $E$. Then the same argument as in the effective case yields the required divisor $\Delta$ (see <cit.>).

We say $(X,B)$ is globally sub-F-split (GsFS) if for all $e\geq 1$ sufficiently divisible, letting $\phi_e\colon \Fr^e_* \cL^{(e)}_B \to k(X)$ be the map associated to $B$ by <ref>, there exists a map of $\cO_X$-modules $\sigma_e\colon \cO_X\to \Fr^e_* \cL^{(e)}_B$ such that $\phi_e\circ \sigma_e=\id_{\cO_X}$. If $B\geq 0$, we say $(X,B)$ is globally F-split (GFS).

We say $(X,B)$ is globally F-regular (GFR) if $B\geq 0$ and for every effective Weil divisor $E$ and all $e\geq 1$ sufficiently divisible, the $\cO_X$-linear map

\[
\Fr^e_*\cL^{(e)}_{B+\frac{E}{p^e-1}}\hookrightarrow\Fr^e_*\cL^{(e)}_B\xrightarrow{T_B^e}\cO_X
\]

admits a splitting

\[
\id_{\cO_X}\colon \cO_X\xrightarrow{\sigma_{E,e}}\Fr^e_*\cL^{(e)}_{B+\frac{E}{p^e-1}}\to\cO_X.
\]

Globally $F$-split and globally $F$-regular pairs should be thought of as pairs of log Calabi-Yau type, resp. log Fano type, with arithmetically well-behaved Frobenius. This is made somewhat more precise by the next statements.

Let $(X,B)$ be a klt pair over $\bC$ such that $-K_X-B$ is ample. Then $(X,B)$ has open GFR type, that is for every model $(\cX,\cB)\to Spec(A)$ of $(X,B)$ the set of primes $\mathfrak{p}\subset A$ such that $(\cX_{\mathfrak{p}},\cB_{\mathfrak{p}})$ is GFR is open and dense in $\Spec(A)$.

A similar statement is expected to hold for Calabi-Yau pairs.

Let $(X,B)$ be a log canonical pair over $\bC$ such that $-K_X-B\sim_{\bQ}0$. Then $(X,B)$ has dense GFS type, that is for every model $(\cX,\cB)\to Spec(A)$ of $(X,B)$ the set of primes $\mathfrak{p}\subset A$ such that $(\cX_{\mathfrak{p}},\cB_{\mathfrak{p}})$ is GFS is dense in $\Spec(A)$.

The next Lemma shows that the class of GFR varieties is stable under small perturbations of the boundary.

Let $(X,B)$ be a GFR projective couple where $B$ is a $\bZ_{(p)}$-divisor and let $D\geq 0$ be a $\bQ$-divisor. Then $(X,B+\epsilon D)$ is GFR for all $0 \leq \epsilon\ll 1$ such that $B+\epsilon D$ is a $\bZ_{(p)}$-divisor.

If $B\geq 0$, then $(X, B)$ is GFS iff $S^0(X, B; \cO_X)= H^0(X, \cO_X)$.

Let $(X,B)$ be a couple such that $B$ is a $\bZ_{(p)}$-divisor. Then $(X,B)$ is GFR if and only if for all $\bQ$-divisors $D\geq 0$ the couple $(X,B+D/(p^e-1))$ is GFS for all $e\gg 1$.

§.§ Calabi-Yau deformations of globally $F$-split varieties

In this subsection we will study how GFS singularities behave in family.

Let $(X,B)$ be a sub-couple over a perfect field $k$, such that $K_X+B$ is a $\bZ_{(p)}$-divisor. Then all the different classes of $F$-singularities can be given by replacing the absolute Frobenius $\Fr_X^e$ by the $k$-linear Frobenius $\Fr^e_{X^e/k^e}$, as the two differ by the automorphism $\Fr^e_{k}$.

Let $k$ be a perfect field, and let $R$ be a smooth local $k$-algebra, essentially of finite type, with fraction field $K$ and residue field $k$. Let $\pi\colon \left( X,B=\sum_i a_iB_i\right)\to \Spec R$ be a pair over $R$, where $\pi$ is a flat contraction with geometrically normal fibres and $\supp(B_i)\to\Spec R$ is flat for all $i$. Assume $(1-p^e)(K_X+B)\sim 0$ and that $(X_{\overline{k}},B_{\overline{k}})$ is GFS. Then $(X_{\overline{K}},B_{\overline{K}})$ is GFS too.

Let $g^e\colon \Spec \overline{k}^e\to\Spec R^e$, and consider the corresponding base-change of the leftmost triangle in <ref>

\begin{equation*}
\begin{tikzcd}
X^e_{\overline{k}^e}\arrow[rr]\arrow[d,"\Fr^e_{X^e_{\overline{k}^e}/\overline{k}^e}",swap] & & X^e\arrow[d,"\Fr^e_{X^e/R^e}"]\\
X_{\overline{k}^e} \arrow[rr,"q_e"]\arrow[d] & & X_{R^e}\arrow[d]\\
\Spec\overline{k}^e\arrow[rr,"g_e"] & & \Spec R^e.
\end{tikzcd}
\end{equation*}

In what follows we will freely and implicitly replace $X$ with the complement of a closed subset having codimension and relative codimension $\geq 2$, since all the sheaves we will be dealing with are reflexive. In particular, as $\pi$ has geometrically normal fibres, we may actually assume that $\pi$ is smooth. By <cit.> the trace map of the relative Frobenius is compatible with base-change, that is we have a natural commutative diagram

\begin{equation*}
\begin{tikzcd}
\Fr^e_{X^e_{\overline{k}^e}/\overline{k}^e,*}\omega_{X^e_{\overline{k}^e}/\overline{k}^e} \arrow[d] & & q_e^*\Fr^e_{X^e/R^e,*}\omega_{X^e/R^e} \arrow[d]\arrow[ll]\\
\omega_{X_{\overline{k}^e}/\overline{k}^e} & & q_e^*\omega_{X_{R^e}/R^e}\arrow[ll]
\end{tikzcd}
\end{equation*}

where the horizontal arrows are isomorphisms. Note that we have

\[
\]

by <cit.>. After twisting the above commutative square by $\omega_{X_{R^e}/R^e}^{-1}$ and composing with the inclusion $\cL_{X^e,B^e}^{(e)} \subseteq \cL_{X^e}^{(e)}$, we obtain

\begin{equation*}
\begin{tikzcd}
\Fr^e_{X^e_{\overline{k}^e}/\overline{k}^e,*}\cL^{(e)}_{X^e_{\overline{k}^e},B^e_{\overline{k}^e}} \arrow[d] & & q_e^*\Fr^e_{X^e/R^e,*}\cL_{X^e,B^e}^{(e)} \arrow[d]\arrow[ll]\\
\cO_{X_{\overline{k}^e}} & & q_e^*\cO_{X_{R^e}}.\arrow[ll]
\end{tikzcd}
\end{equation*}

Hence, by taking global sections we obtain

\begin{equation*}
\begin{tikzcd}
H^0(X_{\overline{k}^e},\cL^{(e)}_{X^e_{\overline{k}^e},B^e_{\overline{k}^e}}) \arrow[d] & & H^0(X^e,\cL_{X^e,B^e}^{(e)}) \arrow[d]\arrow[ll,"q_e^*"]\\
H^0(X_{\overline{k}^e},\cO_{X_{\overline{k}^e}}) & & H^0(X_{R^e},\cO_{X_{R^e}}).\arrow[ll,"q_e^*"]
\end{tikzcd}
\end{equation*}

Now, as $(1-p^e)(K_{X^e/R^e}+B^e)\sim 0$ we have a unique (up to $R^*$) nonzero section $s\in H^0(X_{R^e},\cO_{X^e}((1-p^e)(K_{X^e/R^e}+B^e)))$. As $(X_{\overline{k}},B_{\overline{k}})$ is GFS, by <ref> we have that $s_{\overline{k}^e}\coloneqq q_e^*(s)$ gets mapped to a unit in $\overline{k}^e$ by the (twisted) trace map of $\Fr^e_{X^e_{\overline{k}^e}/\overline{k}^e}$. Hence $s$ gets mapped to a unit in $R^e$ by the (twisted) trace map of $\Fr^e_{X^e/R^e}$. As the trace map is compatible with base-change, by restricting to the geometric generic fibre the same argument yields that $s_{\overline{K}^e}$ gets mapped to a unit in $\overline{K}^e$ by the (twisted) trace map of $\Fr^e_{X^e_{\overline{K}^e}/\overline{K}^e}$.
Thus, <ref> gives the conclusion.

§ STEIN-SEPARABLE MORPHISMS AND CANONICAL BUNDLE FORMULAE

By combining the results of <cit.> with <cit.> one can show that morphisms with $p$-prime Stein degree and GFS general fibre satisfy a canonical bundle formula.

Let $g\colon X\to Z$ be a contraction of normal schemes and let $B\geq 0$ be a $\bQ$-divisor on $X$ such that $(1-p^e)(K_X+B)\sim_Z 0$ for some $e\geq 0$ and $(X_{\overline{\zeta}},B_{\overline{\zeta}})$ is globally $F$-split, where $\zeta \in Z$ is the generic point. Then there exists a canonically defined effective $\bQ$-divisor $B^Z$ on $Z$ such that

* $(1-p^e)(K_X+B)\sim g^*((1-p^e)(K_Z+B^Z))$;

* $(X,B)$ is GFS if and only if $(Z,B^Z)$ is GFS;

* if $\Lambda\geq 0$ is a $\bQ$-Cartier $\bQ$-divisor on $Z$ such that $K_Z+B^Z+\Lambda$ is $\bZ_{(p)}$-Cartier, then $(B+g^*\Lambda)^Z=B^Z+\Lambda$.

Point (a) follows by <cit.> (see also <cit.>). This boils down to the following: write $(1-p^e)(K_X+B)\sim g^*M$ for some Cartier divisor $M$ on $Z$, so that we have $T^e_B\colon \Fr_*^e\cO_X(f^*M)\to\cO_X$. By pushing forward via $g$ and using the projection formula we obtain $\Fr_*^e\cO_Z(M)\xrightarrow{\phi}\cO_Z$. As $(X_{\overline{\zeta}},B_{\overline{\zeta}})$ is GFS we have $\phi\neq 0$ (<cit.>), and <ref> yields a canonically defined $\bQ$-divisor $B^Z\geq 0$ such that $M\sim(1-p^e)(K_Z+B^Z)$ and $\phi=T^e_{B^Z}$.

As for point (b), <ref> implies it is enough to show

\[
S^0(X, B; \cO_X) = S^0(Z, B^Z; \cO_Z).
\]

By the construction in point (a) we have a commutative diagram

\begin{equation}\label{e-pushforward_Fsplitting_contraction}
\begin{tikzcd}
g_*\Fr_*^e\cL^{(e)}_{X,B} \arrow[r,"g_*T_{B}^e"] & g_*\cO_X\\
\Fr_*^e\cL^{(e)}_{Z,B^Z}\arrow[u] \arrow[r,"T_{{B}^Z}^e"] & \cO_Z,\arrow[u]
\end{tikzcd}
\end{equation}

where the vertical arrows are isomorphisms. Taking global sections for $e\geq 1$ large enough yields $S^0(X,B;\cO_X)=S^0(Z,B^Z;\cO_Z)$, thus we conclude by <ref>.

Point (c) follows by the same argument as in (a), noting that <ref> can be completed to

\begin{equation*}
\begin{tikzcd}
g_*\Fr_*^e\cL^{(e)}_{X,B+g^*\Lambda}\arrow[rr,bend left=20,"T_{B+g^*\Lambda}^e"]\arrow[r] & g_*\Fr_*^e\cL^{(e)}_{X,B} \arrow[r,"T_{B}^e",swap] & g_*\cO_X\\
\Fr_*^e\cL^{(e)}_{Z,B^Z+\Lambda}\arrow[r]\arrow[rr,bend right=20,"T_{B^Z+\Lambda}^e",swap]\arrow[u] & \Fr_*^e\cL^{(e)}_{Z,B^Z}\arrow[u] \arrow[r,"T_{B^Z}^e"] & \cO_Z.\arrow[u]
\end{tikzcd}
\end{equation*}

Observe that, since we are twisting by a divisor on $Z$, we still have $T^e_{B+g^*\Lambda,\overline{\zeta}}\neq 0 \neq T^e_{B^Z+\Lambda,\overline{\zeta}}$.

We are now ready to define KGFR pairs.

A projective couple $(X,B)$ is said to be $K$-globally F-regular (KGFR) if

* $-K_X-B$ is semiample with induced contraction $f\colon X\to Y$;

* the geometric generic fibre of $f$, $(X_{\overline{\eta}},B_{\overline{\eta}})$, is GFS;

* $K_X+B\sim_{\bZ_{(p)},Y}0$;

* the pair $(Y,B^Y)$ induced by <ref> is GFR.

Note that $(X,B)$ is GFS a posteriori thanks to <ref>(b).

Let $h\colon Z\to Y$ be a separable finite morphism of integral normal $k$-schemes such that $p\nmid \deg(h)$, and let $B\geq 0$ be a $\bQ$-divisor on $Z$ such that $(Z,B)$ is globally $F$-split and $(1-p^e)(K_Z+B)\sim_Y 0$ for some $e\geq 1$. Then there exists a canonically defined effective $\bQ$-divisor $B^Y$ on $Y$ such that

* $(1-p^e)(K_Z+B)\sim h^*((1-p^e)(K_Y+B^Y))$;

* $(Y,B^Y)$ is GFS;

* if $\Lambda\geq 0$ is a $\bQ$-Cartier $\bQ$-divisor on $Y$ such that $K_Y+B^Y+\Lambda$ is $\bZ_{(p)}$-Cartier and $(Y,B^Y+\Lambda)$ is GFS, then $(Z,B+h^*\Lambda)$ is GFS too, and $(B+h^*\Lambda)^Y=B^Y+\Lambda$;

* If $(Z,B)$ is GFR then $(Y,B^Y)$ is GFR.

Let $M$ be a Cartier divisor such that $(1-p^e)(K_Z+B)\sim h^*(M)$, let $\cL\coloneqq\cO_Y(M)$ and $d\coloneqq\deg(h)$. By <cit.> we have the following commutative diagram

\begin{equation}\label{e-st_finitemor}
\begin{tikzcd}
h_*\Fr^e_*h^*\cL \arrow[rrr,"h_*\phi_Z"] \arrow[dd,"\Fr^e_*\left(\frac{\Tr_{Z/Y}}{d}\otimes\id_{\cL}\right)",bend left=15] & & & h_*\cO_Z \arrow[dd,"\frac{\Tr_{Z/Y}}{d}",bend left=15] \\
\Fr^e_*\cL \arrow[rrr,"\phi_Y"] \arrow[uu,"\Fr^e_*(h^\sharp\otimes\id_{\cL})",bend left=15] & & & \cO_Y, \arrow[uu,"h^\sharp",bend left=15]\\
\end{tikzcd}
\end{equation}

where $\phi_Z$ is defined by $B$ using the correspondence in <ref> and $\phi_Y$ is defined by composition with the natural maps. Note that, for each column, going up and then down yields the identity. By taking global sections and using GFS-ness of $(Z,B)$ we obtain $1\in\Image (H^0(Y,\phi_Y))$. In particular $\phi_Y\neq 0$, thus by <ref> we obtain an effective $\bQ$-divisor $B^Y$ such that $(Y,B_Y)$ is GFS too and $(1-p^e)(K_Z+B)\sim h^*((1-p^e)(K_Y+B^Y))$.

To show point (c), denote by $\lambda\colon \cL((1-p^e)\Lambda)\to\cL$ the natural map, and observe that <ref> can be completed to

\begin{equation*}
\begin{tikzcd}
h_*\Fr^e_*h^*\cL((1-p^e)\Lambda) \arrow[rrrrrr,"\psi_Z",bend left=20] \arrow[rrr,"h_*\Fr^e_*h^*\lambda"] \arrow[dd,"\Fr^e_*\left(\frac{\Tr_{Z/Y}}{d}\otimes\id_{\cL((1-p^e)\Lambda)}\right)",bend left=15] & & & h_*\Fr^e_*h^*\cL \arrow[rrr,"h_*\phi_Z"] \arrow[dd,"\Fr^e_*\left(\frac{\Tr_{Z/Y}}{d}\otimes\id_{\cL}\right)",bend left=15] & & & h_*\cO_Z \arrow[dd,"\frac{\Tr_{Z/Y}}{d}",bend left=15] \\
& & & \\
\Fr^e_*\cL((1-p^e)\Lambda)\arrow[rrrrrr,"\psi_Y",bend right=20] \arrow[rrr,"\Fr^e_*\lambda"]  \arrow[uu,"\Fr^e_*(h^\sharp\otimes\id_{\cL((1-p^e)\Lambda)})",bend left=15] & & &\Fr^e_*\cL \arrow[rrr,"\phi_Y"] \arrow[uu,"\Fr^e_*(h^\sharp\otimes\id_{\cL})",bend left=15] & & & \cO_Y. \arrow[uu,"h^\sharp",bend left=15]\\
\end{tikzcd}
\end{equation*}

As $(Y,B^Y+\Lambda)$ is GFS, we have $1\in \Image(H^0(Y,\psi_Y))$, hence $1\in\Image(H^0(Z,\psi_Z))$. By <ref> we see that the maps

\[
\Fr^e_*h^*\cL((1-p^e)\Lambda)\xrightarrow{\Fr^e_* h^*\lambda}\Fr^e_* h^*\cL\xrightarrow{\phi_Z}\cO_Z\hspace{3mm}\textup{and}\hspace{3mm}\Fr^e_*\cL((1-p^e)\Lambda)\xrightarrow{\Fr^e_*\lambda}\Fr^e_* \cL\xrightarrow{\phi_Y}\cO_Y
\]

correspond to the divisors $B+h^*\Lambda$ and $B^Y+\Lambda$, respectively.

To show point (d) let $D\geq 0$ be a divisor on $Y$. By <ref> it is enough to show that $(Y,B^Y+D/(p^e-1))$ is GFS whenever $e$ is large enough. As $(Z,B)$ is GFR, we have that $(Z,B+h^*\left(D/(p^e-1)\right))$ is GFS provided $e\gg 0$, again by <ref>. Arguing as in point (a) we can push forward the splitting on $Y$ via the trace map of $h$

\begin{equation*}
\begin{tikzcd}
h_*\cO_Z\arrow[r]\arrow[dd,"\frac{\Tr_{Z/Y}}{d}",bend left=15] & h_*\Fr^{e'}_*\cL^{(e')}_{B+h^*\left(D/(p^e-1)\right)} \arrow[rrr,"h_*T_{B+h^*\left(D/(p^e-1)\right)}"] \arrow[dd,"\Fr^e_*\left(\frac{\Tr_{Z/Y}}{d}\otimes\id\right)",bend left=15] & & & h_*\cO_Z \arrow[dd,"\frac{\Tr_{Z/Y}}{d}",bend left=15]\\
& & & \\
\cO_Y\arrow[r]\arrow[uu,"h^\sharp",bend left=15] & \Fr^{e'}_*\cL^{(e')}_{B^Y+D/(p^e-1)} \arrow[rrr,"T_{B^Y+D/(p^e-1)}"] \arrow[uu,"\Fr^e_*(h^\sharp\otimes\id)",bend left=15] & & & \cO_Y, \arrow[uu,"h^\sharp",bend left=15]
\end{tikzcd}
\end{equation*}

hence $(Y,B^Y+D/(p^e-1))$ is GFS.

It might be tempting to try and extend this result to the case of a split finite morphism $h$ (i.e. such that the natural map $\cO_Y\to h_*\cO_Z$ is split). However, this can easily be seen to fail by taking $Z=Y=\bP^1$ and $h=\Fr$. As explained in <cit.> a necessary and sufficient condition for $\psi_Z$ to descend to a nonzero $\psi_Y$ is that the splitting of $\cO_Y\to h_*\cO_Z$ is given by the trace map. In this case it then follows that $B=h^*B^Y+\mathrm{Ram}(h)$, where $\mathrm{Ram}(h)$ denotes the ramification divisor of $h$.

Let $f\colon X \to Y$ be a morphism of integral normal schemes such that $p\nmid\stdeg(f)$, and let $B$ be an effective $\bQ$-divisor on $X$ such that $(X,B)$ is globally $F$-split and $(1-p^e)(K_X + B) \sim_Y 0$. Then there exists a canonically determined effective $\bQ$-divisor $B^Y$ on $Y$ such that

* $(1-p^e)(K_X + B) \sim f^*((1-p^e)(K_Y + B^Y))$;

* $(Y,B^Y)$ is GFS;

* if $\Lambda\geq 0$ is a $\bQ$-Cartier $\bQ$-divisor on $Y$ such that $K_Y+B^Y+\Lambda$ is $\bZ_{(p)}$-Cartier, then $(B+f^*\Lambda)^Y=B^Y+\Lambda$;

* $(X, B+f^*\Lambda)$ is GFS if and only if $(Y,B^Y+\Lambda)$ is GFS.

* $(Y,B^Y)$ is GFR whenever $(X,B)$ is KGFR.

Follows immediately from applying <ref> and <ref> to the Stein factorisation $f\colon X\xrightarrow{g}Z\xrightarrow{h}Y$.

§ F-COMPLEMENTS

Complements were first introduced by Shokurov in <cit.>: given a pair $(X,B)$ a complement is a $\bQ$-divisor $\Gamma\geq 0$ such that $(X,B+\Gamma)$ is lc and $K_X+B+\Gamma\sim_{\bQ}0$. In this subsection we introduce an analogous notion for $F$-singularities in the relative setting.

Let $f\colon (X,B)\to Y$ be a contraction of normal quasi-projective varieties, where $B=B^+-B^-$ is a $\bQ$-divisor such that $\supp (B^-)$ does not dominate $Y$, and whose geometric generic fibre $(X_{\overline{\eta}},B_{\overline{\eta}})$ is globally $F$-split. Let $L$ be a $\bQ$-effective $\bQ$-divisor on $X$. We say $L$ admits an $F$-complement for $(X/Y,B)$ if there exists $\Lambda\in |L|_{\bQ}$ such that $(1-p^e)(K_X+B+\Lambda)\sim_Y 0$ for some $e\geq 1$, and $(X_{\overline{\eta}},B_{\overline{\eta}}+\Lambda_{\overline{\eta}})$ is globally $F$-split. In this case we say $\Lambda$ is an F-complement for $(X/Y,B)$. When $X$ is a $k$-variety and $Y=\Spec(k)$ we just refer to $\Lambda$ as an $F$-complement for $(X,B)$.

By results of Schwede and Smith we have that projective GFS couples admit $F$-complements.

Let $k$ be an $F$-finite field and let $(X, B)$ be a globally F-split quasi-projective normal variety over $k$. Then there exists an F-complement $\Gamma$ for $(X,B)$.

We give a sufficient condition for the existence of $F$-complements on contractions with KGFR fibres.

Let $f \colon X \to Y$ be a contraction of normal quasi-projective varieties with general fibre $X_y$, and let $B$ be a $\bQ$-divisor on $X$ such that $\supp(B^-)$ does not dominate $Y$ and $(X_y,B_{X_y})$ is KGFR. Let $D$ be a $\bQ$-Cartier $\bQ$-divisor on $Y$, set $L\coloneqq-K_X-B-f^*D$, and assume there is $m\in\bN\setminus p\bN$ such that $mL$ is Cartier and $|V|\subseteq |mL|$ such that $\phi_{|V|_{X_y}}$ is a morphism and $p\nmid \stdeg(\phi_{|V|_{X_y}})$. Then $L$ admits an $F$-complement for $(X/Y,B)$.

Before we give a proof, we need the following result.

Let $G$ be a non-normal projective $k$-variety, let $A$ be an ample $\bQ$-divisor on $G$, let $\pi\colon \overline{G}\to G$ be the normalisation morphism and let $\overline{\Delta}\geq 0$ be a $\bQ$-divisor on $\overline{G}$ such that

* $(\overline{G},\overline{\Delta})$ is a GFR pair;

* $-K_{\overline{G}}-\overline{\Delta}\sim_{\bQ}\pi^*A.$

Then there exists an $F$-complement $\overline{\Gamma}$ for $(\overline{G},\overline{\Delta})$ such that $\overline{\Gamma}=\pi^*\Gamma$ for some $\Gamma\in|A|_{\bQ}$.

Let $m\geq 1$ divisible enough, and let $L\coloneqq mA$ and $\overline{L}\coloneqq m(-K_{\overline{G}}-\overline{\Delta})\sim\pi^*L$. Denote by $\overline{\cC}\subset \cO_{\overline{G}}$ and $\cC\subset\cO_G$ the conductor ideals, and by $\overline{C}\subset\overline{G}$ and $C\subset G$ the respective closed subschemes, so that we have an induced finite morphism $\pi|_{\overline{C}}\colon \overline{C}\to C$. We then have the following morphism of short exact sequences (<cit.>) for all $l\geq 1$:

\begin{equation}\label{e-normalisationses}
\begin{tikzcd}
0 \arrow[r] & \cO_G(lL) \otimes_{\cO_G}\cC\arrow[r] \arrow[d,equal] & \cO_G(lL) \arrow[r] \arrow[d,hook] & \cO_C(lL)\arrow[r] \arrow[d,hook] & 0\\
0 \arrow[r] & \pi_*(\cO_{\overline{G}}(l\overline{L})\otimes_{\cO_{\overline{G}}}\overline{\cC}) \arrow[r] & \pi_*\cO_{\overline{G}}(l\overline{L})\arrow[r] & (\pi|_{\overline{C}})_*\cO_{\overline{C}}(l\overline{L})\arrow[r] & 0.
\end{tikzcd}
\end{equation}

Let now $R$ be an effective Cartier divisor on $\overline{G}$ such that $\cO_{\overline{G}}(-R)\subseteq\overline{\cC}$. For $l\gg 0$ we have that $(\overline{G},\overline{\Delta}+R/l)$ is still GFR by <ref>, thus we have an $F$-complement $\overline{\Xi}$ for $(\overline{G},\overline{\Delta}+R/l)$ by <ref>. In particular, $\overline{\Gamma}\coloneqq\overline{\Xi}+R/l$ is an $F$-complement for $(\overline{G},\overline{\Delta})$. We now need to show that $\overline{\Gamma}$ descends to $G$. After multiplying by $lmn$ for some $n \geq 1$ to clear denominators, we obtain that $lmn\overline{\Gamma}$ is the divisor of a section $\overline{\gamma}\in H^0(\overline{G},\cO_{\overline{G}}(ln\overline{L})\otimes_{\cO_{\overline{G}}}\overline{\cC})$, and <ref> shows that $\overline{\gamma}=\pi^*\gamma$ for some $\gamma\in H^0(G,\cO_G(lnL)\otimes_{\cO_G}\cC)$. We conclude by letting $\Gamma\coloneqq (\gamma=0)/lmn$.

Up to replacing $m$ with a multiple (cf. <cit.>) we may assume $m=p^e-1$ for some $e\geq 1$. Throughout the rest of the proof we will always freely implictly replace $m$ with a suitable multiple (equivalently, replace $e$ with a suitable multiple). Consider the following diagram

\begin{equation} \label{e-restrictionmorphism}
\begin{tikzcd}
& \overline{G}\arrow[rd,"\pi"] &\\
X_y \arrow[dd,hook]\arrow[ur,"\overline{\varphi}"]\arrow[rr,"\varphi"] & & G\arrow[dd,hook]\arrow[dr,hook] &    \\
& &  & \bP V\\
X\arrow[rr,dashed,"\Phi"] & & W\arrow[ur,hook] &    \\
\end{tikzcd}
\end{equation}

where notation is as follows: $\Phi\coloneqq \phi_{|V|}$, $\varphi\coloneqq\phi_{|V|_{X_y}}$, $\pi$ is the normalisation morphism, and $\overline{\varphi}$ is the naturally induced morphism. Let $A\coloneqq\cO_W(1)$ and $A_G\coloneqq\cO_{G}(1)$. Then <ref> yields an effective $\bQ$-divisor $(B_{X_y})^{\overline{G}}$ on $\overline{G}$, such that $(\overline{G},(B_{X_y})^{\overline{G}})$ is GFR and $-m(K_{X_y}+B_{X_y})\sim \overline{\varphi}^*(-m(K_{\overline{G}}+(B_{X_y})^{\overline{G}}))$. By <ref> there is an $F$-complement $\Lambda_{\overline{G}}\in |-K_{\overline{G}}-(B_{X_y})^{\overline{G}}|_{\bQ}$ for $(\overline{G},(B_{X_y})^{\overline{G}})$ such that $\Lambda_{\overline{G}}=\pi^*\Lambda_{G}$ for some $\Lambda_G\in |A_G/m|_{\bQ}$. Letting $\Lambda_{X_y}\coloneqq\varphi^*\Lambda_G$ we then have that $(X_y,B_{X_y}+\Lambda_{X_y})$ is GFS and $K_{X_y}+B_{X_y}+\Lambda_{X_y}\sim_{\bZ_{(p)}}0$, by <ref>. The above diagram <ref> induces the following on global sections for all $l\geq 0$:

H^0(X_y,_X_y(mlL_X_y))     H^0(G,A_G^l)[l,"φ^*"]

H^0(X,_X(mlL))[uu]     H^0(W,A^l).[uu][l,"Φ^*"]

As the two rightmost maps are surjective for all $l\gg 0$ by Serre Vanishing, we conclude we can lift $\Lambda_{X_y}$ to $\Lambda\in |L|_{\bQ}$. As $(X_y,B_{X_y}+\Lambda_{X_y})$ is GFS, we have that $(X_{\overline{\eta}},B_{\overline{\eta}}+\Lambda_{\overline{\eta}})$ is also GFS by <ref>.

Thanks to the KGFR condition on the fibres, we can find $F$-complements even after a small perturbation of the boundary.

Let $f \colon X \to Y$ be an equidimensional contraction of normal quasi-projective varieties with general fibre $X_y$, and let $B$ be a $\bQ$-divisor on $X$ such that $\supp(B^-)$ does not dominate $Y$ and $(X_y,B_{X_y})$ is KGFR. Let $D$ be a $\bQ$-divisor on $Y$, set $L\coloneqq-K_X-B-f^*D$, and assume there is $m\in\bN\setminus p\bN$ such that $mL$ is Cartier and $|V|\subseteq |mL|$ such that $\phi_{|V|_{X_y}}$ is a morphism and $p\nmid\stdeg(\phi_{|V|_{X_y}})$. Let $E$ be a $\bQ$-divisor on $Y$ and suppose there exists a $\bQ$-divisor $0\leq \Gamma\sim_{\bQ}L-f^*E$. Then $L_\epsilon\coloneqq(1-\epsilon)L$ admits an $F$-complement for $(X/Y,B+\epsilon\Gamma)$ for all $\epsilon\in \bZ_{(p),>0}$ small enough.

\[
B_\epsilon\coloneqq B+\epsilon\Gamma ,\hspace{3mm} D_\epsilon\coloneqq D+\epsilon E, \hspace{3mm} L_\epsilon\coloneqq -K_X-B_\epsilon-f^*D_\epsilon
\]

so that $L_\epsilon\sim_{\bQ}(1-\epsilon)L$. The corollary will follow from <ref> as soon as we verify that

(1) $(X_y,B_{\epsilon,F})$ is KGFR, and

(2) there is $n\in \bN\setminus p\bN$ s.t. $nL_{\epsilon}$ is Cartier and there exists $|V_\epsilon|\subseteq |nL_\epsilon|$ s.t. $|V_\epsilon|_{X_y}$ induces a morphism with Stein degree not divisible by $p$.

Let $\psi\colon X_y\to H$ be the semiample contraction of $-K_{X_y}-B_{X_y}$. Then <ref> yields an effective $\bQ$-divisor $(B_{X_y})^H$ such that $(H,(B_{X_y})^H)$ is GFR. In particular we can write $\Gamma_{X_y}=\psi^*\Gamma_H$ for some $\Gamma_H\in |-K_H-(B_{X_y})^H|_{\bQ}$. As $(H,(B_{X_y})^H)$ is GFR we have $(H,(B_{X_y})^H+\epsilon\Gamma_H)$ is GFR for $\epsilon$ small enough such that $(B_{X_y})^H+\epsilon\Gamma_H$ is a $\bZ_{(p)}$-divisor by <ref>, and $K_H+(B_{X_y})^H+\epsilon\Gamma_H$ is $\bZ_{(p)}$-Cartier, hence $(X_y,B_{X_y}+\epsilon\Gamma_{X_y})$ is also KGFR by <ref> (b) and (c). This shows (1). To show (2), let $l$ be a $p$-prime positive integer such that $(1-\epsilon)l$ is an integer too. Then we conclude by letting $V_\epsilon\coloneqq V^{\otimes (1-\epsilon)l}\subseteq H^0(X,mlL_{\epsilon})$.

§ INJECTIVITY THEOREM

The main result of this section is the following injectivity theorem (see <cit.>).

Let $f \colon X \to Y$ be an equidimensional contraction between normal quasi-projective varieties over a perfect field. Assume the general fibre $X_y$ is normal.
Let $B=B^+-B^-$ be a $\bQ$-divisor on $X$ such that $\supp(B^-)$ does not dominate $Y$ and $(X_y,B_{X_y})$ is KGFR. Let $D$ be a $\bQ$-divisor on $Y$, set $L\coloneqq -K_X-B-f^*D$ and assume

* there is $m\in\bN\setminus p\bN$ such that $mL$ is Cartier and $|V|\subseteq |mL|$ such that $\phi_{|V|_{X_y}}$ is a morphism with $p\nmid\stdeg(\phi_{|V|_{X_y}})$;

* there exists a $\bQ$-Cartier $\bQ$-divisor $P \geq B^-$ such that $\kappa(X, f^*(-K_Y-D)+P)=0$.

Then, the natural map

\[
H^0(X, mL) \to H^0(X_y,mL_{X_y})
\]

is injective for all $m \geq 0$. In particular we have $\kappa(X, L)\leq \kappa(X_y, L_{X_y})$.

Provided that our contraction admits $F$-complements, we can follow the same proof as in <cit.>.

Let $f \colon X \to Y$ be a contraction of normal quasi-projective varieties, and let $B=B^+-B^-$ be a $\bQ$-divisor such that $\supp(B^-)$ does not dominate $Y$. Let $D$ be a $\bQ$-Cartier $\bQ$-divisor on $Y$, let $L\coloneqq-K_X-B-f^*D$ and suppose $L$ admits an $F$-complement for $(X/Y,B)$. Lastly assume

* $f$ is equidimensional or

* $B\geq 0$ and $Y$ is $\bQ$-Gorenstein.

Then, $f^*(-K_Y-D)+B^-$ is $\bQ$-effective.

Let $0\leq \Lambda\sim_{\bQ}L$ be an $F$-complement for $(X/Y,B)$ and consider $\Delta\coloneqq B+\Lambda$, so that  we have $(X_{\overline{\eta}},\Delta_{\overline{\eta}})$ is globally $F$-split and $K_X+\Delta\sim_{\bZ_{(p)},Y}0$. Then <cit.> yields a canonically defined $\bQ$-divisor $\Delta_Y$ such that

\[
K_X+\Delta\sim_{\bZ_{(p)}}f^*(K_Y+\Delta_Y)\sim_{\bQ} f^*(-D).
\]

Hence it is enough to show that $f^*(\Delta_Y)+B^-$ is $\bQ$-effective. If $B\geq 0$ then $\Delta_Y=\Delta^Y$ (<cit.>, <ref>), in particular, it is effective. Suppose now $f$ is equidimensional: then every component $P$ of $\supp(\Delta^v)$ is mapped to a prime divisor $Q\subset Y$, hence $f^*(\Delta_Y)\geq \Delta^v$. Indeed, <cit.> yields $\coeff_Q(\Delta_Y)=1-d_Q$, where

\[
d_Q\coloneqq\sup\lbrace t \textup{ s.t. } (X,\Delta+f^*(tQ)) \textup{ is globally sub \textit{F}-split over }\eta_Q\rbrace.
\]

In particular, as globally sub-$F$-split sub-couples are sub-log canonical in codimension one (<cit.>), we have:

\[\coeff_P(\Delta^v)\leq 1-d_Q\coeff_P(f^*(Q))\leq \coeff_P(f^*(Q))(1-d_Q)=\coeff_P(f^*(\Delta_Y)).
\]

As $\Delta^v+B^-\geq 0$, we conclude that

\[
f^*(\Delta_Y)+B^-\geq 0.
\]

Let $f \colon X \to Y$ be a contraction of normal quasi-projective varieties, with $Y$ $\bQ$-Gorenstein, and let $B=B^+-B^-$ be a $\bQ$-divisor on $X$.
If $B^- \neq 0$, assume moreover that $f$ is equidimensional and that $\supp(B^-)$ does not dominate $Y$.
Let $D,E$ be $\bQ$-Cartier $\bQ$-divisors on $Y$, set $L\coloneqq-K_X-B-f^*D$, and assume that

* there exists $0\leq\Gamma\sim_{\bQ} L-f^*E$;

* $L_{\epsilon}\coloneqq(1-\epsilon)L$ admits an $F$-complement for $(X,B+\epsilon \Gamma)$, for $\epsilon\in [0,1)\cap\bQ$.

Then, $f^*(-K_Y-D-\epsilon E)+B^-$ is $\bQ$-effective.

\[
B_\epsilon\coloneqq B+\epsilon\Gamma ,\hspace{3mm} D_\epsilon\coloneqq D+\epsilon E,
\]

so that $L_\epsilon=-K_X-B_\epsilon-f^*D_\epsilon$ and all the hypotheses of <ref> are satisfied with respect to $f\colon (X,B_\epsilon)\to Y$, $L_\epsilon$, and $D_\epsilon$. Then <ref> yields $f^*(-K_Y-D_\epsilon)+B_\epsilon^-=f^*(-K_Y-D-\epsilon E)+B^-$ is $\bQ$-effective.

We are now ready to prove our injectivity theorem.

Let $X_y\coloneqq f^{-1}(y)$ for a general $y \in Y$ and note that we may assume $f$ is flat over a neighborhood of $y$.
Let $U \subseteq Y$ be the regular locus of $Y$, its complement has codimension $\geq 2$. We may assume $y \in U$. Since $f$ is equidimensional, the complement of $X_U \coloneqq f^{-1}(U)$ has codimension $\geq 2$.
In particular, it is enough to prove $H^0(X_U, mL|_{X_U}) \to H^0(X_y,mL_{X_y})$ is injective.
By substituting $Y$ with $U$ and $X$ with $X_U$, we can assume $Y$ is smooth.
By contradiction, suppose the map $H^0(X, mL) \to H^0(X_y,mL_{X_y})$ is not injective. Then there exists a $\bQ$-divisor $0\leq N\sim_{\bQ}L$ such that $X_y\subseteq \supp(N)$. Note that, by hypothesis, there exists a unique effective $\bQ$-divisor $M\sim_{\bQ}f^*(-K_Y-D)+P$, hence we may assume $X_y\not\subseteq \supp(M)$. Consider now the diagram

(X',B') [r,"π"] [d,"f'", swap]     (X,B) [d,"f"]

Y' [r,"μ"]     Y,

where notation is as follows:

* $Y'$ is the blowup of $Y$ at $y$ with exceptional divisor $E$, so that $\mu^*K_Y+aE=K_{Y'}$, with $a=\dim Y-1$;

* $X'$ is the fibre product (hence it is also the blowup of $X$ at $X_y$ with exceptional divisor $G$, since blowup and flat base-change commute);

* $f'$ is the induced morphism (note that since $f$ is equidimensional, so is $f'$);

* $B'$ is the strict transform of $B$, so that $\pi^*(K_X+B)+bG=K_{X'}+B'$, with $b\leq a$.

Let also $D'\coloneqq\mu^*D-aE$ and $L'\coloneqq-K_{X'}-B'-f'^*D'\sim_{\bQ}\pi^*L+(a-b)G\geq \pi^*L$. As $\supp(N)\supseteq X_y$ we have $\supp(\pi^*N)\supseteq G$. Thus, letting $N'\coloneqq\pi^*N+(a-b)G$, $\supp(N') \supseteq G$ too. In particular, for $0<\delta\ll 1$, we have an effective divisor

\[
0\leq \Gamma'\coloneqq N'-\delta G\sim_{\bQ}L'-f'^*E',\hspace{3mm} E'\coloneqq\delta E.
\]

Note that the data $f'\colon (X',B')\to Y',D',E',\Gamma'$ satisfy the hypotheses of <ref>, so $L'$ admits an $F$-complement for $(X'/Y',B'+\epsilon \Gamma')$. Applying <ref> to $f'\colon (X',B')\to Y'$, $L',D',E',\Gamma'$ yields the existence of an effective $\bQ$-divisor $\overline{\Gamma}$ such that

\begin{equation*}
\begin{split}
0\leq \overline{\Gamma} \sim_{\bQ} f'^*(-K_{Y'}-D'-\epsilon E')+(B')^-&=f'^*(\mu^*(-K_Y-D)-\epsilon E')+(B')^-\\
&\leq f'^*(\mu^*(-K_Y-D)-\epsilon E')+\pi^*P\\
&=\pi^*(f^*(-K_Y-D)+P)-\epsilon\delta G\\
&\sim_{\bQ}\pi^*M-\epsilon\delta G,
\end{split}
\end{equation*}

contradicting the assumption $X_y\not\subseteq \supp(M)$.

§ PROOF

Now, we have all the ingredients to prove the main theorem.

Let $f \colon X \to Y$ be a contraction of normal quasi-projective varieties over a perfect field $k$ of positive characteristic.
Assume the general fibre $X_y$ is normal and let $B$ be an effective $\bQ$-divisor on $X$ such that $(X_y,B_{X_y})$ is $K$-globally $F$-regular. Assume $Y$ is $\bQ$-Gorenstein and $D$ is a $\bQ$-Cartier $\bQ$-divisor on $Y$. Set $L\coloneqq-K_X-B-f^*D$ and suppose there is $m\in\bN\setminus p\bN$ such that $mL$ is Cartier and $|V|\subseteq |mL|$ such that $\phi_{|V|_{X_y}}$ is a morphism with $p\nmid \stdeg(\phi_{|V|_{X_y}})$. Then, for a general fibre $X_y$,

\[
\kappa(X,L)\leq \kappa(X_y,L_{X_y})+\kappa(Y,-K_Y-D).
\]

By <ref> and <ref> we may assume that $-K_Y-D$ is $\bQ$-effective.
We consider the contraction $g$ induced by a sufficiently divisible power of $-K_Y-D$ and we resolve it:

X'[swap,d,"f'"] [r,"π"]     X [d,"f"]

Y' [swap,d,"g'"] [r,"μ"]     Y [dl, dashed,"g"]

In the above diagram, $X'$ is the normalisation of the main component of $X \times_Y Y'$ and $f'$, $\mu$, $\pi$ are the induced maps.
We can choose $g'$ such that $f'$ is a sufficiently high equidimensional birational model of $f$ and $\Exc(\pi)\subseteq (f')^{-1}(\Exc(\mu))$.
Specifically, $f'$ is constructed by the following diagram

X' [rr,bend left=35,"π"] [r,"π_1"] [swap,d,"f'"]     X_1 [d,"f_1"] [r,"π_2"]     X[d,"f"]

Y'[rr,bend right=35,"μ"] [r,"μ_1"]     Y_1 [r,"μ_2"]     Y,

where $\mu_2$ is the normalised blowup of the base ideal of $|m(-K_Y-D)|$, $\pi_2$ is the normalisation of $(X\times_Y Y_1)_{\textup{main}}$ and $f'$ is an equidimensional model of $f_1$ such that every $\mu$-exceptional divisor is $\bQ$-Cartier.
Note that the condition on the exceptional loci is satisfied by $f_1$ by construction, and by $f'$ too, since $f_1$ is flat over codimension one points of $Y_1$ (<cit.>).

Let $h'\coloneqq g' \circ f'$.
Since the fibres of $g'$ and of $h'$ may be very singular, we consider the base change with a high enough power of the Frobenius morphism on $Z$:

X_e [r, "b"] [swap, d, "f_e "]     X'[d,"f'"] [r,"π"]     X [d,"f"]

Y_e [swap, d, "g_e "] [r, "a"]     Y' [d,"g'"] [r,"μ"]     Y [dl, dashed,"g"]

Z^e [r, "^e"]     Z,

(i) $X_e$ is the normalisation of the reduction of $X'_{Z^e}$;

(ii)  $Y_e$ is the normalisation of the reduction of $Y'_{Z^e}$;

(iii) $f_e$ and $g_e$ are the induced morphisms;

(iv) $a$ and $b$ are the induced morphisms;

(v) $h_e\coloneqq g_e \circ f_e$.

Choose $e \gg 0$ such that, if $\overline{\zeta}$ is the geometric generic point of $Z$, $X_{e,\overline{\zeta}}= (X_{\overline{\zeta},\mathrm{red}})^{\nu}$ and $Y_{e,\overline{\zeta}}= (Y_{\overline{\zeta},\mathrm{red}})^{\nu}$.
Such an $e$ exists by <ref>.
In particular, the general fibres of $f_e$, $g_e$ and $h_e$ are normal and reduced by <cit.>.

By easy additivity applied to $h_e$ we obtain

\begin{equation*}
\begin{split}
\kappa(X_e,(\pi\circ b)^*L) & \leq \kappa(X_{e,z},((\pi \circ b)^*L)|_{X_{e,z}})+\dim (Z)\\
& =\kappa(X_{e,z},((\pi \circ b)^*L)|_{X_{e,z}})+\kappa(Y,-K_Y-D),
\end{split}
\end{equation*}

where $X_{e,z}$ is a general fibre of $h_e$.
As <ref> implies $\kappa(X_e,(\pi \circ b)^*L)=\kappa(X,L)$, we just need to show

\begin{equation*}
\kappa(X_{e,z},((\pi \circ b)^*L)|_{X_{e,z}})\leq \kappa(X_y,(\pi^*L)|_{X_y}).
\end{equation*}

Note that, since $f$ is separable and its general fibre is normal by assumption, the general fibre of $f_e$ is isomorphic to $X_y$ and the same holds for the geometric generic fibres.
In particular, $X'_{\overline{\eta}}$ is reduced, where $\overline{\eta}$ is the geometric generic point of $Y$.
Note that $X_e$ can also be described as the normalisation of the reduction of $X' \times_{Y'} Y_e$.

Since $f'$ is equidimensional, we can apply <ref>: there exist $C_X$, $C_Y$ $\bQ$-divisors on $X_e$ and $Y_e$ respectively, and $C$, $\bQ$-divisor on $X_e$, such that

(1) $K_{X_e}+ C_X \sim b^*(K_{X'})$;

(2) $K_{Y_e}+ C_Y \sim a^*(K_{Y'})$;

(3) $K_{X_e/X'}- f_e^*K_{Y_e/Y'} \sim -C$;

(4) there exists $U \subseteq Z$ such that $C_X|_{X_{e,z}} \geq 0$, $C_Y|_{Y_{e,z}} \geq 0$ and $C|_{X_{e,z}} \geq 0$, where $X_{e,z}$ is a fibre of $h_e$ over any point $z \in U$ and $Y_{e,z}$ is a fibre of $g_e$ over any point $z \in U$.

In particular, for $X_{e,z}$ general fibre of $h_e$,

\[
(C_X-f_e^*C_Y)|_{X_{e,z}} \sim C|_{X_{e,z}} \geq 0.
\]

The goal now is to apply the injectivity <ref> to the contraction induced on the general fibres:

X_y [r] [d]     X_e,z [d,"f_X_e,z"] [r]     X_e[d,"f_e"]

{y }[r]     Y_e,z [d] [r]     Y_e [d,"g_e"]

{z }[r]     Z,

where $z$ is a general closed point of $Z$ and $y$ is a general closed point of $Y_{e,z}$.

In order to do so, we need to carefully define our divisors.
Define $K_{X'}+B'\coloneqq \pi^*(K_X+B)$; note that $B'$ is not necessarily effective.
However, we can find an effective $\mu$-exceptional divisor $\Theta'$ on $Y'$ such that $f'^*(\Theta')\geq (B')^-$. Define:

\begin{equation*}
\begin{split}
D'\coloneqq\mu^*(D)-\Theta'\hspace{3mm}\textup{and}\hspace{3mm}L' &\coloneqq-K_{X'}-(B')^+-f'^*(D')\\
\end{split}
\end{equation*}

so that $L'\geq \pi^*L$. Note that $L'$ is $\bQ$-Cartier since it differs from $\pi^*L$ only by exceptional divisors.
Moreover, we can choose $\Theta'$ so that $mL'$ is Cartier as well.
Now, let

\begin{equation*}
B_e\coloneqq b^*(B')^+ + C_X -f_e^*(C_Y); \hspace{3mm} D_e\coloneqq a^*(D') + C_Y \hspace{3mm}\textup{and}\hspace{3mm} L_e\coloneqq b^*(L'),
\end{equation*}

so that:

* $a^*(K_{Y'}+D')=K_{Y_e}+D_e$;

* $L_e=-K_{X_e}-B_e-f_e^*(D_e)$.

Note that $B_e|_{X_{e,z}} \geq 0$ by bullet $(4)$ in the above discussion.

Condition (a) in <ref> holds on $(X_y, B_{e,{X_y}})$.
Indeed, $\pi$ is an isomorphism over an open subset of $Y$ and the general fibres of $f'$ are isomorphic to the ones of $f_e$.
In particular, if $\overline{\eta}$ is the geometric generic point of $Y_e$ (or of $Y'$), $X_{e,\overline{\eta}} \simeq X'_{\overline{\eta}}$ and

\[
K_{X_{e,\overline{\eta}}} = (b^*(K_{X'})-C_X)|_{X_{e,\overline{\eta}}} = K_{X'_{\overline{\eta}}} -C_X|_{X_{e,\overline{\eta}}},
\]

whence $C_X|_{X_{e,\overline{\eta}}}=0$ and $B_{e,X_y}= B'_{X_y}=B_{X_y}$.

As for condition (b), we need to find $P \geq (B_e)^-_{X_{e,z}} = 0$ such that

\kappa(X_{e,z},(f_{X_{e,z}})^*(-K_{Y_{e,z}}-D_{e,Y_{e,z}})+P)=0.

We claim $P=0$ does the job.

If $\zeta$ is the generic point of $Z$, $\kappa(Y'_\zeta, \mu^*(-K_Y-D)|_{Y'_\zeta})=0$.
Indeed, if $k$ is uncountable, by construction of $g'$, $\kappa(Y'_z, \mu^*(-K_Y-D)|_{Y'_z})=0$, for the very general fibre $Y'_z$ of $g'$.
By <ref>, this implies $\kappa(Y'_\zeta, \mu^*(-K_Y-D)|_{Y'_\zeta})=0$.
If $k$ is countable, let $\overline{k} \supset k$ be a perfect uncountable extension of $k$ and $\overline{g'} \colon \overline{Y'} \to \overline{Z'}$ the base change of $g'$ with $\overline{k}$.
If $\mathcal{D}$ is a $\bQ$-divisor on $Y'$, denote by $\overline{\mathcal{D}}$ its base change with $\overline{k}$.
By flat base change, $H^0(Y', \mathcal{D}) \otimes_{k} \overline{k}= H^0(\overline{Y'}, \overline{\mathcal{D}})$.
By the above discussion, we can thus conclude that $\kappa(\overline{Y'_\zeta}, \overline{\mu^*(-K_Y-D)}|_{\overline{Y'_\zeta}})=0=\kappa(Y'_\zeta, \mu^*(-K_Y-D)|_{Y'_\zeta})$.

\begin{equation*}
\begin{split}
\mu^*(-K_Y-D)&=-K_{Y'}-\Xi'-\mu^*D\\
\end{split}
\end{equation*}

where $\Xi'$ is a $\mu$-exceptional divisor, not necessarily effective.
Note that, after possibly enlarging $\Theta'$ while keeping it $\mu$-exceptional, we can also assume $\Xi'+\Theta'\geq 0$ and $\mu$-exceptional.
Then the projection formula on $\mu$ yields that $\mu^*(-K_Y-D)$ and $-K_{Y'}-D'$ have the same sections, thus, by <ref>:

\begin{equation*}
\begin{split}
0=\kappa(Y'_\zeta,(\mu^*(-K_Y-D))|_{Y'_\zeta})= \kappa(Y'_\zeta, (-K_{Y'}-D')|_{Y'_{\zeta}})=\\
\kappa(Y_{e,\zeta},(a^*(-K_{Y'}-D'))|_{Y_{e,\zeta}})= \kappa(Y_{e,\zeta}, (-K_{Y_e}-D_e)|_{Y_{e,\zeta}}).
\end{split}
\end{equation*}

Since the general fibre of $g_e$ is normal and reduced, by <ref>, for the general fibre $Y_{e,z}$, we have:

\[
0= \kappa(Y_{e,\zeta},(-K_{Y_e}-D_e)|_{Y_{e,\zeta}})= \kappa(Y_{e,z}, (-K_{Y_e}-D_e)|_{Y_{e,z}}) = \kappa(Y_{e,z}, -K_{Y_{e,z}}-D_{e,Y_{e,z}}).
\]

Thus, we can apply <ref>, which yields the inequality:

\[
\kappa(X_{e,z},L_{e,X_{e,z}})\leq \kappa(X_y,L_{e,X_y}).
\]

Since $\pi$ is an isomorphism over the generic point of $Y$ and $b|_{X_y}$ is the identity morphism on the general fibre $X_y$ of $f_e$, we have $\kappa(X_y,L_{e,X_y})=\kappa(X_y, L'_{X_y})=\kappa(X_y,L_{X_y})$.
As $b^*L'\geq b^*\pi^*L$ we conclude

\[
\kappa(X_{e,z},((\pi \circ b)^*L)_{X_{e,z}})\leq \kappa(X_y,L_{X_y}).
\]


#  Comment on ‘Exact solution of the position-dependent effective mass and
angular frequency Schrödinger equation: harmonic oscillator model with
quantized confinement parameter’

C Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de
Bruxelles,
Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
E-mail<EMAIL_ADDRESS>
###### Abstract

In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 J. Phys. A
53 485301), a confined model of the non-relativistic quantum harmonic
oscillator, where the effective mass and the angular frequency are dependent
on the position, was constructed and it was shown that the confinement
parameter gets quantized. By using a point canonical transformation starting
from the constant-mass Schrödinger equation for the Rosen-Morse II potential,
it is shown here that similar results can be easily obtained without
quantizing the confinement parameter. In addition, an extension to a confined
shifted harmonic oscillator directly follows from the same point canonical
transformation.

Keywords: Schrödinger equation, position-dependent mass and angular frequency,
harmonic oscillator, point canonical transformation, Rosen-Morse II potential

PACS Nos.: 03.65.Fd, 03.65.Ge

In an interesting paper [1], Jafarov et al presented an exact solution of a
confined model of the non-relativistic quantum harmonic oscillator, where the
effective mass and the angular frequency are dependent on the position. In
their model, the kinetic energy operator has the BenDaniel-Duke form [2] and
the position-dependency of the mass and the angular frequency is such that the
homogeneous nature of the harmonic oscillator force constant
$k=M(x)\omega^{2}(x)=m_{0}\omega_{0}^{2}$ and hence the regular harmonic
oscillator potential are preserved, thus leading to the Schrödinger equation
111For simplicity’s sake, we use here units wherein $\hbar=2m_{0}=1$.

$\left(-\frac{d}{dx}\frac{1}{M(x)}\frac{d}{dx}+V_{\rm
eff}(x)\right)\psi_{n}(x)=E_{n}\psi_{n}(x)$ (1)

with

$M(x)=\left(1-\frac{x^{2}}{a^{2}}\right)^{-2},\qquad V_{\rm
eff}(x)=\frac{1}{4}\omega_{0}^{2}x^{2},\qquad-a<x<a.$ (2)

The equation is solved by reducing it to the general Legendre equation. As a
consequence, a quantization of the confinement parameter $a$ is observed,

$a=a_{l}=\sqrt{\frac{2}{\omega_{0}}}\,[l(l+1)-2]^{1/4},\qquad l=2,3,\dots,$
(3)

the bound-state wavefunctions being expressed in terms of associated Legendre
polynomials or, equivalently, Gegenbauer polynomials,

$\displaystyle\psi_{n}(x)=N_{n}\left(1-\frac{x^{2}}{a^{2}}\right)^{(l-n-1)/2}C_{n}^{\left(l-n+\frac{1}{2}\right)}\left(\frac{x}{a}\right),$
(4) $\displaystyle
N_{n}=\frac{(2l-2n)!}{2^{l-n}(l-n)!}\left(\frac{(l-n)n!}{a(2l-n)!}\right)^{1/2},$
(5)

with the corresponding $E_{n}$ given by

$E_{n}=\omega_{0}\sqrt{1+\left(\frac{3}{\omega_{0}a^{2}}\right)^{2}}\,\left(n+\frac{1}{2}\right)-\frac{1}{a^{2}}\left(n+\frac{1}{2}\right)^{2}-\frac{5}{4a^{2}},\qquad
n=0,1,\ldots,l-2.$ (6)

The purpose of the present comment is to show that an alternative derivation
of the solution consists in applying a point canonical transformation (PCT) to
the well-known constant-mass Schrödinger equation for the Rosen-Morse II
potential [3, 4, 5, 6]. Such an approach has the advantage of providing us
with a direct generalization of the results.

In the PCT approach to position-dependent mass (PDM) Schrödinger equations [7,
8], one starts from a constant-mass Schrödinger equation

$\left(-\frac{d^{2}}{du^{2}}+U(u)\right)\phi_{n}(u)=\epsilon_{n}\phi_{n}(u),$
(7)

for some potential $U(u)$, defined on a finite or infinite interval, and one
transforms it into an equation such as (1) by making a change of variable

$u(x)=\bar{a}v(x)+\bar{b},\qquad
v(x)=\int^{x}\sqrt{M(x^{\prime})}\,dx^{\prime},$ (8)

and a change of function

$\phi_{n}(u(x))\propto[M(x)]^{-1/4}\psi_{n}(x).$ (9)

Here, $\bar{a}$ and $\bar{b}$ are assumed to be two real parameters. The
potential $V_{\rm eff}(x)$, defined on a possibly different interval, and the
energy eigenvalues $E_{n}$ of the PDM Schrödinger equation are given in terms
of the potential and the energy eigenvalues of the constant-mass one by

$V_{\rm
eff}(x)=\bar{a}^{2}U(u(x))+\frac{M^{\prime\prime}}{4M^{2}}-\frac{7M^{\prime
2}}{16M^{3}}+\bar{c}.$ (10)

and

$E_{n}=\bar{a}^{2}\epsilon_{n}+\bar{c},$ (11)

where a prime denotes derivation with respect to $x$ and $\bar{c}$ is some
additional real constant.

In equation (7), let us assume that $U(u)$ is the Rosen-Morse II potential

$U(u)=-A(A+1)\operatorname{sech}^{2}u+2B\tanh u,\qquad-\infty<u<\infty,\qquad
B<A^{2}.$ (12)

Then the corresponding $\epsilon_{n}$ and $\phi_{n}(u)$ are given by

$\epsilon_{n}=-(A-n)^{2}-\frac{B^{2}}{(A-n)^{2}},\qquad n=0,1,\ldots,n_{\rm
max},\qquad n_{\rm max}<A-\sqrt{|B|},$ (13)

and

$\displaystyle\phi_{n}(u)$ $\displaystyle={\cal N}_{n}(1-\tanh
u)^{\left(A-n+\frac{B}{A-n}\right)/2}(1+\tanh
u)^{\left(A-n-\frac{B}{A-n}\right)/2}$ $\displaystyle\quad\times
P_{n}^{\left(A-n+\frac{B}{A-n},A-n-\frac{B}{A-n}\right)}(\tanh u),$ (14)

with $P_{n}^{(\alpha,\beta)}(z)$ denoting a Jacobi polynomial and ${\cal
N}_{n}$ a normalization coefficient given by [6]

${\cal
N}_{n}=2^{n-A}\left(\frac{n!\Gamma(2A-n+1)[(A-n)^{2}-B^{2}/(A-n)^{2}]}{(A-n)\Gamma(A+1+B/(A-n))\Gamma(A+1-B/(A-n))}\right)^{1/2}$
(15)

and ensuring that

$\int_{-\infty}^{+\infty}|\phi_{n}(u)|^{2}=1.$ (16)

To start with, let us consider the special case where $B=0$ in equation (12).
Then, for $\epsilon_{n}=-(A-n)^{2}$, $n=0$, 1, …, $n_{\rm max}$, $n_{\rm
max}<A$, the wavefunctions $\phi_{n}(u)$ can be expressed in terms of
Gegenbauer polynomials as

$\phi_{n}(u)=\frac{\Gamma(2A-2n+1)}{2^{A-n}\Gamma(A-n+1)}\left(\frac{(A-n)n!}{\Gamma(2A-n+1)}\right)^{1/2}(\operatorname{sech}u)^{A-n}\,C_{n}^{\left(A-n+\frac{1}{2}\right)}(u).$
(17)

by taking into account equation (22.5.20) of Ref. [9], expressing Jacobi
polynomials with equal parameters in terms of Gegenbauer ones.

For the PDM $M(x)$ defined in equation (2), we get

$v(x)=a\operatorname{arctanh}\frac{x}{a}$ (18)

and

$\frac{M^{\prime\prime}}{4M^{2}}-\frac{7M^{\prime
2}}{16M^{3}}=-\frac{2x^{2}}{a^{4}}+\frac{1}{a^{2}}.$ (19)

With the choice

$\bar{a}=\frac{1}{a},\qquad\bar{b}=0$ (20)

for the constants appearing in equation (8), we obtain for the change of
variable

$u(x)=\operatorname{arctanh}\frac{x}{a}.$ (21)

On selecting

$\bar{c}=\frac{1}{4}\omega_{0}^{2}a^{2}+\frac{1}{a^{2}},$ (22)

the transformed potential (10) becomes

$V_{\rm
eff}(x)=\frac{1}{a^{4}}[A(A+1)-2]x^{2}-\frac{1}{a^{2}}[A(A+1)-2]+\frac{1}{4}\omega_{0}^{2}a^{2}$
(23)

and reduces to that given in equation (2) by choosing

$a=\sqrt{\frac{2}{\omega_{0}}}\,[A(A+1)-2]^{1/4}$ (24)

provided $A(A+1)>2$ or, in other words, $A>1$. From equations (9) and (17),
the corresponding bound-state wavefunctions, normalized on $(-a,+a)$, can be
written as

$\displaystyle\psi_{n}(x)$
$\displaystyle=\frac{\Gamma(2A-2n+1)}{2^{A-n}\Gamma(A-n+1)}\left(\frac{(A-n)n!}{a\Gamma(2A-n+1)}\right)^{1/2}\left(1-\frac{x^{2}}{a^{2}}\right)^{(A-n-1)/2}$
$\displaystyle\quad\times
C_{n}^{\left(A-n+\frac{1}{2}\right)}\left(\frac{x}{a}\right),$ (25)

with $E_{n}$ obtained from (11) as

$\displaystyle E_{n}$
$\displaystyle=-\frac{1}{a^{2}}(A-n)^{2}+\frac{1}{4}\omega_{0}^{2}a^{2}+\frac{1}{a^{2}}$
$\displaystyle=\omega_{0}\sqrt{1+\left(\frac{3}{\omega_{0}a^{2}}\right)^{2}}\left(n+\frac{1}{2}\right)-\frac{1}{a^{2}}\left(n+\frac{1}{2}\right)^{2}-\frac{5}{4a^{2}},$
$\displaystyle\quad n=0,1,\ldots,n_{\rm max},\qquad n_{\rm max}<A-1.$ (26)

Note that the condition $n=0,1,\ldots,n_{\rm max}$, $n_{\rm max}<A$, which was
enough to make the functions $\phi_{n}(u)$ of equation (17) normalizable on
$(-\infty,+\infty)$, has been restricted to $n=0,1,\ldots,n_{\rm max}$,
$n_{\rm max}<A-1$, which is imposed by the normalizability condition of
$\psi_{n}(x)$ on $(-a,+a)$.

We conclude that the results of Ref. [1] can be derived by a PCT applied to
the constant-mass Schrödinger equation for the Rosen-Morse II potential with
$B=0$ and any $A=l=2,3,\ldots$. However, no quantization of $A$, and hence of
$a$, is necessary to get valid bound-state wavefunctions: for any positive $A$
greater than one, wa have shown that equation (1) with $M(x)$ and $V_{\rm
eff}(x)$ given in equation (2) has a finite number of bound-state
wavefunctions (25).

Let us now consider the general Rosen-Morse II potential with $B\neq 0$. On
proceeding as above, except for changing $\bar{c}$ of equation (22) into

$\bar{c}=\frac{1}{4}\omega_{0}^{2}a^{2}+\frac{1}{a^{2}}+b^{2},$ (27)

after setting $B=-\frac{1}{2}\omega_{0}a^{3}b$, we get

$V_{\rm
eff}(x)=\frac{1}{4}\omega_{0}^{2}\left(x-\frac{2b}{\omega_{0}}\right)^{2},\qquad-a<x<a,$
(28)

with $a$ still given by equation (24). The confined harmonic oscillator of
equation (2) is therefore replaced by a confined shifted harmonic oscillator.
The latter has a finite number of bound-state wavefunctions

$\displaystyle\psi_{n}(x)$
$\displaystyle=N_{n}\left(1-\frac{x}{a}\right)^{\left(A-n-1-\frac{\omega_{0}a^{3}b}{2(A-n)}\right)/2}\left(1+\frac{x}{a}\right)^{\left(A-n-1+\frac{\omega_{0}a^{3}b}{2(A-n)}\right)/2}$
$\displaystyle\quad\times
P_{n}^{\left(A-n-\frac{\omega_{0}a^{3}b}{2(A-n)},A-n+\frac{\omega_{0}a^{3}b}{2(A-n)}\right)}\left(\frac{x}{a}\right),$
(29) $\displaystyle N_{n}$
$\displaystyle=2^{n-A}\Bigl{(}n!\Gamma(2A-n+1)\\{(A-n)^{2}-\omega_{0}^{2}a^{6}b^{2}/[4(A-n)^{2}]\\}\Bigr{)}^{1/2}$
$\displaystyle\quad\times\Bigl{(}a(A-n)\Gamma(A+1-\omega_{0}a^{3}b/[2(A-n)])\Gamma(A+1+\omega_{0}a^{3}b/[2(A-n)])\Bigr{)}^{-1/2},$
(30)

with $n=0$, 1, …, $n_{\rm max}$, $n_{\rm
max}<A-\frac{1}{2}\left(1+\sqrt{1+2\omega_{0}a^{3}|b|}\right)$, provided $b$
satisfies the condition $\sqrt{1+2\omega_{0}a^{3}|b|}<2A-1$, which on taking
(24) into account, amounts to

$|b|<\sqrt{\frac{\omega_{0}}{2}}\frac{A(A-1)}{[A(A+1)-2]^{3/4}}.$ (31)

The corresponding bound-state energies, as obtained from equation (11), are
given by

$E_{n}=\omega_{0}\sqrt{1+\left(\frac{3}{\omega_{0}a^{2}}\right)^{2}}\left(n+\frac{1}{2}\right)-\frac{1}{a^{2}}\left(n+\frac{1}{2}\right)^{2}-\frac{5}{4a^{2}}+b^{2}\frac{g(n)}{f(n)},$
(32)

where

$\displaystyle f(n)$
$\displaystyle=\left(n+\frac{1}{2}\right)^{2}-\omega_{0}a^{2}\sqrt{1+\left(\frac{3}{\omega_{0}a^{2}}\right)^{2}}\left(n+\frac{1}{2}\right)+\frac{1}{4}\omega_{0}^{2}a^{4}+\frac{9}{4},$
(33) $\displaystyle g(n)$ $\displaystyle=f(n)-\frac{1}{4}\omega_{0}^{2}a^{4}.$
(34)

In conclusion, we have shown that the confined harmonic oscillator with
position-dependent mass and angular frequency of Ref. [1] can be extended to
the case where the confinement parameter is not quantized, its spectrum
remaining finite, non-equidistant, and dependent on the confinement parameter.
A further extension to a confined shifted harmonic oscillator with similar
characteristics has also been constructed.

This work was supported by the Fonds de la Recherche Scientifique - FNRS under
Grant Number 4.45.10.08.

## References

  * [1] Jafarov E I, Nagiyev S M, Oste R and Van der Jeugt J 2020 J. Phys. A: Math. Theor. 53 485301
  * [2] BenDaniel D J and Duke C B 1966 Phys. Rev. 152 683
  * [3] Rosen N and Morse P M 1932 Phys. Rev. 42 210
  * [4] Lévai G 1989 Phys. Rev. A: Math. Gen. 22 689
  * [5] Cooper F, Khare A and Sukhatme U 1995 Phys. Rep. 251 267
  * [6] Lévai G and Magyari E 2009 J. Phys. A: Math. Theor. 42 195302
  * [7] Bagchi B, Gorain P, Quesne C and Roychoudhury R 2004 Mod. Phys. Lett. A 19 2765
  * [8] Quesne C 2009 SIGMA 5 046
  * [9] Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions (New York: Dover)