2.1 Nef reduction map

In this subsection, we recall the notion of a nef reduction map.

1. (1) $\varphi|_{U}: U \to V$ is proper and $\varphi_{\ast}{\mathcal O}_{U} = {\mathcal O}_{V}$ ;

2. (2) $L|_{F}\equiv 0$ for all fibers F of $\varphi$ over V; and

3. (3) if $x \in X$ is a very general point and C is a curve passing through it, then $C\cdot L = 0$ if and only if C is contracted by $\varphi$ .

It is proved that the nef reduction map exists over an uncountable algebraically closed field.

We call $n(X,L)= \operatorname{dim} Z$ the nef dimension of L, where Z is the target of a nef reduction map $\varphi: X \dashrightarrow Z$ . When the base field is countable and algebraically closed, we can define $n(X,L):=n(X_{K},L_{K})$ by [Reference WitaszekWit21, Proposition 2.16], where K is an uncountable algebraically closed field that contains k, and $X_{K}$ and $L_{K}$ are the base changes of X and L to K. Then $\kappa(X, L)\leq n(X, L)$ is satisfied.

2.2 Surface singularities and semi-log canonical surfaces

In this subsection, we recall the notion of the dual graph of a surface singularity and collect some results on lc surface singularities and semi-log canonical (slc) surfaces. The main references for this subsection are [Reference KollárKol13, Reference SatoSat25]. First we recall some basic knowledge on the intersection theory for curves over a field, which is not necessarily perfect. Let C be an integral projective scheme over a field k with $\operatorname{dim} C =1$ (not necessarily regular).

Definition 2.3. We define the arithmetic genus g(C) of C by

\begin{align*}g(C):=\frac{\mathrm{dim}_{k}(H^{1}(C,{\mathcal O}_{C}))}{\mathrm{dim}_{k}(H^{0}(C,{\mathcal O}_{C}))}.\end{align*}

For a 0-cycle $D =\sum_{i=1}^{n} a_{i}P_{i}$ on C, we define the degree of D over k by

\begin{align*}\mathrm{deg}_{C/k}(D):=\sum_{i=1}^{n}a_{i} \operatorname{dim}_{k}(k(P_{i}))\in {\mathbb Z},\end{align*}

where $k(P_{i})$ are the residue fields of C at $P_{i}$ . Note that any line bundle $L\in \mathrm{Pic}(C)$ is isomorphic to ${\mathcal O}_C(B)$ for some 0-cycle B on C. We define

\begin{align*}\mathrm{deg}_{C/k}(L):=\mathrm{deg}_{C/k}(B).\end{align*}

Since rationally equivalent cycles have the same degree, $\mathrm{deg}_{C/k}(L)$ is well-defined. The map $\mathrm{deg}_{C/k}$ gives the homomorphism

\begin{align*}\mathrm{deg}_{C/k}:\mathrm{Pic}(C)\to {\mathbb Z}.\end{align*}

See [Reference FultonFul98, § 1.4] for details.

Let X be a normal integral scheme with a dualizing complex $\omega_{X}^{\bullet}$ . Then we can define resolutions of singularities (desingularizations for short), the canonical divisor $K_{X}$ , and lc (klt, plt) pairs as usual (see [Reference SatoSat25, § 2] for example).

Let (X,x) be a normal surface singularity, and let $f: Y\to X$ be a proper birational morphism from a normal integral scheme Y with the exceptional divisor $\mathrm{Exc}(f)= \sum_{i}E_{i}$ satisfying $f(\mathrm{Exc}(f))=x$ . Note that $E_{i}$ are defined over the residue field k(x) of X at x.

Definition 2.6. For a Cartier divisor D on Y and an exceptional Weil divisor $Z =\sum_{i}b_{i}E_{i}$ on Y, we define the intersection number $D \cdot Z$ by

\begin{align*}D \cdot Z:=\sum_{i}b_{i} \operatorname{deg}_{E_{i}/k(x)}({\mathcal O}_{Y}(D)|_{E_{i}}).\end{align*}

By [Reference SatoSat25, Definition 2.11 and Remark 2.12], the minimal resolution of a normal surface singularity (X,x) can be defined, and it exists uniquely. Now we can give the definition of the dual graph as follows.

Definition 2.7. Let (X,x) be a normal surface singularity, and $f: Y\to X$ be the minimal resolution with the exceptional divisor $\mathrm{Exc}(f) = \sum_{i=1}^{n} E_{i}$ satisfying $f(\mathrm{Exc}(f))=x$ . The dual graph of (X,x) is a graph whose set of vertices is $\{E_{1}, \ldots , E_{n}\}$ , the number of edges between $E_{i}$ and $E_{j}$ is $(E_{i}\cdot E_{j} )\in {\mathbb N}$ , and the weight at $E_{i}$ is

\begin{align*}(\mathrm{dim}_{k(x)} H^{0}(E_{i}, {\mathcal O}_{E_{i}}), g(E_{i}),E_{i}^{2})\in {\mathbb Z}^{3}.\end{align*}

Definition 2.9 [Reference KollárKol13, Corollary 3.31]. Let S be a two-dimensional normal integral scheme with a dualizing complex, $s\in S$ a closed point, and B a reduced Weil divisor on S. Assume that $(s\in S,B)$ is lc and let $(\Gamma,B)$ be its extended dual graph. We denote the residue field of S at s by k(s). We say that $(s\in S,B)$ is a cyclic quotient singularity if $(\Gamma,B)$ is

\begin{align*}\bullet\,\rule[2.7pt]{1cm}{0.05em}\circ\rule[2.7pt]{1cm}{0.05em}\cdots\rule[2.7pt]{1cm}{0.05em}\circ\quad\mathrm{or}\quad \circ\rule[2.7pt]{1cm}{0.05em}\circ\rule[2.7pt]{1cm}{0.05em}\cdots \rule[2.7pt]{1cm}{0.05em}\,\circ,\end{align*}

where the solid bullet stands for the strict transform of B and every circle stands for an exceptional curve over S. Moreover, every exceptional curve is a ${\mathbb P}^1_{k(s)}$ , except in the case $B=0$ with only one exceptional curve, which may instead be a k(s)-irreducible conic.

We use the following result in Corollary 3.8 and § 5.

Proposition 2.10. Let S be a two-dimensional normal integral scheme with a dualizing complex, $s\in S$ a closed point, and B a reduced Weil divisor on S. Assume that $(s\in S,B)$ is lc and let $(\Gamma,B)$ be its extended dual graph. If $(s\in S,B)$ is plt and $s\in B\neq 0$ , then $(\Gamma,B)$ is

\begin{align*}\bullet\,\rule[2.7pt]{1cm}{0.05em}\circ\rule[2.7pt]{1cm}{0.05em}\cdots\rule[2.7pt]{1cm}{0.05em}\,\circ,\end{align*}

where the solid bullet stands for the strict transform of B, every circle stands for ${\mathbb P}^{1}_{k(s)}$ , and k(s) is the residue field of S at s. In particular, $(s\in S,B)$ is a cyclic quotient singularity.

We recall the following adjunction formula of lc surface singularities.

Theorem 2.11 [Reference KollárKol13, Theorem 3.36]. Let S be a two-dimensional normal integral scheme with a dualizing complex, and let $B\subset S$ be a one-dimensional reduced proper scheme over a field such that (S,B) is lc. Then

\begin{align*}(K_{S}+B)\cdot B=\operatorname{deg} \omega_{B}+\sum_{s\,\mathrm{cyclic,\, plt}}\bigg(1-\frac{1}{\mathrm{det}(\Gamma_{s})}\bigg)\mathrm{deg}[s]+\sum_{s\, \mathrm{dihedral}}\mathrm{deg}[s],\end{align*}

where $\mathrm{det}(\Gamma_{s})$ are the determinants of the dual graphs $\Gamma_{s}$ of cyclic quotient singularities (S,s).

Remark 2.12. We also call $\mathrm{det}(\Gamma_{s})$ the index of (S,s). By [Reference KollárKol13, Theorem 3.32] for a cyclic quotient singularity (S,s), if the residue field k(s) of S at s is algebraically closed and S is defined over k(s), then the completion of S at s is isomorphic to

\begin{align*}\widehat{\mathbb{A}}_{k(s)}^{2}/\mu_{m},\end{align*}

where $\mu_{m}$ is the group scheme $\mathrm{Spec}(k(s)[{\mathbb Z}_{m}])$ . By [Reference KollárKol13, Definition 3.33], we have $\mathrm{det}(\Gamma_{s})=m$ . Moreover, if m is not divisible by the characteristic of k(s), then by [Reference KollárKol13, 3.19], the action of $\mu_{m}\cong {\mathbb Z}_{m}$ on $\widehat{\mathbb{A}}_{k(s)}^{2}$ is linear and diagonalized.

Now we turn to the notion of slc surfaces.

Definition 2.13. Let X be a scheme. A coherent ${\mathcal O}_{X}$ -module ${\mathcal F}$ is $S_{i}$ , where i is a positive integer if it satisfies the condition

\begin{align*}\mathrm{depth}_{{\mathcal O}_{X,x} {\mathcal F}_{x}} \geq \mathrm{min}\{i,\mathrm{dim}(\operatorname{Supp} {\mathcal F}_{x})\}\end{align*}

for all $x \in X$ . We say that X is $S_{i}$ if ${\mathcal O}_{X}$ is $S_{i}$ .

1. (1) $X^{\prime}$ is $S_{2}$ and reduced;

2. (2) g is finite and an isomorphism precisely above U; and

3. (3) the normalization $X^{n}\to X$ factorizes through g.

We recall the following Riemann–Roch theorem for demi-normal surfaces.

Theorem 2.16. Let X be a demi-normal surface over an algebraically closed field. Let L be a Cartier divisor on X. Then we have

\begin{align*}\chi({\mathcal O}_{X}(L))=\tfrac{1}{2}L\cdot(L-K_{X})+\chi({\mathcal O}_{X}).\end{align*}

Proof. In characteristic 0, this is proved in [Reference Liu and RollenskeLR16, Theorem 3.1]. Assume that X is defined over an algebraically closed field of arbitrary characteristic. Let $\pi:\overline{X}\to X$ be the normalization of X, and let $D\subseteq X$ and $\overline D\subseteq \overline{X}$ be conductor subschemes. By the proof of [Reference Liu and RollenskeLR16, Theorem 3.1], it suffices to show that

\begin{align*}\chi({\mathcal O}_X)=\chi({\mathcal O}_{\overline X})+\chi({\mathcal O}_D)-\chi({\mathcal O}_{\overline{D}})\end{align*}

and

\begin{align*}\chi({\mathcal O}_X(L))=\chi(\pi^\ast{\mathcal O}_X(L))+\chi({\mathcal O}_X(L)_{|_D})-\chi(\pi^\ast{\mathcal O}_X(L)_{|_{\overline{D}}}).\end{align*}

We consider the exact sequences

\begin{align*}0\to {\mathcal O}_{\overline X}(-\overline{D})\to {\mathcal O}_{\overline X}\to {\mathcal O}_{\overline{D}}\to 0\end{align*}

and

\begin{align*}0\to {\mathcal I}_{D}\to {\mathcal O}_X\to {\mathcal O}_{D}\to 0.\end{align*}

By the definition of conductor subschemes [Reference KollárKol13, 5.2], we have $\pi_\ast {\mathcal O}_{\overline X}(-\overline{D})\cong {\mathcal I}_{D}.$ It follows that

\begin{align*}\chi({\mathcal O}_X)-\chi({\mathcal O}_D)=\chi({\mathcal I}_{D})=\chi(\pi_\ast {\mathcal O}_{\overline X}(-\overline{D}))=\chi({\mathcal O}_{\overline X})-\chi({\mathcal O}_{\overline{D}}).\end{align*}

Similarly, we have

\begin{align*}\chi({\mathcal O}_X(L))-\chi({\mathcal O}_X(L)_{|_D})=\chi(\pi^\ast{\mathcal O}_X(L))-\chi(\pi^\ast{\mathcal O}_X(L)_{|_{\overline{D}}}).\end{align*}

Therefore, the assertion holds.

The abundance for slc surfaces over any field of positive characteristic holds.

In characteristic 0, the reduced part of the boundary of an lc pair of dimension 3 is slc. It fails in positive characteristic, despite several partial positive results (see [Reference ArvidssonArv23, Reference Arvidsson and PosvaAP26, Reference Arvidsson, Bernasconi and PatakfalviABP24] for example). However, we have the following analogous result.

2.3 MMP for threefolds in positive characteristic

In this subsection, we recall the theory of MMP for projective lc pairs of dimension 3 over a perfect field of characteristic $p\gt 3$ . Moreover, we define a partial MMP over algebraically closed fields of characteristic $p\gt 3$ (see Definition 2.23) and we will use this construction to study the abundance in § 5.

We recall the notion of MMP with scaling. Let (X,B) be a projective lc pair of dimension 3 over a perfect field k of characteristic $p\gt3$ and $A\neq 0$ an effective ${\mathbb Q}$ -Cartier ${\mathbb Q}$ -divisor on X. Suppose that there is $t_{0}\gt 0$ such that $(X,B+ t_{0}A)$ is lc and $K_{X}+B+t_{0}A$ is nef. We describe how to run a $(K_{X}+B)$ -MMP with scaling of A as follows.

Let $\lambda_{0}=\mathrm{inf} \{t: K_{X}+B+tA\ \mathrm{is}\ \mathrm{nef} \}$ . Suppose we can find a $(K_{X}+ B)$ -negative extremal ray $R_{0}$ which satisfies $(K_{X}+B+\lambda_{0}A)\cdot R_{0}= 0$ . (In general, it is possible that there is no such extremal ray.) This is the first ray we contract in our MMP. If the contraction is a Mori fiber contraction, we stop. Otherwise, let $X_{1}$ be the result of the divisorial contraction or flip. Then $K_{X_{1}}+ B_{X_{1}}+\lambda_{0}A_{X_{1}}$ is also nef, where $B_{X_{1}}$ and $A_{X_{1}}$ denote the birational transforms on $X_{1}$ of B and A, respectively. We define $\lambda_{1} =\mathrm{inf} \{t: K_{X_{1}}+ B_{X_{1}} + tA_{X_{1}}\\mathrm{is}\ \mathrm{nef} \}$ . The next step in our MMP is chosen to be a $(K_{X_{1}} + B_{X_{1}})$ -negative extremal ray $R_{1}$ which is $(K_{X_{1}}+B_{X_{1}}+ \lambda_{1} A_{X_{1}})$ -trivial. So long as we can always find the appropriate extremal rays, contractions, and flips, we can continue this process.

1. (1) there is a $(K_{X}+B)$ -negative extremal ray which is $(K_{X}+B+W)$ -trivial; or

2. (2) $K_{X}+B+(1-\varepsilon)W$ is nef for any sufficiently small rational $\varepsilon\gt0$ .

In this paper, we use the following construction.

Let $\lambda_{0}=\mathrm{inf} \{t: K_{X}+B+tA\ \mathrm{is}\ \mathrm{nef} \}$ . If $\lambda_{0}\lt1$ , then we stop. Otherwise, by Proposition 2.21 there exists a $(K_{X}+ B)$ -negative extremal ray $R_{0}$ which satisfies $(K_{X}+B+A)\cdot R_{0}= 0$ . We contract this extremal ray. If the contraction is a Mori fiber contraction, we stop. Otherwise, let $\mu_{0}: X\dashrightarrow X_{1}$ be the divisorial contraction or flip. Repeat this process for $(X_{1},\mu_{0\ast}B)$ , $\mu_{0\ast}A$ , and so on.

We call this construction a $(K_{X}+B)$ -MMP which is $(K_{X}+B+A)$ -trivial.

The following lemma tells us what the output of this construction is if it terminates.

If a $(K_{X}+B)$ -MMP which is $(K_{X}+B+A)$ -trivial terminates, then its output is a ${\mathbb Q}$ -factorial projective lc pair $(X^{\prime}, B^{\prime}+A^{\prime})$ , and either:

1. (1) $X^{\prime}$ is a Mori fiber space over a variety Y, $K_{X^{\prime}}+B^{\prime}+A^{\prime}$ is the pullback of a ${\mathbb Q}$ -Cartier ${\mathbb Q}$ -divisor from Y, and $\operatorname{Supp} A^{\prime}$ dominates Y; or

2. (2) $K_{X^{\prime}}+B^{\prime}+(1-\varepsilon)A^{\prime}$ is nef for any sufficiently small rational $\varepsilon\gt0$ .

Moreover, $K_{X^{\prime}}+B^{\prime}+A^{\prime}$ is semi-ample if and only if $K_{X}+B+A$ is.

Proof. By Definition 2.23, if a $(K_{X}+B)$ -MMP which is $(K_{X}+B+A)$ -trivial does not terminate with a Mori fiber space, then we get a ${\mathbb Q}$ -factorial projective lc pair $(X^{\prime}, B^{\prime}+A^{\prime})$ such that

\begin{align*}\lambda:=\mathrm{inf} \{t: K_{X^{\prime}}+B^{\prime}+tA^{\prime}\ \mathrm{is}\ \mathrm{nef} \} \lt1.\end{align*}

That is to say $K_{X^{\prime}}+B^{\prime}+(1-\varepsilon)A^{\prime}$ is nef for any sufficiently small rational $\varepsilon\gt0$ . We are in case (2) above. If the $(K_{X}+B)$ -MMP which is $(K_{X}+B+A)$ -trivial terminates with a Mori fiber space $f:(X^{\prime},B^{\prime}+A^{\prime})\to Y$ , then, by [Reference Hashizume, Nakamura and TanakaHNT17, Theorem 1.4] and [Reference Hacon and WitaszekHW22b], $K_{X^{\prime}}+B^{\prime}+A^{\prime}$ is the pullback of a ${\mathbb Q}$ -Cartier ${\mathbb Q}$ -divisor from Y since f contracts a $(K_{X^{\prime}}+B^{\prime}+A^{\prime})$ -trivial extremal ray. Moreover, $\operatorname{Supp} A^{\prime}$ dominates Y since f only contracts curves which have positive intersections with $A^{\prime}$ . We are in case (1). The last assertion holds since $K_{X^{\prime}}+B^{\prime}+A^{\prime}$ and $K_{X}+B+A$ coincide after being pulled back to a common resolution of X’ and X.

We use the following results on termination of flips.

Proof of Lemma 2.26. Since every step of a $K_{X}$ -MMP which is $(K_{X}+B)$ -trivial is a step of a $K_{X}$ -MMP, the assertion follows from [Reference Kollár and MoriKM98, Theorem 6.17].

2.4 Divisorial log terminal modifications

The following result helps us to reduce some problems for lc pairs to ${\mathbb Q}$ -factorial divisorial log terminal (dlt) pairs.

Proof. The first assertion is proven in the case when $p\gt5$ in [Reference BirkarBir16, Theorem 1.6], and the proof there holds in the case when $p=5$ by [Reference Hacon and WitaszekHW22b]. Let us prove that we can make X terminal. We take a crepant ${\mathbb Q}$ -factorial dlt model $g: (X^{\prime},B^{\prime})\to (X,B)$ by the first assertion. Hence, by replacing (X, B) by $(X^{\prime},B^{\prime})$ , we may assume that (X, B) is ${\mathbb Q}$ -factorial and dlt. Let $U \subseteq X$ be the largest open set such that $(U, B|_{U})$ is a simple normal crossing (snc) pair. Then $\mathrm{codim}_{ X} (X\setminus U) \geq 2$ . Note that (X, 0) is klt. By [Reference BirkarBir16, Theorem 1.7] we take a terminal model $f : (X^{\prime}, \Theta^{\prime})\to (X, 0)$ of (X, 0) such that $K_{X^{\prime}} + \Theta^{\prime}= f^{\ast}K_{X}$ . Then f is an isomorphism over the smooth locus of X; in particular, f is an isomorphism over U. Let $Z = X\setminus U$ . Define $B^{\prime}:= \Theta^{\prime}+ f^{\ast}B$ on $X^{\prime}$ so that

\begin{align*}K_{X^{\prime}} + B^{\prime} = f^{\ast}(K_{X}+ B)\end{align*}

and $(X^{\prime}, B^{\prime})$ is lc.

It remains to show that $(X^{\prime}, B^{\prime})$ is a dlt pair. Let $U^{\prime} = f^{-1}(U)$ and $Z^{\prime} = X^{\prime}\setminus U^{\prime}$ . Then $(U^{\prime}, B^{\prime}|_{U^{\prime}})$ is a snc pair. If E is an exceptional divisor with center in $Z^{\prime}$ , then its center in X is contained in Z. Hence $a(E; X^{\prime}, B^{\prime})= a(E; X, B) \gt -1$ . This completes the proof.

2.5 Some known results on the abundance

The following theorem collects recent results toward the abundance conjecture in positive characteristics. For condition (1) see [Reference WaldronWal17, Theorem 1.3], [Reference ZhangZha19, Theorem 3.1], [Reference Das and WaldronDW19, Theorem A], and [Reference XuXu24, Theorem 5.1]. For (2) see [Reference WitaszekWit21, Theorem 5] and [Reference XuXu24, Theorem 6.2]. For (3) see [Reference ZhangZha20, Theorem 1.1], [Reference WitaszekWit21, Corollary 4.13], and [Reference XuXu24, Theorem 6.3].

1. (1) $\kappa(X,K_{X}+B)\geq 1$ ;

2. (2) the nef dimension $n(X,K_{X}+B)\leq 2$ ; or

3. (3) the Albanese map $a_{X}: X\to\mathrm{Alb}(X)$ is non-trivial.

Then $K_{X}+ B$ is semi-ample.

Moreover, the non-vanishing theorem for threefolds in characteristic $p\gt3$ has been proved. See [Reference Xu and ZhangXZ19, Theorem 1.1], [Reference WitaszekWit21, Theorem 3], and [Reference XuXu24, Theorem 4.4].

2.6 Slope stability in positive characteristic

In this subsection, we recall some results on slope stability in positive characteristic. Let k be an algebraically closed field of characteristic $p\gt0$ .

Definition 2.30. Let X be a smooth projective variety of dimension $n\geq 2$ over k and let $D_{1},\ldots,D_{n-1}$ be nef divisors on X such that the 1-cycle $D_{1}\cdots D_{n-1}$ is not numerically trivial. Let ${\mathcal E}$ be a torsion-free coherent sheaf on X. The slope of ${\mathcal E}$ with respect to $(D_{1},\ldots,D_{n-1})$ is defined by

\begin{align*}\mu({\mathcal E})=\frac{c_{1}({\mathcal E})\cdot D_{1}\cdots D_{n-1}}{\mathrm{rk}({\mathcal E})}.\end{align*}

We say that ${\mathcal E}$ is $(D_{1},\ldots,D_{n-1})$ -semistable (respectively $(D_{1},\ldots,D_{n-1})$ -stable) if for every non-trivial subsheaf ${\mathcal E}^{\prime}\subseteq {\mathcal E}$ we have

\begin{align*}\mu({\mathcal E}^{\prime})\leq \mu({\mathcal E}) \quad (\mathrm{respectively} \mu({\mathcal E}^{\prime}) \lt \mu({\mathcal E})).\end{align*}

We say that ${\mathcal E}$ is strongly $(D_{1},\ldots,D_{n-1})$ -semistable (respectively strongly $(D_{1},\ldots,D_{n-1})$ -stable) if for every integer $e\geq 0$ the pullback $F^{e\ast}{\mathcal E}$ is $(D_{1},\ldots,D_{n-1})$ -semistable (respectively $(D_{1},\ldots,D_{n-1})$ -stable), where F is the Frobenius morphism.

The following theorem is an analogy of the Bogomolov inequality in positive characteristic.

Theorem 2.31 [Reference LangerLan04, Theorem 3.2]. Let X be a smooth projective variety of dimension $n\geq 2$ over k and let $D_{1},\ldots,D_{n-1}$ be nef divisors on X such that the 1-cycle $D_{1}\cdots D_{n-1}$ is not numerically trivial. Let ${\mathcal E}$ be a strongly $(D_{1},\ldots,D_{n-1})$ -semistable torsion-free sheaf on X. Then

\begin{align*}\Delta({\mathcal E})\cdot D_{2}\cdots D_{n-1}\geq 0,\end{align*}

where $\Delta({\mathcal E}):=2\, \mathrm{rk}({\mathcal E}) c_{2}({\mathcal E})-(\mathrm{rk}({\mathcal E})-1)c_{1}({\mathcal E})^{2}$ .

We will use the following result to prove our main result (see Proposition 5.6).

2.7 Properties of 1-foliations

We recall the notion of 1-foliations.

The basic properties of 1-foliations are as follows.

1. (1) The variety $Y:= X/{\mathcal F}=\operatorname{Spec}_{X} \mathrm{Ann}({\mathcal F}) := \{a\in{\mathcal O}_{X} \mid \xi(a)=0,\; \forall\, \xi\in{\mathcal F}\}$ is a normal variety, and we have the commutative diagram

   
   where F is the Frobenius morphism, and $\pi$ is a morphism with $\operatorname{deg} \pi=p^{\operatorname{rk} {\mathcal F}}$ .

2. (2) There is a one-to-one correspondence between 1-foliations and normal varieties between X and $X^{(1)}$ . The correspondence is given by ${\mathcal F} \mapsto X/{\mathcal F}$ and the inverse correspondence by $Y\mapsto \mathrm{Ann}({\mathcal O}_{Y}) := \{\xi\in T_{X} \mid \xi(a)=0,\; \forall\, a\in {\mathcal O}_{Y}{\kern-1pt}\}$ .

3. (3) The variety Y is regular if and only if ${\mathcal F}$ is smooth.

4. (4) If $Y_{0}$ denotes the regular locus of1pt Y and $X_{0}=\pi^{-1}(Y_{0})$ , then

   \begin{align*}K_{X_{0}}\sim \pi^{\ast}K_{Y_{0}}+(p-1)\operatorname{det} {\mathcal F}|_{X_{0}}.\end{align*}

The following criterion helps us to construct 1-foliations.

Lemma 2.35 [Reference EkedahlEke87, Lemma 4.2]. Let X be a smooth variety over an algebraically closed field k of characteristic $p\gt0$ and ${\mathcal F}$ be a saturated ${\mathcal O}_{X}$ -submodule of $T_{X}$ . If

\begin{align*}\mathrm{Hom}(\bigwedge^{2}{\mathcal F},T_{X}/{\mathcal F})=\mathrm{Hom}(F^{\ast}{\mathcal F},T_{X}/{\mathcal F})=0,\end{align*}

then ${\mathcal F}$ is a 1-foliation.

The following theorem tells us how to do ‘bend and break’ by 1-foliations.

Theorem 2.36 [Reference LangerLan15, Theorem 2.1]. Let X be a smooth variety over an algebraically closed field k of characteristic $p\gt0$ and L be a nef ${\mathbb R}$ -divisor on X. Let $f: C \to X$ be a non-constant morphism from a smooth projective curve C. Assume that ${\mathcal F}\subseteq T_{X}$ is a 1-foliation and smooth along f(C). If

\begin{align*}c_{1}({\mathcal F})\cdot C \gt \frac{K_{X}\cdot C}{p-1},\end{align*}

then for every $x\in f(C)$ there is a rational curve $B_{x}\subseteq X$ passing through x such that

\begin{align*}L\cdot B_{x}\leq 2 \operatorname{dim} X\frac{pL\cdot C}{(p-1)c_{1}({\mathcal F})\cdot C-K_{X}\cdot C}.\end{align*}

4.1 Local relative Chern classes

In this subsection, we recall the notion of local relative Chern classes for resolutions of normal surfaces. The main reference of this subsection is [Reference LangerLan25, § 3]. Let k be an algebraically closed field and let A be an excellent normal two-dimensional henselian local k-algebra. Let $X =\operatorname{Spec} A$ and let $x\in X$ be the closed point of X. Let $f: \widetilde{X}\to X$ be a desingularization of X with $\mathrm{Exc}(f)=\sum_{i}E_{i}$ .

Definition 4.1. Let ${\mathcal F}$ be a vector bundle on $\widetilde{X}$ . We define the first relative Chern class of ${\mathcal F}$ with respect to f to be the unique ${\mathbb Q}$ -divisor $c_{1}(f,{\mathcal F})$ supported on E such that, for every irreducible component $E_{i}$ of E, we have

\begin{align*}c_{1}(f,{\mathcal F})\cdot E_{i}=\operatorname{deg} {\mathcal F}|_{E_{i}}.\end{align*}

Let $\widetilde{\pi}: \widetilde{Y}\to \widetilde{X}$ be a generically finite proper morphism from a regular surface $\widetilde{Y}$ . The Stein factorization of $f\circ \widetilde{\pi}$ gives a proper birational morphism $g:\widetilde{Y}\to Y$ . Then we have the following commutative diagram.

Fix a rank-r vector bundle ${\mathcal F}$ on $\widetilde{X}$ . By [Reference KleimanKle69, Theorem 4.7 (ii)], possibly after further blowing up $\widetilde{Y}$ , there exists a filtration

\begin{align*}0={\mathcal F}_{0}\subseteq {\mathcal F}_{1} \subseteq\cdots\subseteq {\mathcal F}_{r}=\widetilde{\pi}^{\ast}{\mathcal F} \end{align*}

such that all quotients ${\mathcal L}_{i} = {\mathcal F}_{i}/{\mathcal F}_{i-1}$ are line bundles on $\widetilde{Y}$ . We define $L_{i}:=c_{1}(g,{\mathcal L}_{i})$ .

Definition 4.3. The second relative Chern class $c_{2}(f,{\mathcal F})$ of ${\mathcal F}$ with respect to f is defined as the real number

\begin{align*}c_{2}(f,{\mathcal F}):=\mathrm{inf}\bigg(\frac{\sum_{i\lt j}L_{i}L_{j}}{\mathrm{deg}\ \widetilde{\pi}} \bigg),\end{align*}

where the infimum is taken over all $\widetilde{\pi}$ and filtrations as above.

Definition 4.4. We define the relative Euler characteristic $\chi(f,{\mathcal F})$ by

\begin{align*}\chi(f,{\mathcal F}):=\operatorname{dim} H^{0}(X,f_{[\ast]}{\mathcal F}/f_{\ast}{\mathcal F})+\operatorname{dim} H^{0}(X,R^{1}f_{\ast}{\mathcal F})\end{align*}

and we set

\begin{align*}a(f,{\mathcal F}):=\chi(f,{\mathcal F})-r\chi(f,{\mathcal O}_{\widetilde{X}})+\tfrac{1}{2}c_{1}(f,{\mathcal F})(c_{1}(f,{\mathcal F})-K_{\widetilde{X}})-c_{2}(f,{\mathcal F}).\end{align*}

Remark 4.5. The invariant $a(f,{\mathcal F})$ is important since it does not depend on the choice of desingularization f (see [Reference LangerLan25, Proposition 3.7]). Hence, for a reflexive sheaf ${\mathcal E}$ on X, we can set

\begin{align*}a(x,{\mathcal E}):=a(f',{\mathcal F}'),\end{align*}

where $f': X' \to X$ is any desingularization of X and ${\mathcal F}'$ is any vector bundle on X’ such that $f'_{[\ast]}{\mathcal F}'\cong {\mathcal E}$ .

We use the following result in the proof of Proposition 5.3.

Proposition 4.6. Assume that the base field k has characteristic $p\gt0$ , and that ${\mathcal F}_{1}$ , ${\mathcal F}_{2}$ , and ${\mathcal F}_{3}$ are vector bundles on $\widetilde{X}$ . If the exact sequence

\begin{align*}0\to {\mathcal F}_{1}\to{\mathcal F}_{2}\to{\mathcal F}_{3}\to 0\end{align*}

exists, then

\begin{align*}\mathrm{ch}_{2}(f,{\mathcal F}_{2})=\mathrm{ch}_{2}(f,{\mathcal F}_{1})+\mathrm{ch}_{2}(f,{\mathcal F}_{3}),\end{align*}

where $\mathrm{ch}_{2}(f,\cdot):=\frac{1}{2}c_{1}(f,\cdot)^{2}-c_{2}(f,\cdot)$ .

Proof. By [Reference LangerLan25, Theorem 3.15], for $i=1,2,3$ and any positive integer m, we have

\begin{align*}\mathrm{ch}_{2}(f,{\mathcal F}_{i})=\frac{1}{2}c_{1}(f,{\mathcal F}_{i})^{2}-c_{2}(f,{\mathcal F}_{i})=-\lim_{m\to\infty}\frac{\chi(f,(F^{m})^{\ast}{\mathcal F}_{i})}{p^{2m}}.\end{align*}

We consider the exact sequence $0\to {\mathcal F}_{1}\to{\mathcal F}_{2}\to{\mathcal F}_{3}\to 0$ . Since $\widetilde{X}$ is smooth, the maps $F^m$ are flat. Hence we have the exact sequence

\begin{align*}0\to(F^{m})^{\ast} {\mathcal F}_{1}\to(F^{m})^{\ast}{\mathcal F}_{2}\to(F^{m})^{\ast}{\mathcal F}_{3}\to 0.\end{align*}

Thus the assertion follows from the additivity of the relative Euler characteristic.

4.2 Chern classes of reflexive sheaves

In this subsection, we recall the definition of Chern classes of reflexive sheaves on normal varieties. The main reference of this subsection is [Reference LangerLan25, §§ 4 and 5]. Let X be a proper normal surface defined over an algebraically closed field k, and let ${\mathcal E}$ be a reflexive sheaf on X. Let $f :\widetilde{X} \to X$ be a desingularization with $\mathrm{Exc}(f)= E$ . For every $x\in f(E)$ we consider the map $\nu_{x}:\operatorname{Spec} {\mathcal O}^{h}_{X,x} \to X$ from the spectrum of the henselization of the local ring of X at x and the base change

(4.1) \begin{equation} f_{x} = \nu^{\ast}_{x}f : \widetilde{X}_{x} \to \operatorname{Spec} {\mathcal O}^{h}_{X,x} \end{equation}

of f via $\nu_{x}$ . Note that the group of divisors on $\widetilde{X}_{x}$ that are supported on the exceptional locus $E_{x}$ of $f_{x}$ embeds into $A_{1}(\widetilde{X})$ , where $A_{1}(\widetilde{X})$ is the group of 1-cycles modulo rational equivalence on $\widetilde{X}$ . For any vector bundle ${\mathcal F}$ on $\widetilde{X}$ we write $c_{1}(f_{x},{\mathcal F})\in A_{1}(\widetilde{X})\otimes {\mathbb Q}$ for the image of $c_{1}(f_{x},{\mathcal F}|_{\widetilde{X}_{x}})$ . Then we have

\begin{align*}c_{1}({\mathcal F})=f^{\ast}c_{1}(f_{[\ast]}{\mathcal F})+\sum_{x\in f(E)}c_{1}(f_{x},{\mathcal F})\in A_{1}(\widetilde{X})\otimes{\mathbb Q}.\end{align*}

We write $a(x,{\mathcal E})$ for $a(x,\nu^{\ast}_{x}{\mathcal E})$ (see Remark 4.5).

Definition 4.7. Let ${\mathcal F}$ be a vector bundle on $\widetilde X$ such that $f_{[\ast]}{\mathcal F}\cong {\mathcal E}$ . Using the homomorphism $f_{\ast}:A_{0}(\widetilde{X})\to A_{0}(X)$ , we define the second Chern class of ${\mathcal E}$ to be

\begin{align*}c_{2}({\mathcal E}):=f_{\ast}c_{2}({\mathcal F})-\sum_{x\in f(E)}c_{2}(f_{x},{\mathcal F})[x]\in A_{0}(X)\otimes{\mathbb R},\end{align*}

where $A_{0}(\widetilde{X})$ and $A_{0}(X)$ denote the groups of 0-cycles modulo rational equivalence on $\widetilde{X}$ and X, respectively. Moreover, we define the discriminant

\begin{align*}\Delta({\mathcal E}):=2r\, c_{2}({\mathcal E})-(r-1)c_{1}({\mathcal E})^{2}\in A_{0}(X)\otimes{\mathbb R}\end{align*}

where r is the rank of ${\mathcal E}$ . For a more detailed explanation of these definitions, see [Reference LangerLan25, 4.1].

In higher dimensions, we can define Chern classes by reducing to the case when X is a surface (see [Reference LangerLan25, § 5]). We will use the following result in Proposition 5.6.

Theorem 4.8 [Reference LangerLan24, Theorem 3.4]. Let X be a normal projective threefold over an algebraically closed field of characteristic $p\gt0$ . Let $L_{1}$ and $ L_{2}$ be nef line bundles on X such that $L_{1}\cdot L_{2}$ is numerically non-trivial, and let ${\mathcal E}$ be a strongly $(L_{1},L_{2})$ -semistable reflexive sheaf on X. Then

\begin{align*}\Delta({\mathcal E})\cdot L_{2}:=2\,\mathrm{rk}({\mathcal E})\int_{X} c_{2}({\mathcal E})\cdot L_{2}-(\mathrm{rk}({\mathcal E})-1)\int_{X} c_{1}({\mathcal E})^{2}\cdot L_{2}\geq 0,\end{align*}

where $\int_{X}$ is the degree map of 0-cycles on X (see [Reference FultonFul98, Definition 1.4]).

4.3 Computations of some relative second Chern classes

In this subsection, we compute relative second Chern classes of some cyclic quotient singularities.

1. (1) $\big|\mathrm{Fix}({\mathbb Z}_{m})\big|\lt +\infty$ , where $\mathrm{Fix}({\mathbb Z}_{m})$ is the set of fixed points for the action of ${\mathbb Z}_{m}$ ; and

2. (2) for any $t\in \mathrm{Fix}({\mathbb Z}_{m})$ we have $\widehat{T}_{t}/{\mathbb Z}_{m}\cong X$ , where $\widehat{T}_{t}$ is the completion of1pt T at t.

Proof. The proof is similar to [Reference ReidRei87, p. 405]. By the definition, $X=\widehat{\mathbb{A}}_{k}^{2}/{\mathbb Z}_{m}$ comes from a linear and diagonalized action of ${\mathbb Z}_{m}$ on ${\mathbb A}^{2}$ . This action gives an action of ${\mathbb Z}_{m}$ on ${\mathbb A}^{5}$ by

\begin{align*}g\in {\mathbb Z}_{m}:(x_{1},x_{2})\in {\mathbb A}^{2}\times {\mathbb A}^{3}\cong{\mathbb A}^{5} \mapsto (g(x_{1}),x_{2}).\end{align*}

It induces a natural action of ${\mathbb Z}_{m}$ on ${\mathbb P}^{4}$ . Then it is clear that the fixed point set

\begin{align*}\mathrm{Fix}({\mathbb Z}_{m})\subseteq (x_{1}=0)\cup(x_{2}=0)\subset{\mathbb P}^{4}.\end{align*}

Consider the quotient map $\pi:{\mathbb P}^{4}\to{\mathbb P}^{4}/{\mathbb Z}_{m}$ . We use two general hyperplane sections to cut out a subvariety $Y\subseteq{\mathbb P}^{4}/{\mathbb Z}_{m}$ . By Proposition 3.1, $T:=\pi^{-1}(Y)$ is smooth. Since $\mathrm{dim}(\mathrm{Fix}({\mathbb Z}_{m}))=2$ , we have that assertion (1) holds. Moreover, (2) holds by our construction.

In order to compute $a(0,\Omega_{X}^{[1]})$ , we need to apply the Lefschetz fixed point formula in positive characteristic. Let W(k) be the Witt ring of k. For $c\in k$ , we set $w(c):=(c,0,0,\ldots)\in W(k)$ . Let $\psi$ be an endomorphism of a finite-rank k-vector space. Recall that the Brauer trace ${\operatorname{tr}_{\mathrm{B}}}(\psi)$ of $\psi$ is defined to be $\sum_{i} w(c_{i})$ , where $c_{i}$ are eigenvalues of $\psi$ , with the appropriate multiplicities. We refer to [Reference SerreSer77, 18.1] for more details.

Lemma 4.11. Let T be a smooth projective surface over k with an action of ${\mathbb Z}_{m}$ on T. Assume that $\big|\mathrm{Fix}({\mathbb Z}_{m})\big|\lt +\infty$ , where $\mathrm{Fix}({\mathbb Z}_{m})$ is the set of fixed points for the action of ${\mathbb Z}_{m}$ . For any ${\mathbb Z}_{m}$ -equivariant locally free sheaf ${\mathcal F}$ on T and $0\neq g\in {\mathbb Z}_{m}$ , we have

\begin{align*}\sum_{i}(-1)^{i} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(T,{\mathcal F})})=\sum_{t\in\mathrm{Fix}({\mathbb Z}_{m})}\ \frac{ {\operatorname{tr}_{\mathrm{B}}} (g|_{{\mathcal F}_{t}})}{(1-w(c_{g,1}))(1-w(c_{g,2}))},\end{align*}

where ${\mathcal F}_{t}$ is the fiber of ${\mathcal F}$ at t and $c_{g,i}$ are the eigenvalues of g on ${\mathbb A}^{2}$ .

Lemma 4.12 [Reference SerreSer77, 18.1]. Let V be a vector space over k and let ${\mathbb Z}_{m}\to \mathrm{GL}(V)$ be a representation. Then we have

\begin{align*}\mathrm{dim}(V^{{\mathbb Z}_{m}})=\frac{1}{m}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g),\end{align*}

where $V^{{\mathbb Z}_{m}}$ is the subspace of V which is fixed by ${\mathbb Z}_{m}$ .

The following result is an adaptation of [Reference LangerLan00, Theorem 5.4].

Theorem 4.13. Let X be as stated in Setting 4.9. Then we have

\begin{align*}a(0,\Omega_{X}^{[1]})=\frac{1}{m}-1.\end{align*}

Proof. Let T be a smooth projective surface as constructed in Lemma 4.10, and let $\pi:T\to Y:=T/{\mathbb Z}_{m}$ be the quotient map. We set $N:=\big|\mathrm{Fix}({\mathbb Z}_{m})\big|$ and $y:=\pi(t)$ , where $t\in \mathrm{Fix}({\mathbb Z}_{m})$ is a fixed point for the action of ${\mathbb Z}_{m}$ . Note that by Lemma 4.10 (2), we have that $a(y,\Omega_{Y}^{[1]})=a(0,\Omega_{X}^{[1]})$ is independent of the choice of t. It follows from [Reference LangerLan25, Proposition 4.2(4)] that $\int_{Y}c_{2}(\Omega_{Y}^{[1]})=(1/m)\int_{Y}c_{2}(\Omega_{T})$ . Moreover, by [Reference LangerLan25, Theorem 4.4], we have

\begin{align*}\chi(Y,\Omega_{Y}^{[1]})=-\int_{Y}c_{2}(\Omega_{Y}^{[1]})+2\chi(Y,{\mathcal O}_{Y})+N a(y,\Omega_{Y}^{[1]})\end{align*}

and

\begin{align*}\int_{Y}c_{2}(\Omega_{T})=-\chi(T,\Omega_{T})+2\, \chi(T,{\mathcal O}_{T}).\end{align*}

This implies that

\begin{equation*}\begin{aligned}a(y,\Omega_{Y}^{[1]})&=\frac{1}{N}\bigg( \chi(Y,\Omega_{Y}^{[1]})+\int_{Y}c_{2}(\Omega_{Y}^{[1]})-2\,\chi(Y,{\mathcal O}_{Y}) \bigg)\\&=\frac{1}{N}\bigg(\sum_{i}(-1)^{i}\,\mathrm{dim}(H^{i}(Y,\Omega_{Y}^{[1]}))+\frac{1}{m}\int_{Y}c_{2}(\Omega_{T})-2\sum_{i}(-1)^{i}\,\mathrm{dim}(H^{i}(Y,{\mathcal O}_{Y})) \bigg)\\&=\frac{1}{N}\bigg(\sum_{i}(-1)^{i}\,\mathrm{dim}(H^{i}(T,\Omega_{T})^{{\mathbb Z}_{m}})+\frac{1}{m}(-\chi(T,\Omega_{T})+2\,\chi(T,{\mathcal O}_{T}))\\&\quad -2 \sum_{i}(-1)^{i}\,\mathrm{dim}(H^{i}(T,{\mathcal O}_{T})^{{\mathbb Z}_{m}}) \bigg),\end{aligned}\end{equation*}

where $H^{i}(T,\cdot)^{{\mathbb Z}_{m}}$ is the subspace of $H^{i}(T,\cdot)$ fixed by ${\mathbb Z}_{m}$ . The third equality holds since $\pi_{\ast}(\Omega_{T})^{{\mathbb Z}_{m}}=\Omega_{Y}^{[1]}$ and $\pi_{\ast}({\mathcal O}_{T})^{{\mathbb Z}_{m}}={\mathcal O}_{Y}$ , where $\pi_{\ast}(\Omega_{T})^{{\mathbb Z}_{m}}$ and $\pi_{\ast}({\mathcal O}_{T})^{{\mathbb Z}_{m}}$ are the subsheaves of $\pi_{\ast}(\Omega_{T})$ and $\pi_{\ast}({\mathcal O}_{T})$ , respectively, fixed by ${\mathbb Z}_{m}$ . Note that, by Lemma 4.12,

\begin{align*}\mathrm{dim}(H^{i}(T,\Omega_{T})^{{\mathbb Z}_{m}}) =\frac{1}{m}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(\Omega_{T})}),\quad\mathrm{dim}(H^{i}(T,{\mathcal O}_{T})^{{\mathbb Z}_{m}}) =\frac{1}{m}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}({\mathcal O}_{T})}).\end{align*}

Then by Lemma 4.11, we have

\begin{align*}a(y,\Omega_{Y}^{[1]})&=\frac{1}{mN}\bigg(\sum_{i}(-1)^{i}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(\Omega_{T})})-\chi(T,\Omega_{T})+2\, \chi(T,{\mathcal O}_{T})\\&\quad -2 \sum_{i}(-1)^{i}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}({\mathcal O}_{T})}) \bigg)\\&=\frac{1}{mN}\bigg(\sum_{i}(-1)^{i}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(\Omega_{T})})-\bigg(\sum_{i}(-1)^{i}\sum_{g=\mathrm{id}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(\Omega_{T})})\bigg)\\&\quad+2\bigg(\sum_{i}(-1)^{i}\sum_{g=\mathrm{id}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}({\mathcal O}_{T})})\bigg)-2 \sum_{i}(-1)^{i}\sum_{g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}({\mathcal O}_{T})}) \bigg)\\&= \frac{1}{mN}\bigg(\sum_{i}(-1)^{i}\sum_{\mathrm{id}\neq g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}(\Omega_{T})})-2\sum_{i}(-1)^{i}\sum_{\mathrm{id}\neq g\in{\mathbb Z}_{m}} {\operatorname{tr}_{\mathrm{B}}} (g^{\ast}|_{H^{i}({\mathcal O}_{T})}) \bigg) \\\end{align*}

\begin{align*}\hphantom{a(y,\Omega_{Y}^{[1]})} &=\frac{1}{mN}\bigg(N\sum_{\mathrm{id}\neq g\in{\mathbb Z}_{m}}\frac{ {\operatorname{tr}_{\mathrm{B}}} (g|_{\Omega_{T,t}})}{(1-w(c_{g,1}))(1-w(c_{g,2}))}-2N\sum_{\mathrm{id}\neq g\in{\mathbb Z}_{m}}\frac{1}{(1-w(c_{g,1}))(1-w(c_{g,2}))}\bigg)\\&=\frac{1}{m}\sum_{\mathrm{id}\neq g\in{\mathbb Z}_{m}}\frac{ {\operatorname{tr}_{\mathrm{B}}} (g|_{\Omega_{T,t}})-2}{(1-w(c_{g,1}))(1-w(c_{g,2}))},\end{align*}

where the fourth equality follows from Lemma 4.10 (2). Then by [Reference LangerLan00, Example 5.6 and Corollary 5.9], we have $a(0,\Omega_{X}^{[1]})=a(y,\Omega_{Y}^{[1]})={1}/{m}-1$ , and hence the assertion holds.

Corollary 4.14. Let X and f be as stated in Setting 4.9. We have

\begin{align*}c_{2}(f,\Omega_{\widetilde{X}})\geq 2- \frac{1}{m}.\end{align*}

Proof. By Theorem 4.13, it suffices to show that $c_{2}(f,\Omega_{\widetilde{X}})\geq 1- a(0,\Omega_{X}^{[1]})$ . Note that, by Definition 4.4,

\begin{align*}c_{2}(f,\Omega_{\widetilde{X}})=\chi(f,\Omega_{\widetilde{X}})-2\,\chi(f,{\mathcal O}_{\widetilde{X}})+\tfrac{1}{2}c_{1}(f,\Omega_{\widetilde{X}})\cdot(c_{1}(f,\Omega_{\widetilde{X}})-K_{\widetilde{X}})-a(0,\Omega_{X}^{[1]}).\end{align*}

Since $c_{1}(f,\Omega_{\widetilde{X}})$ is supported on E and $c_{1}(f,\Omega_{\widetilde{X}})-K_{\widetilde{X}}$ is f-trivial, we have

\begin{align*}c_{1}(f,\Omega_{\widetilde{X}})\cdot(c_{1}(f,\Omega_{\widetilde{X}})-K_{\widetilde{X}})=0.\end{align*}

Moreover, since X has rational singularities, we have

\begin{align*}\chi(f,{\mathcal O}_{\widetilde{X}})=h^{0}(f_{[\ast]}{\mathcal O}_{\widetilde{X}}/{\mathcal O}_{X})+h^{0}(R^{1}f_{\ast}{\mathcal O}_{\widetilde{X}})=0.\end{align*}

Therefore,

\begin{align*}c_{2}(f,\Omega_{\widetilde{X}})=\chi(f,\Omega_{\widetilde{X}})-a(0,\Omega_{X}^{[1]}).\end{align*}

Since

\begin{align*}\chi(f,\Omega_{\widetilde{X}})=h^{0}(f_{[\ast]}\Omega_{\widetilde{X}}/f_{\ast}\Omega_{\widetilde{X}})+h^{0}(R^{1}f_{\ast}\Omega_{\widetilde{X}}),\end{align*}

we only need to show that $h^{0}(R^{1}f_{\ast}\Omega_{\widetilde{X}})\geq 1$ . By the exact sequence

\begin{align*}0\to\Omega_{\widetilde{X}}(-E)\to\Omega_{\widetilde{X}}\to\Omega_{\widetilde{X}}|_{E}\to 0,\end{align*}

and $R^{2}f_{\ast}\Omega_{\widetilde{X}}(-E)=0$ since $\mathrm{Exc}(f)$ is one-dimensional, we know that the map

\begin{align*}R^{1}f_{\ast}\Omega_{\widetilde{X}}\to R^{1}f_{\ast}(\Omega_{\widetilde{X}}|_{E})\cong H^{1}(E,\Omega_{\widetilde{X}}|_{E})\end{align*}

is surjective. Moreover, there exists the following exact sequence:

\begin{align*}{\mathcal O}_{E}(-E)\to\Omega_{\widetilde{X}}|_{E}\to\Omega_{E}\to 0.\end{align*}

It follows that the map

\begin{align*}H^{1}(E,\Omega_{\widetilde{X}}|_{E})\to H^{1}(E,\Omega_{E})\end{align*}

is surjective. By Serre duality, $h^{1}(E,\Omega_{E})=h^{0}(E,{\mathcal O}_{E})=1$ . Therefore $h^{0}(R^{1}f_{\ast}\Omega_{\widetilde{X}})\geq 1$ , and hence the assertion holds.

5.1 A key inequality

Setting 5.1. Let (X,B) be an lc projective pair of dimension 3 over an algebraically closed field k of characteristic $p\gt3$ . We assume that the following hold:

1. (1) X is ${\mathbb Q}$ -factorial, B is reduced, and $X\setminus B$ is terminal;

2. (2) there exists a divisor $D\in |m(K_{X}+B)|$ such that $D_{\mathrm{red}}=B$ , where m is a positive integer such that $L:=m(K_{X}+B)$ is Cartier;

3. (3) $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$ ; and

4. (4) if C is a curve in X with $C\cdot (K_{X}+B)\gt0$ , then (X,B) is plt at the generic point of C.

1. (1) The Iitaka fibration f associated to $K_{S^{\lambda}}+\Theta$ is a morphism to a smooth curve.

2. (2) Let $\Theta_{h}$ be the horizontal part of $\Theta$ with respect to f. We write $\Theta_{h}=\sum c_{q}\Gamma_{q}+\sum d_{l}\Delta_{l}$ , where the $\Gamma_{q}$ map under $\lambda$ to the singular locus of X and the $\Delta_{l}$ to the non-normal locus of B. Then we have that f is a generically smooth map and one of the following holds.

   1. (i) The generic fiber of $\lambda$ is a smooth elliptic curve and $\Theta_{h}=0$ ,

   2. (ii) The generic fiber of $\lambda$ is ${\mathbb P}^{1}$ , $d_{l}=1$ , and $c_{q}=1-({1}/{m_{q}})$ , where $m_{q}$ are the indices of cyclic quotient singularities of X at the generic point of $\Gamma_{q}$ (see Theorem 2.11). Moreover, $(d_{1},\ldots;m_{1},\ldots)$ must be one of the following: $(1,1;)$ , $(1;2)$ , $(1;2,2)$ , $(;2)$ , $(;2,2)$ , $(;2,2,2)$ , $(;2,2,2,2)$ , $(;2,3,6)$ , $(;2,4)$ , $(;2,4,4)$ , $(;3)$ , $(;3,3)$ , or $(;3,3,3)$ .

Proof. For assertion (1), note that

\begin{align*}0\leq (K_{S^{\lambda}}+\Theta)^{2} = S\cdot (K_{X}+B)^{2}=0,\end{align*}

where the last equality holds since, by (2) and (3) in Setting 5.1, $(K_{X}+B)^{2}$ is numerically trivial on B. Then $(S^{\lambda},\Theta)$ is an slc pair with $\nu(K_{S^{\lambda}}+\Theta)=1$ since L is not numerically trivial on S. It follows from Theorem 2.18 that $K_{S^{\lambda}}+\Theta$ is semi-ample, and hence the Iitaka fibration f associated to $K_{S^{\lambda}}+\Theta$ is a morphism to a smooth curve. For assertion (2), since $p\gt3$ , by [Reference WitaszekWit21, Proposition 2.2], f is a generically smooth map. Let F be the generic fiber of f and consider the adjunction

\begin{align*}(K_{S^{\lambda}}+\Theta)|_{F}\sim_{{\mathbb Q}}K_{F}+\Theta_{h}|_{F}.\end{align*}

If $\Theta_{h}|_{F}=0$ , then $\Theta_{h}=0$ . We get (i). Otherwise, we have $\operatorname{deg} K_{F}\lt0$ . Since F is smooth, $F={\mathbb P}^{1}$ . This implies that $\operatorname{deg} \Theta_{h}|_{F}=2$ . Now by (4) in Setting 5.1, Proposition 2.10, and Theorem 2.11, we have that (ii) holds.

The following proposition is the key inequality for proving Proposition 5.6.

Proposition 5.3. Let $\rho: V\to X$ be a desingularization. Then

\begin{align*}\rho^{\ast}L \cdot(K^{2}_{V}+c_{2}(V))\geq L\cdot(K_{X}^{2}+c_{2}(\Omega_{X}^{[1]})).\end{align*}

Moreover, if the equality holds, then we have $(K_{X}+B)|_{B^{\prime}}\sim_{{\mathbb Q}}K_{B^{\prime}}+\Delta_{B^{\prime}}$ such that $(K_{X}+B)|_{B^{\prime}}$ is numerically trivial on $\operatorname{Supp} \Delta_{B^{\prime}}$ where $B^{\prime}$ is the $S_{2}$ -fication of B.

Proof. By [Reference ZhangZha23, Corollary 2.2] we can choose a sufficiently ample divisor A such that $nL+A$ are very ample for all $n\gg 0$ . We take a general section $H_{n}\in |nL+A|$ . Note that $\rho^{\ast}L \cdot(K^{2}_{V}+c_{2}(V))$ does not depend on the choice of $\rho$ . Hence by Proposition 3.9, we can assume that $\rho: V\to X$ is a desingularization such that $\widetilde{H}_{n}:=\rho^{-1}(H_{n})=\rho^{\ast}H_{n}$ is smooth, and $\rho|_{\widetilde{H}_{n}}:\widetilde{H}_{n}\to H_{n}$ is the minimal resolution of $H_{n}$ . Then by [Reference LangerLan25, Theorem 0.4] we have

\begin{equation*}\begin{split}\rho^{\ast}(nL+A) \cdot(K^{2}_{V}+c_{2}(V))&=\widetilde{H}_{n}\cdot(K^{2}_{V}+c_{2}(V)) \\[3pt]&= (K_{V}|_{\widetilde{H}_{n}})^{2}+c_{2}(\Omega_{V}|_{\widetilde{H}_{n}})\end{split}\end{equation*}

and

\begin{equation*}\begin{split}(nL+A)\cdot(K_{X}^{2}+c_{2}(\Omega_{X}^{[1]}))&= H_{n}\cdot (K_{X}^{2}+c_{2}(\Omega_{X}^{[1]}))\\[3pt]&=(K_{X}|_{H_{n}})^{2}+c_{2}(\Omega^{[1]}_{X}|_{H_{n}}).\end{split}\end{equation*}

We claim that

\begin{align*} \rho_{\ast}(K_{V}|_{\widetilde{H}_{n}})^{2}+\rho_{\ast}c_{2}(\Omega_{V}|_{\widetilde{H}_{n}})-(K_{X}|_{H_{n}})^{2}-c_{2}(\Omega^{[1]}_{X}|_{H_{n}})=\rho_{\ast}K_{\widetilde{H}_{n}}^{2}+\rho_{\ast}c_{2}(\Omega_{\widetilde{H}_{n}})-K_{H_{n}}^{2}-c_{2}(\Omega^{[1]}_{H_{n}})\end{align*}

as 0-cycles. Consider the exact sequence

(5.1) \begin{equation} 0\to {\mathcal O}_{\widetilde{H}_{n}}(-\widetilde{H}_{n})\to \Omega_{V}|_{\widetilde{H}_{n}}\to \Omega_{\widetilde{H}_{n}}\to 0. \end{equation}

Let x be a point contained in the singular locus of $H_{n}$ . Let $\rho_{x}$ be the map over $\operatorname{Spec} {\mathcal O}_{H_{n},x}^{h}$ defined by (4.1). By Proposition 4.6 and [Reference LangerLan25, Proposition 3.2(1)], we have

(5.2) \begin{equation}\begin{aligned}&\tfrac{1}{2}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})^{2}-c_{2}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})\\[3pt]&\quad=\tfrac{1}{2}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}-c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})+\tfrac{1}{2}c_{1}(\rho_{x},{\mathcal O}_{\widetilde{H}_{n}}(-\widetilde{H}_{n}))^{2}-c_{2}(\rho_{x},{\mathcal O}_{\widetilde{H}_{n}}(-\widetilde{H}_{n}))\\[3pt]&\quad=\tfrac{1}{2}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}-c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})+\tfrac{1}{2}c_{1}(\rho_{x},{\mathcal O}_{\widetilde{H}_{n}}(-\widetilde{H}_{n}))^{2}\\&\quad=\tfrac{1}{2}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}-c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}}),\end{aligned}\end{equation}

where the last equality holds since

\begin{align*}{\mathcal O}_{\widetilde{H}_{n,x}}(\widetilde{H}_{n,x})\sim \rho^{\ast}_{x}{\mathcal O}_{H_{n}}(H_{n})\end{align*}

and by Definition 4.1

(5.3) \begin{equation} c_{1}(\rho_{x},{\mathcal O}_{\widetilde{H}_{n}}(\widetilde{H}_{n}))\sim_{{\mathbb Q}}{\mathcal O}_{\widetilde{H}_{n,x}}(\widetilde{H}_{n,x})-\rho^{\ast}_{x}{\mathcal O}_{H_{n}}(H_{n})\sim_{{\mathbb Q}}0.\end{equation}

By Definitions 4.1 and 4.7, we have

\begin{align*}\sum_{x\in H_{n}\,\mathrm{singular}}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})&=K_{V}|_{\widetilde{H}_{n}}-\rho^{\ast} c_{1}(\Omega^{[1]}_{X}|_{H_{n}}),\\\sum_{x\in H_{n}\,\mathrm{singular}}c_{2}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})[x]&=\rho_{\ast}c_{2}(\Omega_{V}|_{\widetilde{H}_{n}})-c_{2}(\Omega^{[1]}_{X}|_{H_{n}}),\\\sum_{x\in H_{n}\,\mathrm{singular}}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})&=K_{\widetilde{H}_{n}}-\rho^{\ast} c_{1}(\Omega^{[1]}_{H_{n}}),\end{align*}

and

\begin{align*}\\[-21pt]\sum_{x\in H_{n}\,\mathrm{singular}} c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})[x]&=\rho_{\ast}c_{2}(\Omega_{\widetilde{H}_{n}})-c_{2}(\Omega^{[1]}_{H_{n}}).\end{align*}

These imply that

\begin{equation*}\begin{aligned}&\rho_{\ast}(K_{V}|_{\widetilde{H}_{n}})^{2} +\rho_{\ast}c_{2}(\Omega_{V}|_{\widetilde{H}_{n}}) -(K_{X}|_{H_{n}})^{2}-c_{2}(\Omega^{[1]}_{X}|_{H_{n}})\\&\quad=\rho_{\ast}\bigg(\rho^{\ast}(K_{X}|_{H_{n}})+\sum_{x\in H_{n}\,\mathrm{singular}}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})\bigg)^{\!2}-(K_{X}|_{H_{n}})^{2}+\sum_{x\in H_{n}\,\mathrm{singular}} c_{2}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})[x]\\&\quad= \sum_{x\in H_{n}\,\mathrm{singular}}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})^{2}[x]+\sum_{x\in H_{n}\, \mathrm{singular}} c_{2}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})[x]\\&\quad=\sum_{x\in H_{n}\,\mathrm{singular}}\frac{3}{2}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})^{2}[x]+\sum_{x\in H_{n}\,\mathrm{singular}} c_{2}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})[x]\\&\qquad -\sum_{x\in H_{n}\, \mathrm{singular}}\frac{1}{2} c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}})^{2}[x].\end{aligned}\end{equation*}

Note that by (5.1), (5.3), and Definition 4.1, we have

\begin{align*}c_{1}(\rho_{x},\Omega_{V}|_{\widetilde{H}_{n}}) =c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})+c_{1}(\rho_{x},{\mathcal O}_{\widetilde{H}_{n}}(-\widetilde{H}_{n})) =c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}}).\end{align*}

Hence by (5.2) we have

\begin{equation*}\begin{aligned}&\rho_{\ast}(K_{V}|_{\widetilde{H}_{n}})^{2} +\rho_{\ast}c_{2}(\Omega_{V}|_{\widetilde{H}_{n}})-(K_{X}|_{H_{n}})^{2} -c_{2}(\Omega^{[1]}_{X}|_{H_{n}})\\&\quad=\sum_{x\in H_{n}\, \mathrm{singular}}\frac{3}{2} c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}[x]+\sum_{x\in H_{n}\, \mathrm{singular}} c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})[x]-\sum_{x\in H_{n}\, \mathrm{singular}}\frac{1}{2} c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}[x]\\&\quad=\sum_{x\in H_{n}\, \mathrm{singular}} c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}[x]+\sum_{x\in H_{n}\, \mathrm{singular}} c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})[x]\\&\quad=\rho_{\ast}\bigg(\rho^{\ast}K_{H_{n}}+\sum_{x\in H_{n}\,\mathrm{singular}}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})\bigg)^{\!2}+\rho_{\ast}c_{2}(\Omega_{\widetilde{H}_{n}})-K_{H_{n}}^{2}-c_{2}(\Omega^{[1]}_{H_{n}})\\&\quad=\rho_{\ast}K_{\widetilde{H}_{n}}^{2}+\rho_{\ast}c_{2}(\Omega_{\widetilde{H}_{n}})-K_{H_{n}}^{2}-c_{2}(\Omega^{[1]}_{H_{n}}).\end{aligned}\end{equation*}

Hence, the claim holds. Note that

\begin{align*}a_{n}&:=\rho_{\ast}K_{\widetilde{H}_{n}}^{2}+\rho_{\ast}c_{2}(\Omega_{\widetilde{H}_{n}})-K_{H_{n}}^{2}-c_{2}(\Omega^{[1]}_{H_{n}})\\&=\sum_{x\in H_{n}\, \mathrm{singular}}c_{1}(\rho_{x},\Omega_{\widetilde{H}_{n}})^{2}[x]+\sum_{x\in H_{n}\, \mathrm{singular}} c_{2}(\rho_{x},\Omega_{\widetilde{H}_{n}})[x]\end{align*}

is a 0-cycle supported on the singular locus of $H_{n}$ . Since $X\setminus B$ is terminal by (1) in Setting 5.1, $a_{n}$ is supported on $H_{n}\cap C$ , where $C:=C_{1}+C_{2}$ , $C_{1}\subseteq B$ is the union of curves contained in the singular locus of X on which L are numerically trivial, and $C_{2}\subseteq B$ is the union of curves contained in the singular locus of X on which L are positive.

We claim that the degrees of $a_{n}$ on $C_{1}$ are bounded from below. Note that, for any irreducible component $\Gamma$ of $C_{1}$ ,

\begin{align*}\Gamma\cdot H_{n}=\Gamma\cdot (nL+A)=\Gamma\cdot A\end{align*}

are independent of n. Let $z_{1,n}$ be a point in $H_{n}\cap\Gamma$ . It suffices to show that the degrees of $a_{n}$ at $z_{1,n}$ are bounded from below. By Theorem 3.6, the dual graphs of $(H_{n},z_{1,n})$ are independent of n. Hence, $c_{1}(\rho_{z_{1,n}},\Omega_{\widetilde{H}_{n}})^{2}$ are bounded from below since $\rho|_{\widetilde{H}_{n}}$ are minimal resolutions and $c_{1}(\rho_{z_{1,n}},\Omega_{\widetilde{H}_{n}})^{2}$ are determined by the dual graphs of $(H_{n},z_{1,n})$ . By [Reference LangerLan25, Proposition 3.2],

\begin{align*}c_{1}(\rho_{z_{1,n}},\Omega_{\widetilde{H}_{n}})^{2}+c_{2}(\rho_{z_{1,n}},\Omega_{\widetilde{H}_{n}})\geq\tfrac{5}{4}c_{1}(\rho_{z_{1,n}},\Omega_{\widetilde{H}_{n}})^{2}.\end{align*}

Hence the claim follows. Therefore, to prove that $\rho^{\ast}L \cdot(K^{2}_{V}+c_{2}(V))- L\cdot(K_{X}^{2}+c_{2}(\Omega_{X}^{[1]}))\geq 0,$ it suffices to show that $(nL+A) \cdot (\rho_{\ast}(K^{2}_{V}+c_{2}(V))-(K_{X}^{2}+c_{2}(\Omega_{X}^{[1]})))=a_{n}$ are of non-negative degrees on $C_{2}$ for all $n\gg 0$ . Fix an $n\gg 0$ . Let $z_{2}\in C_{2}$ be a singular point of $H_{n}$ . Note that $z_{2}$ is in an irreducible component S of B, on which L is not numerically trivial. Then we can apply Lemma 5.2 and Corollary 3.8 to conclude that $(H_{n},z_{2})$ is a cyclic quotient singularity whose index m is one of 2, 3, 4, or 6. Then it follows from Corollary 4.14 that, if $z_{2}$ is a Du Val singularity, the degree of $a_{n}$ at $z_{2}$ , which is equal to $c_{2}(\rho_{z_{2}},\Omega_{\widetilde{H}_{n}})$ , is at least $\frac{3}{2}$ , $\frac{5}{3}$ , $\frac{7}{4}$ , or $\frac{11}{6}$ when m is 2, 3, 4, or 6, respectively. Otherwise, by [Reference KollárKol13, Definition 3.33], m is one of 3, 4, or 6, and $c_{1}(\rho_{z_{2}},\Omega_{\widetilde{H}_{n}})^{2}$ is $-\frac{1}{3}$ , $-1$ , or $-\frac{8}{3}$ when m is 3, 4, or 6, respectively. Hence, by Corollary 4.14, the degree of $a_{n}$ at $z_{2}$ is at least $\frac{4}{3}$ , $\frac{3}{4}$ , or $-\frac{5}{6}$ when m is 3, 4, or 6, respectively. By Lemma 5.2 and Corollary 3.8, an index-6 point is always accompanied by an index-2 point and an index-3 point. Moreover, different index-6 points give different index-2 points and index-3 points. Hence we always have that $a_{n}$ is of positive degree on $C_{2}$ . Therefore, the inequality holds. Moreover, if the equality holds, then there is no such singular point $z_{2}$ . It is to see that (ii) holds in (2) of Lemma 5.2. Hence, the second assertion holds by Lemma 2.19.

5.2 A special case

In this subsection, we prove a special case of the abundance conjecture in the case of $\nu(K_{X}+B)=2$ (see Proposition 5.6). Before proving Proposition 5.6, we give some results which help us to calculate some cohomology.

Lemma 5.4. Let B be a reduced equidimensional quasi-projective scheme of dimension 2 over a field and $g:B^{\prime}\to B$ be its $S_{2}$ -fication. Assume that L is a Cartier divisor on B. Then

\begin{align*}h^{i}(B^{\prime},{\mathcal O}_{B^{\prime}}(ng^{\ast}L))-h^{i}(B,{\mathcal O}_{B}(nL))\end{align*}

are bounded for all i and $n\geq 0$ .

Proof. Since g is a finite morphism by Proposition 2.14, we have

\begin{align*}h^{i}(B^{\prime}, {\mathcal O}_{B^{\prime}}(ng^{\ast}L))=h^{i}(B,g_{\ast}{\mathcal O}_{B^{\prime}}\otimes {\mathcal O}_{B}(nL)).\end{align*}

Note that there exists an exact sequence

\begin{align*}0\to{\mathcal O}_{B}\to g_{\ast}{\mathcal O}_{B^{\prime}}\to {\mathcal Q}\to 0\end{align*}

such that ${\mathcal Q}$ is a coherent sheaf supported on some points by Proposition 2.14. Tensoring it with ${\mathcal O}_{B}(nL)$ , we get

\begin{align*}0\to{\mathcal O}_{B}(nL)\to g_{\ast}{\mathcal O}_{B^{\prime}}\otimes{\mathcal O}_{B}(nL)\to {\mathcal Q}\otimes{\mathcal O}_{B}(nL)\to 0.\end{align*}

They give the exact sequence

\begin{align*}H^{i-1}(B,{\mathcal Q}\otimes{\mathcal O}_{B}(nL))\to H^{i}(B,{\mathcal O}_{B}(nL))\to H^{i}(B,g_{\ast}{\mathcal O}_{B^{\prime}}\otimes{\mathcal O}_{B}(nL))\to H^{i}(B,{\mathcal Q}\otimes{\mathcal O}_{B}(nL)).\end{align*}

Since both $h^{i-1}(B,{\mathcal Q}\otimes{\mathcal O}_{B}(nL))$ and $h^{i}(B,{\mathcal Q}\otimes{\mathcal O}_{B}(nL))$ are constants for $n\geq 0$ , the assertion holds.

Proof. We take a general hyperplane section A on X and consider the exact sequence

\begin{align*}0\to{\mathcal O}_{X}(K_{X}+nM+lA)\to {\mathcal O}_{X}(K_{X}+nM+(l+1)A)\to {\mathcal Q}_{n,l+1}\to 0, \end{align*}

where $l\geq 0$ is an integer. Since both ${\mathcal O}_{X}(K_{X}+nM+lA)$ and ${\mathcal O}_{X}(K_{X}+nM+(l+1)A)$ are $S_{2}$ , by [Reference KollárKol13, Lemma 2.60] ${\mathcal Q}_{n,l+1}$ are $S_{1}$ and supported on A. Hence they are torsion-free. Note that the sheaves ${\mathcal O}_{A}((K_{X}+nM+(l+1)A)|_{A})$ are reflexive. They give the exact sequence

\begin{align*}0\to{\mathcal Q}_{n,l+1}\to {\mathcal O}_{A}((K_{X}+nM+(l+1)A)|_{A})\to {\mathcal G}_{n,l+1}\to 0,\end{align*}

where ${\mathcal G}_{n,l+1}$ are supported on some points. Moreover, since M is ${\mathbb Q}$ -Cartier, ${\mathcal G}_{n,l+1}$ are of bounded rank at every point. Hence there is the exact sequence

\begin{align*}H^{0}(A,{\mathcal G}_{n,l+1})\to H^{1}(A,{\mathcal Q}_{n,l+1})\to H^{1}(A,{\mathcal O}_{A}((K_{X}+nM+(l+1)A)|_{A})).\end{align*}

Since A is a klt surface by Theorem 3.6, for $n\gg 0$ and any fixed l, we have

\begin{align*}H^{1}(A,{\mathcal O}_{A}((K_{X}+nM+(l+1)A)|_{A}))=0\end{align*}

by [Reference TanakaTan15, Theorem 2.11]. Thus, both $h^{0}(A,{\mathcal G}_{n,l+1})$ and $h^{1}(A,{\mathcal O}_{A}((K_{X}+nM+(l+1)A)|_{A}))$ are bounded for $n\geq 0$ and any fixed l. This implies that there exists a constant $C_{l+1}$ such that

\begin{align*}h^{1}(X,{\mathcal Q}_{n,l+1})\leq C_{l+1}\end{align*}

for $n\geq 0$ and any l. By [Reference KeelerKee03, Theorem 1.5], there exists an integer $l_{0}\gt 0$ such that, for $l\geq l_{0}$ and all $n\geq 0$ , we have $h^{2}(X,{\mathcal O}_{X}(K_{X}+nM+(l+1)A))=0$ . It follows that

\begin{align*}h^{2}(X,{\mathcal O}_{X}(K_{X}+nM))&\leq h^{1}(X,{\mathcal Q}_{n,1})+h^{2}(X,{\mathcal O}_{X}(K_{X}+nM+A))\\ &\leq \cdots\leq \sum_{i=1}^{l_{0}+1}h^{1}(X,{\mathcal Q}_{n,i})\leq\sum_{i=1}^{l_{0}+1}C_{i}. \end{align*}

1. (1) X is ${\mathbb Q}$ -factorial, B is reduced, and $X\setminus B$ is terminal;

2. (2) there exists a divisor $D\in |m(K_{X}+B)|$ such that $D_{\mathrm{red}}=B$ , where m is a positive integer such that $L:=m(K_{X}+B)$ is Cartier;

3. (3) $K_{X}+(1-\varepsilon)B$ is nef for any sufficiently small $\varepsilon \geq 0$ ;

4. (4) $\nu(K_{X}+B)=2$ ; and

5. (5) if C is a curve in X with $C\cdot (K_{X}+B)\gt0$ , then (X,B) is plt at the generic point of C.

Then $K_{X}+B$ is semi-ample.

Proof. By Theorem 2.28 it suffices to show that $\kappa(K_{X}+B)\gt 0$ . Since $K_{X}+(1-\varepsilon)B$ is nef for any sufficiently small $\varepsilon\geq 0$ , by Theorem 2.29 there exists an effective ${\mathbb Q}$ -divisor $\Delta_{\varepsilon}\sim_{{\mathbb Q}}K_{X}+(1-\varepsilon)B$ for any sufficiently small rational $\varepsilon\geq 0$ . Moreover, by assumption (2) we can assume that $B\subseteq \operatorname{Supp} \Delta_{\varepsilon}$ and $\Delta_{\varepsilon}$ have the same support for all sufficiently small rational $\varepsilon\geq 0$ . Therefore by [Reference KollárKol92, Lemma 11.3.3] we have

\begin{align*}\nu(K_{X}+(1-\varepsilon)B)=\nu(K_{X}+B)=2\end{align*}

for any sufficiently small rational $\varepsilon\gt0$ . This implies that

\begin{align*}(K_{X}+(1-\varepsilon)B)^{3}=(K_{X}+B)^{3}-3\varepsilon(K_{X}+B)^{2}\cdot B+3\varepsilon^{2}(K_{X}+B)\cdot B^{2}-\varepsilon^{3}B^{3}=0.\end{align*}

Hence, we have

(5.4) \begin{equation} (K_{X}+B)^{3}=K_{X}\cdot (K_{X}+B)^{2}=K_{X}^{2}\cdot (K_{X}+B)=B^{3}=0.\end{equation}

Now let $\rho:V\to X$ be a desingularization of X. By the Riemann–Roch theorem, we have

\begin{align*}\chi(V,{\mathcal O}_{V} (n\rho^{\ast}L))=\frac{n^{3}}{6}(\rho^{\ast}L)^{3} -\frac{n^{2}}{4}K_{V}\cdot(\rho^{\ast}L)^{2} +\frac{n}{12}(c_{2}(V)+K_{V}^{2})\cdot \rho^{\ast}L+\chi({\mathcal O}_{V}). \end{align*}

Note that by the projection formula and (5.4),

\begin{align*}(\rho^{\ast}L)^{3}=K_{V}\cdot(\rho^{\ast}L)^{2}=0.\end{align*}

Hence we have

\begin{align*}\chi(V,{\mathcal O}_{V}(n\rho^{\ast}L))=\frac{n}{12}(K_{V}^{2}+c_{2}(V))\cdot \rho^{\ast}L+\chi({\mathcal O}_{V}).\end{align*}

If $T_{V}$ is not strongly $(\rho^{\ast}L,\rho^{\ast}L)$ -semistable, then by Theorem 2.32 there is a 1-foliation ${\mathcal F}\subseteq T_{V}$ such that $\mu({\mathcal F})\gt0=\mu(T_{V})$ with respect to $(\rho^{\ast}L,\rho^{\ast}L)$ . By [Reference ZhangZha23, Corollary 2.2] we can take a sufficiently ample line bundle H on V such that $a\rho^{\ast}L+H$ are very ample for all integers $a\geq 0$ . Then we can use two general divisors in $|a\rho^{\ast}L+H|$ to cut out a smooth curve $C_{a}$ such that ${\mathcal F}$ is smooth along $C_{a}$ . Note that

\begin{align*}c_{1}({\mathcal F})\cdot C_{a}&=c_{1}({\mathcal F}) \cdot(a\rho^{\ast}L+H)^{2}\\&=a^{2}c_{1}({\mathcal F})\cdot(\rho^{\ast}L)^{2} +2a c_{1}({\mathcal F}) \cdot\rho^{\ast}L\cdot H+c_{1}({\mathcal F})\cdot H^{2}, \\K_{V}\cdot C_{a}&=K_{V}\cdot(a\rho^{\ast}L+H)^{2}\\&=2a K_{V}\cdot\rho^{\ast}L\cdot H+K_{V}\cdot H^{2},\end{align*}

and

\begin{align*}\rho^{\ast}L\cdot C_{a}&=\rho^{\ast}L\cdot(a\rho^{\ast}L+H)^{2}\\&=2a (\rho^{\ast}L)^{2}\cdot H+\rho^{\ast}L\cdot H^{2}.\end{align*}

Since $\mu({\mathcal F})=({c_{1}({\mathcal F})\cdot(\rho^{\ast}L)^{2}})/{\mathrm{rk}\,{\mathcal F}}\gt0$ , $c_{1}({\mathcal F})\cdot C_{a}$ grows with $a^2$ , while $K_{V}\cdot C_{a}$ and $\rho^{\ast}L\cdot C_{a}$ grow at most linearly in a. In particular, for $a\gg 0$ ,

\begin{align*}c_{1}({\mathcal F})\cdot C_{a} \gt \frac{K_{V}\cdot C_{a}}{p-1}.\end{align*}

By Theorem 2.36, for general $y\in C_{a}$ , we know that there exists a rational curve $B_{y}$ such that $B_{y}$ passes through y and

\begin{align*}\rho^{\ast}L\cdot B_{y}\leq \frac{6p\rho^{\ast}L\cdot C_{a}}{(p-1)c_{1}({\mathcal F})\cdot C_{a}-K_{V}\cdot C_{a}}.\end{align*}

Note that $\rho^{\ast}L\cdot B_{y}$ is a non-negative integer, while the denominator of the right-hand side above grows with $a^2$ , and the numerator grows at most linearly in a. Hence $\rho^{\ast}L\cdot B_{y}=0$ for $a\gg 0$ . Since $C_{a}$ and y are general, we obtain a family of curves $B_{y}$ covering V such that $\rho^{\ast}L\cdot B_{y}=0$ . It follows that

\begin{align*}n(K_{X}+B)=n(L)=n(\rho^{\ast}L)\leq 2.\end{align*}

By Theorem 2.28, we have that $K_{X}+B$ is semi-ample.

From now on, we can assume that $T_{V}$ is strongly $(\rho^{\ast}L,\rho^{\ast}L)$ -semistable. Then we have that $\Omega^{[1]}_{X}$ is strongly (L,L)-semistable since $\rho$ is birational. Hence by Theorem 4.8,

\begin{align*}\Delta(\Omega^{[1]}_{X})\cdot L=(6c_{2}(\Omega^{[1]}_{X})-2c_{1}(\Omega^{[1]}_{X})^{2})\cdot L\geq 0.\end{align*}

Since $K_{X}^{2}\cdot L=K_{X}^{2}\cdot (K_{X}+B)=0$ by (5.4), we have

(5.5) \begin{equation} c_{2}(\Omega^{[1]}_{X})\cdot L\geq 0.\end{equation}

Note that by assumptions (1)–(5) of the proposition, the assumptions in Setting 5.1 are satisfied. Hence we can apply Proposition 5.3 and (5.4) to conclude that

(5.6) \begin{align}\chi(V,{\mathcal O}_{V}(n\rho^{\ast}L))&=\frac{n}{12}(K_{V}^{2}+c_{2}(V))\cdot \rho^{\ast}L+\chi({\mathcal O}_{V})\nonumber\\[3pt]&\geq \frac{n}{12}(K_{X}^{2}+c_{2}(\Omega^{[1]}_{X}))\cdot L+\chi({\mathcal O}_{V})\nonumber\\[3pt]&= \frac{n}{12}c_{2}(\Omega^{[1]}_{X})\cdot L+\chi({\mathcal O}_{V}).\end{align}

We claim that there exists a constant $t_{1}$ such that

(5.7) \begin{equation}\chi(X,{\mathcal O}_{X}(nL))\geq \chi(V,{\mathcal O}_{V}(n\rho^{\ast}L))+t_{1}.\end{equation}

Note that $R^{i}\rho_{\ast}{\mathcal O}_{V}$ are supported on the closed subsets of X which are of dimension at most $2-i$ for $i=1$ or 2. By the Leray spectral sequence, we have

\begin{equation*}\begin{aligned}&\chi(V,{\mathcal O}_{V}(n\rho^{\ast}L))\\&\quad=h^{0}(V,{\mathcal O}_{V}(n\rho^{\ast}L))-h^{1}(V,{\mathcal O}_{V}(n\rho^{\ast}L))+h^{2}(V,{\mathcal O}_{V}(n\rho^{\ast}L))-h^{3}(V,{\mathcal O}_{V}(n\rho^{\ast}L))\\&\quad=h^{0}(X,{\mathcal O}_{X}(nL))-(h^{1}(X,{\mathcal O}_{X}(nL))+h^{0}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL)))\\&\qquad+(h^{2}(X,{\mathcal O}_{X}(nL))+ h^{1}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))+h^{0}(X,R^{2}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL)))\\&\qquad-h^{3}(X,{\mathcal O}_{X}(nL))\\&\quad=\chi(X,{\mathcal O}_{X}(nL))-h^{0}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))+h^{1}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))\\&\qquad+h^{0}(X,R^{2}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL)).\end{aligned}\end{equation*}

It follows that

\begin{align*}&\chi(X,{\mathcal O}_{X}(nL))+h^{1}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))+h^{0}(X,R^{2}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))\\&\quad\geq \chi(V,n\rho^{\ast}L)+h^{0}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL)).\end{align*}

Note that $h^{0}(X,R^{2}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))$ is bounded for $n\geq 0$ since $R^{2}\rho_{\ast}{\mathcal O}_{V}$ are supported on some points. Since $R^{1}\rho_{\ast}{\mathcal O}_{V}$ is a coherent sheaf supported on a closed subset $Z_{1}$ of X which is of dimension at most 1, $h^{1}(X,R^{1}\rho_{\ast}{\mathcal O}_{V}\otimes{\mathcal O}_{X}(nL))$ is bounded for $n\geq 0$ by [Reference LazarsfeldLaz04, Theorem 1.4.40]. Hence (5.7) holds.

Now let us consider the exact sequence

(5.8) \begin{equation} 0\to {\mathcal O}_{X}(nL-B)\to{\mathcal O}_{X}(nL)\to{\mathcal O}_{B}(nL|_{B})\to 0.\end{equation}

We claim that $h^{2}(X,nL)$ is bounded for $n\geq 0$ . We write $g:B^{\prime}\to B$ for the $S_{2}$ -fication of B. By Lemma 2.19, we have $(K_{X}+B)|_{B^{\prime}}\sim_{{\mathbb Q}}K_{B^{\prime}}+\Delta_{B^{\prime}}$ and $(B^{\prime},\Delta_{B^{\prime}})$ is an slc pair, where $|_{B'}$ denotes the pullback to B’. Note that by (5.4) we have

\begin{align*}(K_{B^{\prime}}+\Delta_{B^{\prime}})^{2}=B\cdot (K_{X}+B)^{2}=0,\end{align*}

and by assumptions (2) and (4) of the proposition we have

\begin{align*}(K_{B^{\prime}}+\Delta_{B^{\prime}})\cdot A|_{B^{\prime}}=B\cdot (K_{X}+B)\cdot A\gt0,\end{align*}

where A is an ample divisor on X. Hence $\nu(K_{B^{\prime}}+\Delta_{B^{\prime}})=1$ . By Theorem 2.18,

\begin{align*}\kappa(L|_{B'})=\kappa(K_{B^{\prime}}+\Delta_{B^{\prime}})=\nu(K_{B^{\prime}}+\Delta_{B^{\prime}})=1.\end{align*}

Hence

\begin{align*}h^{0}(B^{\prime},{\mathcal O}_{B^{\prime}}(nm(K_{B^{\prime}} +\Delta_{B^{\prime}})))=h^{0}(B^{\prime},{\mathcal O}_{B^{\prime}}(n L|_{B'}))\end{align*}

grows with n. By Serre duality,

\begin{equation*}\begin{split}h^{2}(B^{\prime},{\mathcal O}_{B^{\prime}}(n L|_{B'}))&=h^{0}(B^{\prime},{\mathcal O}_{B^{\prime}}(K_{B^{\prime}}-n L|_{B'}))\\&=h^{0}(B^{\prime},{\mathcal O}_{B^{\prime}}(K_{B^{\prime}}-nm(K_{X}+B)|_{B^{\prime}}))\\&= h^{0}(B^{\prime},{\mathcal O}_{B^{\prime}}(K_{B^{\prime}}-nm(K_{B^{\prime}}+\Delta_{B^{\prime}})))\\&=0\end{split}\end{equation*}

for $n\gg 0$ . It follows that

\begin{align*}h^{2}(B,{\mathcal O}_{B}(nL|_{B}))\end{align*}

is bounded for $n\geq 0$ by Lemma 5.4. Moreover, since

\begin{align*}nL-B\sim_{{\mathbb Q}}K_{X}+(nm-1)(K_{X}+B)\end{align*}

and $K_{X}+B$ is a nef Weil divisor such that $\nu(K_{X}+B)=2$ , by Lemma 5.5 we have that $h^{2}(X,{\mathcal O}_{X}(nL-B))$ is bounded for $n\geq 0$ . Therefore, by (5.8), $h^{2}(X,{\mathcal O}_{X}(nL))$ is bounded for $n\geq 0$ .

If the inequality in (5.6) is strict, then by (5.5)

\begin{align*}(K_{V}^{2}+c_{2}(V))\cdot \rho^{\ast}L \gt c_{2}(\Omega^{[1]}_{X})\cdot L\geq 0.\end{align*}

Hence $\chi(V,{\mathcal O}_{V}(n\rho^{\ast} L))$ grows with n. By (5.7), $\chi(X,{\mathcal O}_{X}(nL))$ , and hence $h^{0}(X,{\mathcal O}_{X}(nL))$ , grow with n. This implies that $\kappa(K_{X}+B)\gt0$ . The assertion holds. Otherwise, we have $(K_{X}+B)|_{B^{\prime}}$ is numerically trivial on $\operatorname{Supp} \Delta_{B^{\prime}}$ by Proposition 5.3. Since $h^{2}(X,{\mathcal O}_{X}(nL))$ is bounded for $n\geq 0$ , by (5.5), (5.6), and (5.7) there exists a constant $t_{2}$ such that