Differential forms on log canonical spaces in positive characteristic

TL;DR

This paper proves extension properties of differential forms on log canonical surfaces in positive characteristic, establishing new residue and restriction sequences, and providing a new proof of the characteristic zero theorem.

Contribution

It extends the understanding of differential forms on log canonical spaces in positive characteristic, including new residue and restriction sequences for tamely dlt pairs.

Findings

Differential forms extend with logarithmic poles on log canonical surfaces in characteristic p ≥ 7.

Residue and restriction sequences are established for tamely dlt pairs.

Results are sharp in surfaces and demonstrate failure in higher dimensions.

Abstract

Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous statement for regular differential forms, under an additional tameness hypothesis. In addition, residue and restriction sequences for tamely dlt pairs are established. We give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. On the other hand, our techniques yield a new proof of the characteristic zero Logarithmic Extension Theorem in any dimension.

Equations

$${\pi }_{\ast }{\Omega }_{Y/k}^{[q]}\hookrightarrow {\Omega }_{X/k}^{[q]}$$

$$\pi_{*}\Omega_{Y/k}^{[q]}\big{(}\!\log\,\lfloor{D_{Y}}\rfloor+E\big{)}\xhookrightarrow{\quad}\Omega_{X/k}^{[q]}\big{(}\!\log\,\lfloor{D}\rfloor\big{)}$$

$$0\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D-P\big{)}\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\xrightarrow{\;\operatorname{res}_{P}\;}\mathscr{O}_{P}\longrightarrow 0$$

$$0\longrightarrow\Omega_{P}^{1}\big{(}\!\log\lfloor{P^{c}}\rfloor\big{)}\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\big{|}_{P}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}\xrightarrow{\;\operatorname{res}_{P}^{P}\;}\mathscr{O}_{P}\longrightarrow 0.$$

$$\operatorname{res}_{P}^{m}\colon\operatorname{Sym}^{[m]}\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\longrightarrow\mathrel{\mkern-25.0mu}\longrightarrow\mathscr{O}_{P}$$

$$0\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D-P\big{)}\xrightarrow{\;\operatorname{restr}_{P}\;}\Omega_{P}^{1}\big{(}\!\log\lfloor{P^{c}}\rfloor\big{)}\longrightarrow 0$$

$$\operatorname{restr}_{P}^{m}\colon\operatorname{Sym}^{[m]}\Omega_{X}^{[1]}\big{(}\!\log D-P\big{)}\longrightarrow\mathrel{\mkern-25.0mu}\longrightarrow\mathscr{O}_{P}\big{(}mK_{P}+\lfloor{mP^{c}}\rfloor\big{)}$$

$$\mathrm{Div}(X){\otimes }_{\mathbb{Z}}{\mathbb{Z}}_{(p)}⟶\mathrm{WDiv}(X){\otimes }_{\mathbb{Z}}{\mathbb{Z}}_{(p)}.$$

$$\mathscr{E}(*D)\coloneqq\varinjlim\big{(}\mathscr{E}\otimes\mathscr{O}_{X}(mD)\big{)}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$$

$$Y=Y_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K_Y+{\tilde{D}}_0={\pi }^{\ast }(K_X+D)+\sum _{i=1}^n(a_i+1)F_i.$$

$$K_Z+{\tilde{D}}_r=f^{\ast }(K_X+D)+\underset{\text{f-nef by construction}}{\underbrace{{\phi }_{\ast }\left(\sum _{i=1}^n(a_i+1)F_i\right)}}.$$

$$(K_{X}+B+B^{\prime})\big{|}_{\overline{B}}\;\sim_{\mathbb{Q}}\;K_{\overline{B}}+\mathrm{Diff}_{\overline{B}}(B^{\prime}).$$

$${\mathrm{Diff}}_B(B^{\prime })=\sum _{x\in B}{\delta }_x\cdot [x],$$

$${\mathrm{Diff}}_P(D-P)=\sum _{x\in P}{\delta }_x\cdot [x]$$

$${\gamma }_{\alpha }:V_{\alpha }\to U_{\alpha }\text{finite quasi-eˊtale separable cyclic Galois}$$

$$D_V$$

$${\gamma }_D$$

$$\sigma$$

$${\sigma }_V$$

$$\omega$$

$$\mathrm{res}({\sigma }_V)=\mathrm{res}({\omega }^{[m]})=\mathrm{res}(\omega {)}^m={\omega }^{\prime m},$$

$$\mathrm{res}({\sigma }_V)={\gamma }_D^{\ast }\mathrm{res}(\sigma ).$$

$${\epsilon }^k{u}^{km}(\epsilon m{u}^{m-1}\mathrm{d}u+u^m\mathrm{d}\epsilon {)}^m$$

$$=(\mathrm{d}u{)}^m,$$

$$\operatorname{Sym}^{[m]}\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\longrightarrow\mathscr{O}_{P}(*\lceil{P^{c}}\rceil)$$

$$\operatorname{res}_{P_{V}}^{m}\colon\operatorname{Sym}^{[m]}\Omega_{V}^{[1]}\big{(}\!\log D_{V}\big{)}=\operatorname{Sym}^{m}\Omega_{V}^{1}\big{(}\!\log D_{V}\big{)}\longrightarrow\operatorname{Sym}^{m}\mathscr{O}_{P_{V}}=\mathscr{O}_{P_{V}}.$$

$$\gamma^{*}\sigma\in\mathit{\Gamma}\!\left(V,\;\operatorname{Sym}^{[m]}\Omega_{V}^{[1]}\big{(}\!\log D_{V}\big{)}\right).$$

$$(\gamma {|}_{P_V}{)}^{\ast }(\tilde{\sigma })\in \Gamma \left(P_V,{\mathcal{O}}_{P_V}(\ast \mathrm{supp}\gamma {|}_{P_V}^{\ast }\lceil P^c\rceil )\right)$$

$${\mathrm{res}}_{P_V}^m({\gamma }^{\ast }\sigma )\in \Gamma \left(P_V,{\mathcal{O}}_{P_V}\right)$$

$$\operatorname{res}_{P_{V}}^{m}\colon\operatorname{Sym}^{m}\Omega_{V}^{1}\big{(}\!\log D_{V}\big{)}\longrightarrow\mathscr{O}_{P_{V}}$$

$$0\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\longrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\big{|}_{P}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}\longrightarrow 0.$$

$$0\longrightarrow\Omega_{X}^{1}\big{(}\!\log D\big{)}(-P)\longrightarrow\Omega_{X}^{1}\big{(}\!\log D-P\big{)}\xrightarrow{\;\operatorname{restr}_{P}\;}\Omega_{P}^{1}\big{(}\!\log P^{c}\big{)}\longrightarrow 0$$

Full text

The author was supported in full by a DFG Research Fellowship.

1. 1 Introduction
2. 2 Notation and conventions
3. 3 Factorizing resolutions
4. 4 Adjunction and the different on dlt surface pairs
5. 5 Residues and restriction on dlt surfaces
6. 6 Lifting forms along a non-positive map
7. 7 Proof of 1.2
8. 8 Proof of 1.3
9. 9 The characteristic zero Extension Theorem revisited
10. 10 Sharpness of results
11. 11 Counterexamples in higher dimensions

1. Introduction

Differential forms play an essential rôle in the study of algebraic varieties. Given an algebraic variety $X$ over a field $k$ and a resolution of singularities $\pi\colon Y\to X$, it is natural to ask whether any $p$-form on the regular locus ${X}_{\mathrm{reg}}$ extends to a regular $p$-form on $Y$. There is also a version of this question which concerns pairs and allows certain logarithmic poles. In order to fix our terminology once and for all, we introduce the following language. (For notation, see Section 2).

Definition** (Extension properties for differential forms).**

Let $(X,D)$ be a pair (i.e. $X$ is normal and $D$ is a Weil $\mathbb{Q}$-divisor with coefficients in $[0,1]\cap\mathbb{Q}$) defined over a field $k$, and $1\leq q\leq\dim X$ an integer.

• $\circ$

We say that $(X,D)$ satisfies the Regular Extension Theorem for $q$-forms if for any proper birational map $\pi\colon Y\to X$ from a normal variety $Y$, the natural inclusion

[TABLE]

is an isomorphism. Equivalently, the sheaf $\pi_{*}\Omega_{Y/k}^{[q]}$ is reflexive. It is sufficient to check this for a resolution of singularities $Y\to X$ (if available): cf. [GKK10, Lemma 2.13] and note that the proof given there is independent of the base field.

• $\circ$

We say that $(X,D)$ satisfies the Logarithmic Extension Theorem for $q$-forms if for any map $\pi$ as above, with $D_{Y}$ the strict transform of $D$ and $E\subset Y$ the reduced divisorial part of the exceptional set $\operatorname{Exc}(\pi)$, the natural inclusion

[TABLE]

is an isomorphism. Equivalently, the sheaf \pi_{*}\Omega_{Y/k}^{[q]}\big{(}\!\log\,\lfloor{D_{Y}}\rfloor+E\big{)} is reflexive. Again, it is sufficient to check this for a log resolution $Y\to X$ of $(X,D)$.

• $\circ$

We say that $(X,D)$ satisfies the Regular Extension Theorem if it satisfies the Regular Extension Theorem for $q$-forms, for all values of $q$. Ditto for the logarithmic variant.

Over the complex numbers, the problem of when the Extension Theorems hold has a long history. It has been studied by several people using different methods—the following list is not exhaustive: [SvS85, Fle88, Nam01, dJS04, GKK10, GKKP11]. The paper mentioned last, [GKKP11], can in many ways be seen as the culmination111Very recently, it has been generalized further in [KS21], using perverse sheaves. of this line of research. It proved the following:

1. (1.1.1)

Any complex klt (= Kawamata log terminal) pair $(X,D)$ satisfies the Regular Extension Theorem [GKKP11, Thm. 1.4]. 2. (1.1.2)

Any complex log canonical pair $(X,D)$ satisfies the Logarithmic Extension Theorem [GKKP11, Thm. 1.5].

Given the importance of these results, it is not free of interest to ask whether similar results also hold in positive characteristic. Curiously enough, no research in this direction has been conducted so far. We have identified two main reasons for this:

• $\circ$

It has been known to experts for some time that (1.1.1) fails in a strong sense in positive characteristic. In fact, over any field of nonzero characteristic, there exists a strongly $F$-regular (in particular, klt) surface $X$ violating the Regular Extension Theorem (10.2).

• $\circ$

The proof of (1.1.2) relies on rather subtle Hodge-theoretic vanishing theorems for Du Bois spaces. These are either false or not known in positive characteristic, inextricably linking the proof to the complex numbers. The same can be said of the techniques in [KS21].

The purpose of this article is to overcome these obstacles, at least for surfaces (but see 1.6 for higher dimensions). Concerning the first issue, our approach is pretty straightforward: as (1.1.1) fails, we instead concentrate on (1.1.2). (Cf. however 1.3, which explores the failure of (1.1.1) more thoroughly.) To deal with the second problem, we develop a completely novel and much more hands-on approach to extension. Our first main result is as follows.

Theorem 1.2** (Logarithmic Extension for surfaces).**

Let $(X,D)$ be a log canonical surface pair over a perfect field $k$ of characteristic $p\geq 7$. Then $(X,D)$ satisfies the Logarithmic Extension Theorem.

Our second main result explains when the Logarithmic Extension Theorem does imply the Regular Extension Theorem.

Theorem 1.3** (Regular Extension for surfaces).**

Let $\left(0\in X,D\right)$ be a surface singularity over a field $k$ of characteristic $p>0$. Assume that for some (not necessarily log) resolution $\pi\colon Y\to X$, with exceptional curves $E_{1},\dots,E_{\ell}$, the determinant of the intersection matrix $(E_{i}\cdot E_{j})$ is not divisible by $p$. Then if $\left(0\in X,D\right)$ satisfies the Logarithmic Extension Theorem for $1$-forms, it also satisfies the Regular Extension Theorem for $1$-forms.

We would like to emphasize the advantages of our approach over the existing techniques. First of all, we feel that our proof offers a new level of both transparency and tangibility, as it does not explicitly use any Hodge theory (it does, however, rely on the Minimal Model Program). Secondly, this very same feature also makes it, to a large extent, insensitive to the characteristics of the ground field. In fact, aside from some effortless changes our approach also yields a new proof of the characteristic zero extension theorem [GKKP11, Thm. 1.5]—the details are worked out in Section 9. Thirdly and maybe most importantly, we obtain a lucid explanation of why the Logarithmic Extension Theorem fails in low characteristics, even for surface rational double points.

Further results in this paper

Apart from the above extension results, we establish residue and restriction sequences for reflexive differential forms on dlt pairs in positive characteristic, and symmetric powers thereof. This is analogous to known results in characteristic zero [GKKP11, Gra15]. However, it is important to note that actually a slightly stronger notion is required, called tamely dlt in this paper. A dlt pair $(X,D)$ is tamely dlt if $D$ is reduced and the Cartier index of $K_{X}+D$ is not divisible by $p$ (4.2).

The precise statement is as follows. Even though we only use it as a technical tool in the proof of our main result, we believe that it is of independent interest.

Theorem 1.4** (Residue sequence).**

Let $(X,D)$ be a tamely dlt surface pair (in particular, $D$ is reduced), and let $P\subset D$ be an irreducible component. Set $P^{c}\coloneqq\mathrm{Diff}_{P}(D-P)$, so that (K_{X}+D)\big{|}_{P}=K_{P}+P^{c}. Then there is a short exact sequence

[TABLE]

which on the snc locus of $(X,D)$ agrees with the usual residue sequence. Its restriction to $P$ induces a short exact sequence222Here, of course, in the middle term we are taking the double dual on $P$ and not on $X$ (the latter would be zero).

[TABLE]

More generally, for every $m\in\mathbb{N}$ there is a surjective map

[TABLE]

which generically coincides with the $m$-th symmetric power of the residue map.

Theorem 1.5** (Restriction sequence).**

Notation as above. Then there is a short exact sequence333By definition, \Omega_{X}^{[1]}\big{(}\!\log D\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{}} \rotatebox{90.0}{{}}}}} means the double dual of \Omega_{X}^{[1]}\big{(}\!\log D\big{)}\otimes\mathscr{O}_{X}(-P). Taking the reflexive hull is necessary because $P\subset X$ is in general not a Cartier divisor.

[TABLE]

which on the snc locus of $(X,D)$ agrees with the usual restriction sequence. More generally, for every $m\in\mathbb{N}$ there is a surjective map

[TABLE]

which generically coincides with the $m$-th symmetric power of the restriction map.

Sharpness of results

In Section 10, we have gathered a number of examples to show that our results are sharp. First of all, 1.2 does fail in characteristic less than seven, even if $k$ is algebraically closed, $D=0$ and $X$ is a rational double point (RDP). More precisely, we show by explicit calculation that the singularity given by the equation $z^{2}+x^{3}+y^{5}=0$ violates the Logarithmic Extension Theorem over any field of characteristic $p\leq 5$. In the terminology of Artin’s classification of RDPs [Art77], this is the $E_{8}^{0}$ singularity. This failure also occurs for some singularities of types $D_{n}$ ($p=2$) and $E_{6},E_{7}$ ($p=2,3$). We have omitted those calculations, as they are very similar in spirit to the $E_{8}$ case.

Turning to 1.3, its statement is sharp too, as shown by the example of contracting a smooth rational curve with self-intersection $-p$ in any characteristic $p>0$. In this case, the Logarithmic Extension Theorem holds for $1$-forms, but the Regular Extension Theorem does not. Again, this can be seen via explicit computation.

The latter example can also be elaborated upon to show that 1.5 fails for dlt pairs that are not tamely dlt. If one tries to run the proof of 1.2 on, say, a $D_{n}$ singularity in characteristic two, the lack of a suitable restriction map is exactly where the argument breaks down: already the first contraction performed by the MMP produces a pair that is not tamely dlt. This should be seen as the deeper reason for the failure of 1.2 in low characteristics.

Higher dimensions

For the majority of readers, a most pressing question will be to what extent 1.2 carries over to higher dimensions. As we will see in Section 9, in characteristic zero the higher-dimensional Logarithmic Extension Theorem is intimately linked to the fact that on a projective snc pair $(X,D)$, a line bundle \mathscr{L}\subset\Omega_{X}^{p}\big{(}\!\log D\big{)} cannot be big unless $p=\dim X$. This is the content of the Bogomolov–Sommese vanishing theorem [EV92, Cor. 6.9], while the weaker statement that $\mathscr{L}$ cannot be ample is a special case of (Kodaira–Akizuki–)Nakano vanishing [AN54, Thm. 1*′′*]. Both results fail badly in positive characteristic and in fact there are counterexamples strong enough to show that 1.2 itself does not hold. The precise statement is as follows and the details of the construction can be found in Section 11.

Theorem 1.6** (Failure of the higher-dimensional Logarithmic Extension Theorem).**

Fix an algebraically closed field $k$ of characteristic $p>0$.

1. (1.6.1)

In any dimension $n\geq p-1$, there exists a log canonical pair $(X,\emptyset)$ over $k$ that violates the Logarithmic Extension Theorem for $(n-2)$-forms. 2. (1.6.2)

If $n\geq 2p-1$, there exists a canonical pair $(X,\emptyset)$ for which the Logarithmic Extension Theorem fails as above. 3. (1.6.3)

If $n\geq 3p-1$, there even exists a terminal pair $(X,\emptyset)$ as above.

Furthermore, the above examples admit log resolutions.

As 1.2 already fails for surfaces if the characteristic is low, 1.6 becomes interesting only for $p\geq 7$. In this sense, the lowest-dimensional example it provides is a $6$-dimensional singularity in characteristic $7$. The following conjecture hence remains open.

One Sacrilegious Conjecture**.**

Over a perfect field of characteristic $p\geq 7$, the Logarithmic Extension Theorem holds for log canonical pairs of dimension $\leq p-2$.

Of course, we do not believe in the Sacrilegious Conjecture. Rather, our inability to disprove it is caused by a lack of techniques to produce meaningful counterexamples.

Relation to $F$-singularities

The examples in 1.6 are (un-)fortunately not $F$-pure. On the other hand, using classification results [Har98, Thm. 1.1] one can show that all normal $F$-regular surface singularities over a perfect field satisfy the Logarithmic Extension Theorem. The same is probably true for $F$-pure surfaces, but the case distinctions get much more tedious. These observations have led us to the following intriguing question:

Question 1.7**.**

Is there a version of the Logarithmic Extension Theorem for strongly $F$-regular/$F$-pure singularities that does not exclude low characteristics and works in any dimension?

The following line of attack appears to be quite promising. By [Wat91, Thm. 3.3], the affine cone over a smooth projective variety $X$ is $F$-pure if and only if $X$ is (globally) $F$-split. Hence one would need to investigate whether $F$-split varieties satisfy Nakano vanishing. Since at least Kodaira vanishing obviously holds for these, chances may not be that bad. This would immediately provide a positive answer to 1.7 for cones. On the other hand, if Nakano vanishing failed, we would obtain an $F$-pure counterexample to the Logarithmic Extension Theorem.

Outline of proof

We would like to explain the general strategy for the proof of 1.2. It can be broken up into three major steps.

Step 1: Baby case

The basic idea is quite simple. Consider an lc surface pair $(X,D)$ as in the theorem, where for simplicity we will assume $D=0$. Also, fix a log resolution $\pi\colon Y\to X$, with exceptional divisor $E$. Given any reflexive $1$-form $\sigma\in\mathrm{H}^{0}\!\left(X,\Omega_{X}^{[1]}\right)$, we may extend $\sigma$ to a rational logarithmic $1$-form $\widetilde{\sigma}$ on $Y$, i.e. a global section of \Omega_{Y}^{1}\big{(}\!\log E\big{)}(G), where $G\neq 0$ is effective and $\operatorname{supp}(G)\subset E$. This is equivalent to giving a map \mathscr{O}_{Y}(-G)\to\Omega_{Y}^{1}\big{(}\!\log E\big{)}. We use the fact that $G^{2}<0$ combined with the residue sequence on the snc pair $(Y,E)$ to show that this map factors as \mathscr{O}_{Y}(-G)\to\Omega_{Y}^{1}\big{(}\!\log E-P\big{)}, for some component $P\subset E$. The same argument repeated, but this time using the restriction sequence, then shows that our map actually even factors as \mathscr{O}_{Y}(-G)\to\Omega_{Y}^{1}\big{(}\!\log E\big{)}(-P). This means that $G$ may be replaced by the strictly smaller divisor $G-P$. Repeating this procedure finitely many times, namely as long as $G$ is nonzero, we finally obtain $G=0$ and hence \widetilde{\sigma}\in\mathrm{H}^{0}\!\left(Y,\Omega_{Y}^{1}\big{(}\!\log E\big{)}\right) as desired.

It turns out that for this approach to work smoothly, $-(K_{Y}+E)$ needs to be $\pi$-nef. By adjunction, (K_{Y}+E)\big{|}_{P}=K_{P}+(E-P)\big{|}_{P} for every component $P\subset E$ and hence the nefness condition in practice means that $E$ is either a chain or a cycle of rational curves, or a single elliptic curve. Curiously, this already implies 1.2 (in any characteristic!) for two extreme cases: the $A_{n}$ singularities on the one hand and Gorenstein log canonical singularities that are not canonical on the other hand.

Step 2: General case

For e.g. a $D_{4}$ singularity, the baby case argument breaks down and this is where the technical complications, as well as the restrictions on the ground field, start. Indeed, the crucial idea is to use the Minimal Model Program to factor the resolution $\pi$ into a series of steps each of which satisfies the nefness condition from Step 1, as detailed in Section 3. As is well-known, if run on an snc pair, the MMP will produce intermediate steps and an end result that are only dlt. This is where Theorems 1.4 and 1.5 come into play. They are proved in Section 5, with preparations in Section 4.

With these technical generalizations in place, the ideas from Step 1 apply to show that $1$-forms extend along each step in the factorization of $\pi$ provided by the MMP. The precise statement may be found in 6.1. Piecing all the steps together, we can prove our result if $(X,D)$ has a tame resolution, i.e. one that factors in such a way that all intermediate pairs are tamely dlt. It is easy to see that this implies the following weak form of 1.2: For every extended dual graph $\Gamma$ (i.e. incorporating self-intersection numbers as well as the boundary components), there is a prime number $p_{0}=p_{0}(\Gamma)$ such that every log canonical surface pair with dual graph $\Gamma$ and defined over a field of characteristic $p\geq p_{0}$ satisfies the Logarithmic Extension Theorem.

Step 3: Effective bounds

Two things remain to be done: First, to eliminate the dependency of $p_{0}$ on $\Gamma$ and second, to give an effective value for $p_{0}$. In order to achieve this, we resort to the classification of log canonical surface pairs over an algebraically closed field. (This is also where the perfectness hypothesis on $k$ comes from: the base change to the algebraic closure needs to be separable.) We stress that this is the only place in the whole paper where classification is used. It turns out that we may choose $p_{0}=7$, finishing the proof of 1.2. The details are contained in Section 7.

Acknowledgements

I would like to thank Karl Schwede and Thomas Polstra for helpful discussions and answering many of my questions. Jorge Vitório Pereira has brought the paper [Kol95] to my attention via MathOverflow. The Department of Mathematics at the University of Utah has provided support and excellent working conditions.

I am particularly grateful to the anonymous referee for extremely diligent proofreading, which has not only greatly improved the presentation of certain details, but also led to some simplifications.

2. Notation and conventions

Base field

Throughout this paper, we work over a field $k$, which except for Section 9 will be assumed to be of positive characteristic $p>0$. Further assumptions (perfect, algebraically closed, …) will be expressly stated whenever necessary.

Pairs and divisors

A pair $(X,D)$ consists of a normal variety $X$ and a Weil $\mathbb{Q}$-divisor $D=\sum a_{i}D_{i}$ with coefficients $0\leq a_{i}\leq 1$. The pair is called reduced if $D$ is reduced. The round-down of $D$ is denoted by $\lfloor{D}\rfloor\coloneqq\sum\lfloor{a_{i}}\rfloor D_{i}$, and similarly for the round-up $\lceil{D}\rceil$. The fractional part $\{D\}$ is, by definition, $D-\lfloor{D}\rfloor$. For a uniform definition of the singularities of the MMP (klt, plt, dlt, lc, …), we refer to [Kol13, Def. 2.8].

The regular and singular loci of a variety $X$ are denoted ${X}_{\mathrm{reg}}$ and ${X}_{\mathrm{sg}}$, respectively. We say that a closed subset $Z\subset X$ is small if $\mathrm{codim}_{X}(Z)\geq 2$, and that an open subset $U\subset X$ is big if $X\setminus U$ is small.

A Weil divisor $D$ on a normal variety $X$ is said to be $\mathbb{Z}_{(p)}$-Cartier if it has a multiple not divisible by $p$ which is Cartier. Equivalently, $D$ is in the image of the natural map

[TABLE]

Since $\mathbb{Z}_{(0)}=\mathbb{Q}$, in characteristic zero we recover the usual notion of being $\mathbb{Q}$-Cartier. More generally, the Cartier index of $D$ is the smallest integer $m>0$ with $mD$ Cartier (or $+\infty$ if no such $m$ exists).

Reflexive sheaves

Let $X$ be a normal variety and $\mathscr{E}$ a coherent sheaf on $X$. The $\mathscr{O}_{X}$-double dual (or reflexive hull) of $\mathscr{E}$ is denoted by $\mathscr{E}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$. The sheaf $\mathscr{E}$ is called reflexive if the canonical map $\mathscr{E}\to\mathscr{E}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ is an isomorphism. A Weil divisorial sheaf is a reflexive sheaf of rank one. A coherent subsheaf $\mathscr{A}\subset\mathscr{E}$ of a reflexive sheaf is said to be saturated if the quotient \left.\raise 2.0pt\hbox{\mathscr{E}}\right/\hskip-2.0pt\raise-2.0pt\hbox{\mathscr{A}} is torsion-free. We use square brackets [-] as an abbreviation for taking the double dual, e.g. $\mathscr{E}^{[k]}=(\mathscr{E}^{\otimes k})^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ and $f^{[*]}\mathscr{E}=(f^{*}\mathscr{E})^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ for a map $f\colon Y\to X$ with $Y$ normal.

Let $D\subset X$ be a reduced divisor. Then we denote by

[TABLE]

the quasi-coherent sheaf of sections of $\mathscr{E}$ with arbitrarily high order poles along $D$. If $i\colon U\hookrightarrow X$ is the inclusion of the snc locus of $(X,D)$, the sheaf of reflexive differential $q$-forms is defined to be \Omega_{X/k}^{[q]}\big{(}\!\log D\big{)}\coloneqq i_{*}\Omega_{U/k}^{q}\big{(}\!\log D\big{|}_{U}\big{)}. The base field $k$ will usually be dropped from notation.

Following are some useful properties of reflexive sheaves which will be used implicitly or explicitly. For proofs, we refer to [Gra15, Sec. 3].

Lemma 2.1**.**

Let $\mathscr{E}$ be a reflexive sheaf on the normal variety $X$ and $\mathscr{A},\mathscr{B}\subset\mathscr{E}$ coherent subsheaves, with $\mathscr{A}$ saturated.

1. (2.1.1)

The sheaf $\mathscr{A}$ is reflexive. 2. (2.1.2)

Let $s$ be a rational section of $\mathscr{A}$ which is regular as a section of $\mathscr{E}$. Then $s$ is also regular as a section of $\mathscr{A}$. 3. (2.1.3)

Suppose that for some dense open subset $U\subset X$, the subsheaves \mathscr{A}\big{|}_{U} and \mathscr{B}\big{|}_{U} of \mathscr{E}\big{|}_{U} are equal. Then it follows that $\mathscr{B}\subset\mathscr{A}$. ∎

3. Factorizing resolutions

It is well-known that in characteristic zero, the MMP can be used to obtain log crepant partial resolutions for log canonical pairs (called “minimal dlt models”, “dlt blowups”, or “dlt modifications”). See for example [KK10, Thm. 3.1]. Here we would like to point out that the same argument also works for surfaces over arbitrary fields. The reason is that the MMP for log canonical surfaces is very well developed [Tan18]. In fact, our proof is even simpler than the one in [KK10] because we do not have to perturb the dlt pair of interest into a linearly equivalent klt pair.

Unlike [KK10], we are not only interested in the end product of the MMP (in the notation below, the map $f$), but also in the intermediate steps. Note that since we are on a surface, we can use Mumford’s pullback to get the same result also for numerically log canonical pairs [KM98, Notation 4.1]. This will be important later.

Theorem 3.1**.**

Let $(X,D)$ be a numerically log canonical surface pair and $\pi\colon Y\to X$ a log resolution, with exceptional divisor $E$. Then $\pi$ can be factored into a sequence of maps as follows:

[TABLE]

such that, setting $\widetilde{D}_{0}\coloneqq\pi^{-1}_{*}D+E$ and $\widetilde{D}_{i+1}\coloneqq(\varphi_{i})_{*}\widetilde{D}_{i}$, the following properties hold:

1. (3.1.1)

For any $0\leq i\leq r$, the pair $(Y_{i},\widetilde{D}_{i})$ is dlt and $Y_{i}$ is $\mathbb{Q}$-factorial. 2. (3.1.2)

For any $0\leq i\leq r-1$, the exceptional locus of $\varphi_{i}$ is irreducible. 3. (3.1.3)

The map $f$ is (numerically) log crepant, that is, $K_{Z}+\widetilde{D}_{r}=f^{*}(K_{X}+D)$.

Proof.

Let $F_{1},\dots,F_{n}$ be all the irreducible components of $E$, and consider the ramification formula $K_{Y}+\pi^{-1}_{*}D=\pi^{*}(K_{X}+D)+\sum_{i=1}^{n}a_{i}F_{i}$, where $\pi^{*}(-)$ denotes Mumford’s pullback. We then have

[TABLE]

We may run the MMP on the dlt pair $(Y,\widetilde{D}_{0})$ and obtain a minimal model $\varphi\colon Y\to Z$ over $X$ [Tan18, Thm. 1.1]. This provides the maps in the statement to be proven. Also, (3.1.1) and (3.1.2) are clear by construction. It remains to show (3.1.3). To this end, push forward 3.1.1 to $Z$:

[TABLE]

The Negativity Lemma [KM98, Lemma 3.40] implies that the underbraced term in the above formula is zero. Hence 3.1.2 simplifies to (3.1.3). ∎

4. Adjunction and the different on dlt surface pairs

The different is a correction term that makes the adjunction formula work in the presence of singularities. For a general treatment of the different, including the case of positive characteristic, see [Kol13, Ch. 4]. On a surface, things are somewhat simpler, as explained in [Kol13, Def. 2.34].

Proposition/Definition 4.1** (Different on surfaces).**

Let $X$ be a normal $\mathbb{Q}$-factorial surface and $B\subset X$ a reduced irreducible curve with normalization $\overline{B}\to B$. Let $B^{\prime}$ be a $\mathbb{Q}$-divisor that has no common components with $B$. Then there is a canonically defined $\mathbb{Q}$-divisor $\mathrm{Diff}_{\overline{B}}(B^{\prime})$ on $\overline{B}$, called the different, such that

[TABLE]

We will mostly be interested in the case where $(X,B)$ is dlt, in which case $B$ is regular by 4.4 below. Hence $\overline{B}=B$ and we may write

[TABLE]

where $\delta_{x}\neq 0$ only for points $x$ that are singular on $X$ or contained in $\operatorname{supp}B^{\prime}$. We need to compute the coefficients $\delta_{x}$ in relation to the singularities of $(X,B+B^{\prime})$. In positive characteristic this is only possible under the following additional tameness hypothesis:

Definition 4.2** (Tamely and fiercely dlt pairs).**

A pair $(X,D)$ over a field of characteristic $p$ is called tamely dlt if the following hold:

1. (4.2.1)

$(X,D)$ is reduced and dlt, 2. (4.2.2)

$K_{X}+D$ is $\mathbb{Z}_{(p)}$-Cartier (see Section 2).

If Condition (4.2.1) is satisfied but (4.2.2) is not, the pair is said to be fiercely dlt.

In the case $p=0$, we recover the usual notion of a reduced dlt pair. The main result concerning the different is then as follows. The reader may like to compare this to [Kol13, Thm. 3.36], where a similar formula is proven under slightly different assumptions.

Theorem 4.3** (Computation of the different).**

Let $(X,D)$ be a tamely dlt surface pair, and let $P\subset D$ be an irreducible component. Write

[TABLE]

as above. Then, referring to the dichotomy in 4.4 below:

1. (4.3.1)

If locally at $x$, (4.4.1) holds, then $\delta_{x}=1$. 2. (4.3.2)

If locally at $x$, (4.4.2) holds, then $\delta_{x}=1-\frac{1}{m}$, where $m$ is the Cartier index of $K_{X}+D$ at $x$.

4.A. The local structure of dlt surfaces

Locally, dlt surface pairs are in some sense quite simple (even if they are fierce):

Proposition 4.4** (Dichotomy for dlt surfaces).**

Let $(X,D)$ be a reduced dlt surface pair, and let $x\in\operatorname{supp}D$ be any point. Then either one of the following holds:

1. (4.4.1)

The pair $(X,D)$ is snc at $x$, and $x$ is contained in exactly two components of $D$. 2. (4.4.2)

The divisor $D$ is regular at $x$ and the pair $(X,D)$ is plt at $x$.

In particular, every irreducible component of $D$ is regular.

Proof.

Assume that we are not in case (4.4.1). Then either $(X,D)$ is snc at $x$, but $D$ has only one component at $x$. In this case, (4.4.2) clearly holds. Or the pair $(X,D)$ is not snc at $x$, in which case it is plt at $x$ by definition. Regularity of $D$ at $x$ then follows from [Kol13, 3.35]. ∎

In the following corollary, the crucial point is the separability of the maps $\gamma_{\alpha}$. Note that the $U_{\alpha}$ cover only $\operatorname{supp}D$ and not all of $X$.

Corollary 4.5** (Dlt surfaces as quotients).**

Let $(X,D)$ be a tamely dlt surface pair. Then there exist finitely many Zariski-open subsets $\left\{U_{\alpha}\right\}_{\alpha\in I}$ of $X$ that cover $\operatorname{supp}D$ and admit maps

[TABLE]

such that the pairs $(V_{\alpha},\gamma_{\alpha}^{*}D)$ are snc for all indices $\alpha\in I$.

Proof.

Let $x\in\operatorname{supp}D$ be any point, and apply 4.4. If we are in case (4.4.1), we may take $\gamma_{\alpha}=\mathrm{id}$ and there is nothing to show. In case (4.4.2), let $\gamma_{\alpha}$ be a local index one cover with respect to $K_{X}+D$. Then $\gamma_{\alpha}$ by construction has all the properties claimed, except separability. But separability is also clear because of our assumption that $K_{X}+D$ is $\mathbb{Z}_{(p)}$-Cartier. It remains to see that $(V_{\alpha},\gamma_{\alpha}^{*}D)$ is snc. To this end, note that this pair is again plt [Kol13, Cor. 2.43]. Furthermore, as $K_{V_{\alpha}}+\gamma_{\alpha}^{*}D$ is Cartier, the discrepancies are actually integral and hence non-negative. The pair $(V_{\alpha},\gamma_{\alpha}^{*}D)$ is therefore canonical. Let $y\in V_{\alpha}$ be the unique point in $\gamma_{\alpha}^{-1}(x)$. Then $y\in\operatorname{supp}\gamma_{\alpha}^{*}D$. The claim now follows from [Kol13, Thm. 2.29]. ∎

4.B. Proof of 4.3

Case (4.3.1) is clear, hence we concentrate on Case (4.3.2). We follow the local computational approach as illustrated in [Kol13, Ex. 4.3]. Let $\gamma\colon V\to U$ be a map as in 4.5, where $x\in U$, and put

[TABLE]

Then $\sigma_{V}=\omega^{[m]}$ up to a unit, that is, for a suitable choice of $\omega$. It follows that

[TABLE]

where $\omega^{\prime}$ is a local generator for $\omega_{D_{V}}$. On the other hand, as $\gamma$ is quasi-étale and the residue map (in the snc case) commutes with étale pullback, we have

[TABLE]

Let $t\in\mathscr{O}_{D,x}$ be a local parameter of $D$ at $x$, and let $u\in\mathscr{O}_{D_{V},y}$ be a local parameter of $D_{V}$ at the unique point $y$ lying over $x$ such that $\omega^{\prime}=\mathrm{d}u$. Then $\gamma_{D}^{*}(t)=\varepsilon u^{m}$ for some unit $\varepsilon\in\mathscr{O}_{D_{V},y}^{\times}$. Hence, writing $\operatorname{res}(\sigma)=t^{k}(\mathrm{d}t)^{m}$ up to a unit, with $k$ to be determined, combining 4.5.1 with 4.5.2 gives

[TABLE]

where the dots stand for terms involving higher powers of $u$. By the tameness assumption, $m\neq 0$ in the ground field and we obtain $m(k+m-1)=0$. So $k=1-m$ and $\delta_{x}=-k/m=1-1/m$, as claimed. ∎

5. Residues and restriction on dlt surfaces

In this section, we prove Theorems 1.4 and 1.5.

5.A. Proof of 1.4

The proof is divided into four steps.

Step 1: Symmetric residue maps

First we will construct the maps $\operatorname{res}_{P}^{m}$. So fix a natural number $m$ and consider the $m$-th symmetric power of the residue map on the snc locus of $(X,D)$. Pushing it forward to all of $X$ yields a map

[TABLE]

to the sheaf of rational functions on $P$ with arbitrarily high order poles along $\operatorname{supp}\lceil{P^{c}}\rceil$. We need to show that 5.1.1 factorizes via \operatorname{Sym}^{[m]}\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\to\mathscr{O}_{P}, for this will be the desired map $\operatorname{res}_{P}^{m}$. So let $\sigma$ be an arbitrary local section of \operatorname{Sym}^{[m]}\Omega_{X}^{[1]}\big{(}\!\log D\big{)}, defined on an open set $U\subset X$. Let $\widetilde{\sigma}$ be its image under 5.1.1, and (after possibly shrinking $U$) pick a map $\gamma\colon V\to U$ as in 4.5.

We will employ the following regularity criterion: $\widetilde{\sigma}\in\mathit{\Gamma}\!\left(U,\mathscr{O}_{P}\right)$ if and only if $\gamma^{*}(\widetilde{\sigma})\in\mathit{\Gamma}\!\left(V,\mathscr{O}_{P_{V}}\right)$, where $P_{V}\coloneqq\gamma^{*}P$ and $D_{V}\coloneqq\gamma^{*}D$. This criterion holds because $P$ is regular, in particular normal. (Recall that if $A\subset B$ is a finite extension of normal domains and $Q(A)$ is the fraction field of $A$, then $B\cap Q(A)=A$. In our situation, $A$ is a local ring of $P$ and $B$ is a suitable local ring of $P_{V}$.)

By 4.5, the pair $(V,D_{V})$ is snc, hence \Omega_{V}^{1}\big{(}\!\log D_{V}\big{)} is locally free and we obtain a residue map

[TABLE]

Furthermore, note that $\gamma\colon(V,D_{V})\to(U,D|_{U})$ is a “morphism of logarithmic pairs” in the sense of [GKK10, Def. 2.4] (this simply means that $\gamma^{-1}(D|_{U})=D_{V}$ set-theoretically). Therefore, by [GKK10, Remark 2.10] we can pullback $\sigma$ to a regular section of \operatorname{Sym}^{m}\Omega_{V}^{1}\big{(}\!\log D_{V}\big{)}, at least off the preimage of the non-snc locus ${(U,D|_{U})}_{\mathrm{sg}}$. But $\gamma$ is finite (in particular equidimensional), so this preimage still has codimension two in $V$. Hence $\gamma^{*}\sigma$ is regular on all of $V$. In other words,

[TABLE]

Recall that the standard residue map commutes with étale pullback, and that $\gamma$ is étale over the general point of $P$. So the two functions

[TABLE]

and

[TABLE]

agree on an open subset of $P_{V}$, hence everywhere. This shows that $\widetilde{\sigma}$ is a regular function on $P_{V}$, as desired.

Step 2: Surjectivity

It remains to show surjectivity of the maps $\operatorname{res}_{P}^{m}$. This is a local question, so we may restrict ourselves to an open set $U\subset X$ admitting a map $\gamma\colon V\to U$ as in 4.5. Let $G=\operatorname{Gal}(\gamma)$ be the Galois group of $\gamma$. Start with the map

[TABLE]

as before and note that we can also construct $\operatorname{res}_{P}^{m}$ by applying the functor $\gamma_{*}(-)^{G}$ to $\operatorname{res}_{P_{V}}^{m}$. This means that we consider U=\left.\raise 2.0pt\hbox{V}\right/\hskip-2.0pt\raise-2.0pt\hbox{G} with the trivial $G$-action and look at the invariant sections of the relevant push-forward sheaves (which are $G$-sheaves in a natural way). For more details, cf. [GKKP11, Appendix A].

The claim now follows from the surjectivity of $\operatorname{res}_{P_{V}}^{m}$ (which is due to the fact that the pair $(V,D_{V})$ is snc) and the exactness of the functor $\gamma_{*}(-)^{G}$. This exactness holds because the order of $G$ is prime to $p$ by the “tamely dlt” assumption, and therefore we have the usual Reynolds operator argument at our disposal. Cf. the characteristic zero version of this argument [GKKP11, Lemma A.3].

Step 3: Residue sequence on $X$

Next we prove the existence of sequence 1.4.1. The map $\operatorname{res}_{P}$ is of course nothing but the special case $m=1$ of the maps just constructed. By what we already know, we thus only need to show that its kernel is isomorphic to \Omega_{X}^{[1]}\big{(}\!\log D-P\big{)}. But that kernel is a reflexive sheaf by [Har80, Cor. 1.5]. Furthermore it is isomorphic to \Omega_{X}^{[1]}\big{(}\!\log D-P\big{)} on ${(X,D)}_{\mathrm{snc}}$, by the usual residue sequence for snc pairs. The isomorphism then extends to all of $X$ by reflexivity.

Step 4: Residue sequence on $P$

Finally we turn to sequence 1.4.2. Clearly, the reflexive restriction of $\operatorname{res}_{P}$ to $P$ is a surjective map \operatorname{res}_{P}^{P}\colon\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\big{|}_{P}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}\longrightarrow\mathscr{O}_{P}, and it remains to show that its kernel is isomorphic to \Omega_{P}^{1}\big{(}\!\log\lfloor{P^{c}}\rfloor\big{)}. To this end, first note that there is a short exact sequence

[TABLE]

In fact, the second map is surjective because on a regular curve, taking the double dual really just amounts to dividing out the torsion. And by the same argument as in the previous step, the kernel is reflexive and thus isomorphic to \Omega_{X}^{[1]}\big{(}\!\log D\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}.

Consider now the commutative diagram with exact rows and columns depicted in Figure 1 1.

The first row is the restriction sequence 1.5.1444Needless to say, the proof of 1.5 does not rely on 1.4—see Section 5.B below., while the second row is 5.1.2. The middle column is 1.4.1, the residue sequence on $X$. The Snake Lemma then shows that the dotted arrow \Omega_{P}^{1}\big{(}\!\log\lfloor{P^{c}}\rfloor\big{)}\dashrightarrow\Omega_{X}^{[1]}\big{(}\!\log D\big{)}\big{|}_{P}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}} exists, is injective, and that its image is exactly the kernel of $\operatorname{res}_{P}^{P}$. The column on the right-hand side is therefore likewise exact, and it is precisely sequence 1.4.2. ∎

5.B. Proof of 1.5

The proof of 1.5 is analogous to the proof of 1.4, hence we will only provide an outline, with most details omitted. To begin with, if $(X,D)$ is snc then sequence 1.5.1 reads

[TABLE]

and this exists by [EV92, 2.3(c)]. In particular, we already have $\operatorname{restr}_{P}$ and its $m$-th symmetric power on the snc locus ${(X,D)}_{\mathrm{snc}}$. Pushing forward this symmetric power to all of $X$, we obtain a map

[TABLE]

and we have to show that it factors via a map

[TABLE]

for this will be the desired map $\operatorname{restr}_{P}^{m}$. To this end, we have the following criterion:

Claim 5.2*.*

Notation as in the previous proof. A local section $\widetilde{\sigma}$ of $\mathscr{O}_{P}(mK_{P})(*\lceil{P^{c}}\rceil)$ is contained in $\mathscr{O}_{P}(mK_{P}+\lfloor{mP^{c}}\rfloor)$ if and only if $\gamma^{*}(\widetilde{\sigma})$ is a regular section of $\mathscr{O}_{P_{V}}(mK_{P_{V}}+mP_{V}^{c})$, where P_{V}^{c}\coloneqq\mathrm{Diff}_{P_{V}}(D_{V}-P_{V})=(D_{V}-P_{V})\big{|}_{P_{V}}.

Proof of 5.2.

The proof of this criterion is done by a local computation similar to the one in the proof of 4.3, whose notation we adopt. If locally at $x\in\operatorname{supp}\lceil{P^{c}}\rceil$, Case (4.4.1) holds, the claim is clear. Therefore we focus on Case (4.4.2). Note that in this case $D_{V}$ is smooth, which implies $P_{V}=D_{V}$ and thus $P_{V}^{c}=0$.

Write $\widetilde{\sigma}=t^{k}\,(\mathrm{d}t)^{m}$ with $k\in\mathbb{Z}$ (locally and up to units), and let $\ell$ be the Cartier index of $K_{X}+D$ at $x$. The coefficient of $\lfloor{mP^{c}}\rfloor$ at $x$ is $\lfloor{m(1-\frac{1}{\ell})}\rfloor$, therefore $\widetilde{\sigma}$ is contained in $\mathscr{O}_{P}(mK_{P}+\lfloor{mP^{c}}\rfloor)$ if and only if

[TABLE]

On the other hand, $\gamma^{*}(t)=\varepsilon u^{\ell}$ and hence

[TABLE]

where the dots stand for terms involving higher powers of $u$. We see that $\gamma^{*}(\widetilde{\sigma})$ is regular if and only if $k\ell+m(\ell-1)\geq 0$. This is equivalent to 5.2.1, proving the claim. ∎

Once the maps $\operatorname{restr}_{P}^{m}$ are constructed, their surjectivity follows from the right-exactness of $\gamma_{*}(-)^{G}$, as before. Finally, to obtain sequence 1.5.1 we set $\operatorname{restr}_{P}\coloneqq\operatorname{restr}_{P}^{1}$. On the snc locus ${(X,D)}_{\mathrm{snc}}$, the kernel agrees with \Omega_{X}^{[1]}\big{(}\!\log D\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}} by the snc case mentioned in the beginning of the proof. Since both sheaves are reflexive, they agree everywhere. ∎

6. Lifting forms along a non-positive map

The following theorem, while technical in nature, is at the heart of the paper. The “non-positivity” in the title refers to property (6.1.2) below.

Theorem 6.1** (Lifting forms).**

Let $g\colon Y\to X$ be a proper birational map of normal surfaces over a field $k$, with $E=\operatorname{Exc}(g)$ the reduced exceptional divisor. Furthermore let $D$ be a reduced divisor on $X$, and set $D_{Y}\coloneqq g^{-1}_{*}D+E$. Assume the following:

1. (6.1.1)

The pair $(Y,D_{Y})$ is tamely dlt, and 2. (6.1.2)

the anticanonical divisor $-(K_{Y}+D_{Y})$ is $g$-nef.

Then the natural map

[TABLE]

is an isomorphism.

Step 0: Setup of notation and outline of proof strategy

Let

[TABLE]

be a nonzero reflexive logarithmic $1$-form, and let $g^{*}\sigma$ be its pullback to $Y$, considered as a rational section of the sheaf \Omega_{Y}^{[1]}\big{(}\!\log D_{Y}\big{)}. We want to show that $g^{*}\sigma$ is in fact a regular section of that sheaf. To this end, first pick an effective $g$-exceptional divisor $G$ such that

[TABLE]

For example, $G$ may be taken to be the pole divisor of the rational section $g^{*}\sigma$. We will show that whenever $G$ is nonzero, there is a curve $P\subset\operatorname{supp}G$ such that 6.1.1 continues to hold with $G$ replaced by $G-P$. Iterating this argument finitely often, we arrive at $G=0$, hence g^{*}\sigma\in\mathrm{H}^{0}\!\left(Y,\Omega_{Y}^{[1]}\big{(}\!\log D_{Y}\big{)}\right) as desired.

Step 1: Residue sequence

Assume that 6.1.1 holds for some $G\neq 0$. Then $G^{2}<0$ by the Negativity Lemma (applied on some resolution of $Y$) and consequently, $G\cdot P<0$ for some exceptional curve $P\subset\operatorname{supp}G\subset E$. Twisting by $\mathscr{O}_{Y}(-G)$ and taking the reflexive hull, 6.1.1 induces a map i\colon\mathscr{O}_{Y}(-G)\to\Omega_{Y}^{[1]}\big{(}\!\log D_{Y}\big{)}. As $g^{*}\sigma\neq 0$, this map is nonzero and hence injective. On the tamely dlt pair $(Y,D_{Y})$, we have the residue sequence 1.4.1

[TABLE]

Claim 6.2*.*

The composition ${\operatorname{res}_{P}}\circ i$ is zero, and hence $i$ factors via a map $j$ as indicated by the dashed arrow in the above diagram.

Proof of 6.2.

Let $m\geq 1$ be sufficiently divisible so that $mG$ is Cartier (recall that $Y$ is $\mathbb{Q}$-factorial). The $m$-th reflexive symmetric power of $i$, composed with the map $\operatorname{res}_{P}^{m}$ from 1.4, yields a map

[TABLE]

which is nothing but the $m$-th reflexive symmetric power of ${\operatorname{res}_{P}}\circ i$. Hence in order to show that ${\operatorname{res}_{P}}\circ i$ vanishes, it is sufficient to prove the vanishing of 6.2.1. As the target of the latter map is supported on $P$, it is zero if and only if its restriction to $P$ is zero. But that restriction is a map $\mathscr{O}_{P}(-mG)\to\mathscr{O}_{P}$, or in other words, an element of $\mathrm{H}^{0}\!\left(P,\mathscr{O}_{P}(mG)\right)$. As $G\cdot P<0$ and $mG$ is Cartier, the latter space is zero. ∎

Step 2: Restriction sequence

We essentially repeat Step 1, but with the residue sequence replaced by the restriction sequence 1.5.1:

[TABLE]

Claim 6.3*.*

The composition ${\operatorname{restr}_{P}}\circ j$ is zero, and hence $j$ factors via a map $\iota$ as indicated by the dashed arrow in the above diagram.

Proof of 6.3.

Let $m$ be as in the proof of 6.2, so that $mG$ is Cartier. The $m$-th reflexive symmetric power of $j$, composed with the map $\operatorname{restr}_{P}^{m}$ from 1.5, is the $m$-th reflexive symmetric power of ${\operatorname{restr}_{P}}\circ j$:

[TABLE]

As in 6.2, it suffices to show that the restriction of 6.3.1 to $P$ vanishes. This is a map $\mathscr{O}_{P}(-mG)\to\mathscr{O}_{P}(mK_{P}+\lfloor{mP^{c}}\rfloor)$, or in other words, an element of $\mathrm{H}^{0}\!\left(P,\mathscr{O}_{P}(mK_{P}+{\lfloor{mP^{c}}\rfloor}+mG)\right)$. As

[TABLE]

the latter space is zero. This ends the argument. ∎

The proof of 6.1 is now easily finished: the existence of the map $\iota$ is equivalent to giving a global section of the sheaf \Omega_{Y}^{[1]}\big{(}\!\log D_{Y}\big{)}(G-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}, which of course is exactly the form $g^{*}\sigma$ we started with. This shows that 6.1.1 holds with $G-P$ in place of $G$, as desired. ∎

7. Proof of 1.2

The aim of this section is to prove our first main result: any log canonical surface pair $(X,D)$ over a perfect field of characteristic $\geq 7$ satisfies the Logarithmic Extension Theorem. The following notion will play a key role.

Definition 7.1** (Tame resolutions).**

Let $(X,D)$ be a reduced log canonical surface pair over a field $k$. A tame resolution of $(X,D)$ is a log resolution $\pi\colon Y\to X$ together with a factorization of $\pi$ as in 3.1 such that

1. (7.1.1)

for any $0\leq i\leq r-1$, the pair $(Y_{i},\widetilde{D}_{i})$ is tamely dlt, and 2. (7.1.2)

if $f$ is not an isomorphism (this can happen only if $(X,D)$ is not plt), then also $(Z,\widetilde{D}_{r})$ is required to be tamely dlt.

7.A. Auxiliary results

First we show that when dealing with log canonical surface pairs, there is no loss of generality in assuming them to be reduced. We also prove that having a tame resolution implies the Logarithmic Extension Theorem and that the Logarithmic Extension Theorem is invariant under separable base change. The latter property is used for reducing to the case of an algebraically closed ground field, where the classification of surface singularities becomes simpler.

Proposition 7.2** (Rounding down).**

Let $(X,D)$ be a log canonical surface pair. Then also $(X,\lfloor{D}\rfloor)$ is log canonical.

Proposition 7.3** (Tameness is sufficient).**

Let $(X,D)$ be a reduced log canonical surface pair admitting a tame resolution. Then $(X,D)$ satisfies the Logarithmic Extension Theorem for $1$-forms.

Proposition 7.4** (Base change).**

Let $(X,D)$ be a pair defined over a field $k$, and consider a separable field extension $k^{\prime}/k$. Set $X^{\prime}\coloneqq X\times_{k}k^{\prime}$ and $D^{\prime}\coloneqq D\times_{k}k^{\prime}$.

1. (7.4.1)

If $(X^{\prime},D^{\prime})$ satisfies the Regular Extension Theorem for $q$-forms, for some value of $q$, then so does $(X,D)$. 2. (7.4.2)

If $(X,D)$ admits a log resolution, the converse of (7.4.1) also holds.

Ditto for the Logarithmic Extension Theorem.

Proof of 7.2.

If $(X,D)$ is numerically log canonical, then so is $(X,\lfloor{D}\rfloor)$. Thus it suffices to show that $K_{X}+\lfloor{D}\rfloor$ is $\mathbb{Q}$-Cartier. The question is local, so we may concentrate attention on a point $x\in\operatorname{supp}{\{D\}}$, the fractional part of $D$. At such a point, the pair $(X,\lfloor{D}\rfloor)$ is even numerically dlt and then $(X,\emptyset)$ is numerically klt. Applying 3.1 to the latter pair, we get that $f\colon Z\to X$ is an isomorphism, as there are no exceptional divisors of discrepancy $-1$, and hence $X$ is even $\mathbb{Q}$-factorial because $Z$ is.

An alternative (yet closely related) argument goes by noting that the characteristic zero proof of [KM98, Prop. 4.11] still works if we replace the use of the basepoint-free theorem [KM98, Thm. 3.3] by [Tan18, Thm. 4.2]. ∎

Proof of 7.3.

Let $\pi\colon Y\to X$ be a tame resolution of $(X,D)$, where we keep notation from 3.1. It suffices to extend $1$-forms along each step of the given factorization separately. That is, we will prove the following two statements:

Claim 7.5*.*

The sheaf f_{*}\Omega_{Z}^{[1]}\big{(}\!\log\widetilde{D}_{r}\big{)} is reflexive.

Claim 7.6*.*

For any $0\leq i\leq r-1$, the sheaf (\varphi_{i})_{*}\,\Omega_{Y_{i}}^{[1]}\big{(}\!\log\widetilde{D}_{i}\big{)} is reflexive.

Proof of 7.5.

If $(X,D)$ is plt, then $f$ is an isomorphism and there is nothing to prove. Otherwise, we would like to apply 6.1. The tameness condition (6.1.1) is satisfied by (7.1.2). It remains to check (6.1.2), i.e. that $-(K_{Z}+\widetilde{D}_{r})$ is $f$-nef. To this end, let $P\subset Z$ be any $f$-exceptional curve and note that

[TABLE]

So $K_{Z}+\widetilde{D}_{r}$ is even $f$-numerically trivial. 7.5 is proved. ∎

Proof of 7.6.

Again, we will apply 6.1 and only Condition (6.1.2), the $\varphi_{i}$-nefness of $-(K_{Y_{i}}+\widetilde{D}_{i})$, needs to be checked. Let $P\subset Y_{i}$ be the unique $\varphi_{i}$-exceptional curve. Since $(Y_{i+1},\widetilde{D}_{i+1})$ is dlt (in particular, log canonical) and \widetilde{D}_{i}=(\varphi_{i}^{-1})_{*}\big{(}\widetilde{D}_{i+1}\big{)}+P, we have

[TABLE]

where \lambda=a\big{(}P,Y_{i+1},\widetilde{D}_{i+1}\big{)}+1\geq 0. On the other hand, $P^{2}<0$ by the Negativity Lemma. Hence

[TABLE]

7.6 now follows from 6.1. ∎

By 7.5 and 7.6, also the sheaf

[TABLE]

is reflexive. The proof of 7.3 is thus finished. ∎

Proof of 7.4.

For any object (variety, map, sheaf, …) over $k$, we denote the base change to $k^{\prime}$ by $(-)^{\prime}$. Concerning (7.4.1), let $\pi\colon Y\to X$ be proper birational, with $Y$ normal. Then there is a commutative diagram

[TABLE]

Since $k^{\prime}/k$ is a separable field extension, the horizontal maps are étale and faithfully flat. In particular, $X^{\prime}$ and $Y^{\prime}$ are still normal, and after possibly replacing them by suitable connected components, $\pi^{\prime}$ is proper birational. By assumption, $\pi^{\prime}_{*}\Omega_{Y^{\prime}/k^{\prime}}^{[q]}$ is reflexive. But

[TABLE]

hence the claim follows from (7.7.3).

For (7.4.2), keep notation but assume additionally that $\pi$ is a resolution of singularities. By the above argument and (7.7.2), the sheaf $\pi^{\prime}_{*}\Omega_{Y^{\prime}/k^{\prime}}^{[q]}$ is reflexive. Because $Y^{\prime}\to Y$ is étale, $\pi^{\prime}$ is in fact a resolution and it follows that $X^{\prime}$ satisfies the Regular Extension Theorem for $q$-forms.

The proof in the logarithmic case is similar, and therefore omitted. ∎

Lemma 7.7** (Dual commutes with base change).**

Let $R$ be a noetherian ring, $M$ a finitely generated $R$-module, and $R\subset S$ a flat ring extension. Set $M_{S}\coloneqq M\otimes_{R}S$. Then:

1. (7.7.1)

The natural map $\mathrm{Hom}_{R}\!\left(M,R\right)\otimes_{R}S\to\mathrm{Hom}_{S}\!\left(M_{S},S\right)$ is an isomorphism. 2. (7.7.2)

If $M$ is reflexive, then so is $M_{S}$. 3. (7.7.3)

If $R\subset S$ is faithfully flat, the converse of (7.7.2) also holds.

Proof.

Let $F_{1}\to F_{0}\to M$ be a finite presentation of $M$. Dualizing and tensorizing with $S$, we get the first row in the following commutative diagram. First tensorizing and then dualizing gives the second row.

[TABLE]

The first row is exact because $S$ is flat over $R$, and the second row is exact for general reasons. The leftmost vertical arrow is the map in question, while the other two are isomorphisms because the $F_{i}$ are free. (7.7.1) now follows from the Snake Lemma.

For (7.7.2), consider the natural isomorphism $M\to M^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$. By (7.7.1), after tensorizing with $S$ it becomes the natural map $M_{S}\to M_{S}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}$, and it obviously stays an isomorphism. Hence $M_{S}$ is reflexive, too. If $S$ is faithfully flat, we may run the argument backwards, proving (7.7.3). ∎

7.B. Proof of 1.2

By 7.2, we may assume that $(X,D)$ is reduced. Furthermore, since our ground field $k$ is assumed to be perfect, its algebraic closure $\bar{k}$ is separable over $k$ and hence by 7.4, we may assume that $k=\bar{k}$. The singularities of reduced log canonical surface pairs over an algebraically closed ground field have been classified in [Kol13, Cor. 3.31, 3.39, 3.40]. According to this classification, there are seven cases to be considered. Their dual graphs are depicted in Figures 7–7 7 (the first case is not shown since it has only one exceptional curve). Here we use the following color and labeling pattern. The extra information thus contained in the figures is easily verified.

Notation 7.8*.*

A plain circle denotes an exceptional curve with discrepancy equal to $-1$. A node shaded in gray denotes an exceptional curve with discrepancy $>-1$. All exceptional curves are smooth rational. The components of $\pi^{-1}_{*}D$ are shown in black. A negative number attached to a vertex denotes the self-intersection of the corresponding curve. A leaf is a curve intersecting at most one other curve, while a fork intersects at least three other curves.

Since 1.2 is local, we may shrink $X$ and assume that $(X,D)$ has only one singular point. We use notation from 3.1, applied to the minimal resolution $\pi\colon Y\to X$ of $(X,D)$. In particular, $E$ is the exceptional locus of $\pi$ and $r$ is the number of contractions performed by the MMP before the minimal dlt model is reached. The classification is then as follows. (The names are actually valid only in characteristic zero. Here they are only meant for easier reference and should not be taken literally.)

1. (7.9.1)

(Simple elliptic, [Kol13, (3.39.1)]) Here $D=0$ and $E$ consists of a single smooth elliptic curve, which has discrepancy $-1$. So $r=0$ and the tameness condition on $\pi$ is automatically satisfied. In this case, 1.2 thus follows directly from 7.3. 2. (7.9.2)

(Cusp, Fig. 7) Again, there are no curves of discrepancy $>-1$, so $r=0$ and we conclude as before. 3. (7.9.3)

($\mathbb{Z}/2$-quotient of cusp or simple elliptic, Fig. 7) Here $r=4$ and each step $\varphi_{i}$ contracts a curve $C_{i}\subset U_{i}\subset Y_{i}$, where $U_{i}$ is a smooth open subset of $Y_{i}$ and $C_{i}^{2}=-2$. By [Kol13, Thm. 3.32], the resulting singularity is étale locally555As stated, [Kol13, Thm. 3.32] gives the result only up to completion (which would also be sufficient), but the proof shows that there is a map $\varphi_{i}(U_{i})\to\operatorname{Spec}k[u^{2},uv,v^{2}]$ which is étale at the point $\varphi_{i}(C_{i})$. isomorphic to the vertex of $\operatorname{Spec}k[u^{2},uv,v^{2}]$. Since $\operatorname{char}k\geq 7>2$, this singularity is actually a $\mathbb{Z}/2$-quotient of a smooth surface. Then by the usual norm argument, $2\cdot\Delta_{i}$ is Cartier for every integral Weil divisor $\Delta_{i}$ on $Y_{i}$, where $0\leq i\leq 4$. Applying this to $K_{Y_{i}}+\widetilde{D}_{i}$, we see that $\pi$ is a tame resolution and we conclude by 7.3. 4. (7.9.4)

(Other quotient of simple elliptic, Fig. 7) The three chains of rational curves $\Gamma_{i}$ can obviously be treated independently of each other, hence we will concentrate on, say, $\Gamma_{1}$. The first curve contracted has to be the leaf, since otherwise there would be two components of $\widetilde{D}_{1}$ intersecting at a singular point of $Y_{1}$, contradicting 4.4. Repeating this argument, we see that the curves in $\Gamma_{1}$ are contracted in sequence, starting from the leaf and proceeding towards the fork. In particular, at each step there is only one singular point and it is obtained by contracting a subchain of $\Gamma_{1}$. But $\det\Gamma_{1}\leq 6$ and by the recurrence relation in [Kol13, (3.33.1)], the same also holds for all its subchains. By [Kol13, Thm. 3.32] and the assumption $\operatorname{char}k\geq 7$, the singular point of each $Y_{i}$ is a quotient by a finite group of order $\leq 6$. As in the previous case (7.9.3), this implies that $\pi$ is tame and hence 1.2 holds also in this case. 5. (7.9.5)

(Cyclic quotient, Fig. 7) There are three subcases, according to whether the boundary $D$ has zero, one or two components. In the first two cases, it actually not true that $(X,D)$ has a tame resolution, since the chain $E$ can be arbitrarily long and hence infinitely many (in fact, all) primes would have to be excluded. So we cannot apply 7.3. But note that for every exceptional curve $P\subset Y=Y_{0}$, we have $\deg(K_{P}+P^{c})\leq 0$, where $P^{c}\coloneqq\mathrm{Diff}_{P}(\widetilde{D}_{0}-P)$. (The degree is $-1$ for the leaves and [math] for the other curves, since there is no fork.) Also the pair $(Y_{0},\widetilde{D}_{0})$ is clearly tamely dlt, since it is even snc. Hence in these cases, 1.2 is a direct consequence of 6.1 applied to $\pi$. In the third subcase, we may follow the same argument or else note that $r=0$, so $\pi$ is tame—it boils down to the same thing. 6. (7.9.6)

(Dihedral quotient, Fig. 7) We have two subcases: either $D=0$ or $D\neq 0$. If $D=0$, again there may not be a tame resolution. Instead, the MMP needs to be chosen in such a way that first the two $(-2)$-curves intersecting the fork are contracted. The resulting pairs $(Y_{i},\widetilde{D}_{i})$, $i=1,2$, are tamely dlt by the same reasoning as in case (7.9.3). If $P\subset Y_{2}$ is the image of the fork, both singular points of $Y_{2}$ appear in $P^{c}\coloneqq\mathrm{Diff}_{P}(\widetilde{D}_{2}-P)$ with coefficient either666As $K_{Y_{2}}+\widetilde{D}_{2}$ is not Cartier, the coefficient is actually $1/2$, but we only need an upper bound on the different. zero or $\frac{1}{2}$, by 4.3. Hence

[TABLE]

If $P\subset Y_{2}$ is any other exceptional curve, the above inequality also holds, as in case (7.9.5). We can therefore apply 6.1 to the map $Y_{2}\to X$ to conclude.

If $D\neq 0$, then $r=2$ and only the two $(-2)$-curves are contracted. The resolution $\pi$ is then tame by exactly the same argument as in case (7.9.3). 7. (7.9.7)

(Other quotient of a smooth surface, Fig. 7) The argument is similar to case (7.9.4). First the chains $\Gamma_{i}$ are contracted, starting from the leaves and progressing towards the fork. As $\det\Gamma_{i}\leq 5<7\leq\operatorname{char}k$, this implies that $(Y_{i},\widetilde{D}_{i})$ is tamely dlt for $0\leq i\leq r-1$. Furthermore $X$ is log terminal and $D=0$, so $(X,D)$ is plt and case (7.1.2) of the definition of tameness applies. So $\pi$ is tame and 7.3 gives the result.

Since we have now worked our way through all the cases, the proof of 1.2 is finished. ∎

8. Proof of 1.3

This section contains the proof of our second main result, 1.3. The argument proceeds in three steps.

8.A. Passing to a log resolution

First of all, by blowing up $Y$ further we may turn $\operatorname{Exc}(\pi)+D_{Y}$ into an snc divisor. We need to show that this does not change $\det(E_{i}\cdot E_{j})$ up to sign. Indeed, after renumbering we may assume that we are blowing up a point $p\in Y$ which is contained exactly in the exceptional curves $E_{1},\dots,E_{k}$, where $k\leq\ell$. Let $r_{i}$ be the multiplicity of $E_{i}$ at $p$. Set $A=(a_{ij})=(-E_{i}\cdot E_{j})$, the negative of the intersection matrix on $Y$, and $\widetilde{A}$ the analogous matrix after blowing up $p$, with the new exceptional curve put first. Also, let $A_{0}=\left(\begin{smallmatrix}0&0\\ 0&A\end{smallmatrix}\right)$ with one additional zero row and column. Then

[TABLE]

where there are $\ell-k$ zero rows and columns, respectively. By adding $r_{i}\cdot\text{(first column)}$ to the $(i+1)$-st column for all $1\leq i\leq k$, the matrix $\widetilde{A}$ is transformed into

[TABLE]

while keeping the determinant unchanged. Expanding $\det\widetilde{A}^{\prime}$ by the first row, we see that $\det\widetilde{A}^{\prime}=\det A$. Hence we may make the following

Additional Assumption 8.1*.*

The map $\pi\colon Y\to X$ is a log resolution of $(X,D)$.

8.B. Dropping the non-exceptional divisor

Pick an irreducible component $P\subset\lfloor{D}\rfloor$, and let $P_{Y}$ be its strict transform on $Y$. Then consider the short exact sequence given by the residue map [EV92, 2.3(b)]

[TABLE]

Pushing it forward via $\pi$ yields

[TABLE]

where $\mathscr{Q}\subset\pi_{*}\mathscr{O}_{P_{Y}}$. In particular, $\mathscr{Q}$ is supported on $P$ and it is torsion-free as an $\mathscr{O}_{P}$-module. Hence $\mathscr{Q}$ has only one associated prime, which is of height $1$. It then follows from [Har80, Cor. 1.5] that the sheaf \pi_{*}\Omega_{Y}^{1}\big{(}\!\log\lfloor{D_{Y}}\rfloor+E-P_{Y}\big{)} is likewise reflexive. Repeating this argument for all components $P\subset\lfloor{D}\rfloor$, we arrive at the conclusion that

[TABLE]

is reflexive, and hence isomorphic to $\Omega_{X}^{[1]}$. In other words, $(X,\emptyset)$ satisfies the Logarithmic Extension Theorem.

8.C. Dropping the exceptional divisor

Set $E=E_{1}+\cdots+E_{\ell}$, and consider the residue sequence [EV92, 2.3(a)]

[TABLE]

We need to show that

[TABLE]

is an isomorphism. It suffices to show that the connecting homomorphism

[TABLE]

is injective. To this end, consider the restriction map

[TABLE]

We will show that the composition

[TABLE]

is an isomorphism. In fact, on the left-hand side choose the basis consisting of the constant functions $\mathbf{1}_{E_{i}}$, and on each summand of the right-hand side, choose the basis canonically determined by the trace map. It is easy to see777For more details, the reader is advised to consult the proof of [GK14, Prop. 3.2], which is independent of the characteristic. that with respect to these bases, 8.1.1 is given by the intersection matrix $A\coloneqq(E_{i}\cdot E_{j})$. By the Negativity Lemma [KM98, Lemma 3.40], $A$ is negative definite (in particular, invertible) when considered as an integer matrix. Here, of course, we have to regard $A$ as defined over our ground field $k$ instead. However, by our assumption, the characteristic $p$ of $k$ does not divide $\det A$. Hence the matrix $A$ remains invertible when reduced modulo $p$. In other words, $r\circ\delta$ is an isomorphism, and then $\delta$ is injective. It follows that the sheaf $\pi_{*}\Omega_{Y}^{1}$ is reflexive. ∎

9. The characteristic zero Extension Theorem revisited

The purpose of this section is to explain how the ideas in this paper yield a new proof of [GKKP11, Thm. 1.5], repeated below as A. Even though we are ultimately only interested in that statement, in order to give a self-contained argument we have to set up an inductive procedure involving B below. The latter statement has already been proven in much greater generality in [Gra15, Thm. 1.2], but we must not use that result in our proof in order to avoid a circular dependence on [GKKP11].

For a very brief run-down of $\mathcal{C}$-pairs and $\mathcal{C}$-differentials, we refer to Section 9.A below. In the whole section, all varieties are assumed to be defined over the complex numbers.

Theorem A**.**

Let $(X,D)$ be a complex log canonical pair. Then $(X,D)$ satisfies the Logarithmic Extension Theorem.

Theorem B** (Bogomolov–Sommese vanishing).**

Let $(X,D)$ be a complex-projective dlt $\mathcal{C}$-pair and \mathscr{A}\subset\operatorname{Sym}_{\mathcal{C}}^{[1]}\Omega_{X}^{r}\big{(}\!\log D\big{)}=\Omega_{X}^{[r]}\big{(}\!\log\lfloor{D}\rfloor\big{)} a rank one reflexive subsheaf. Then the $\mathcal{C}$-Kodaira dimension $\kappa_{\mathcal{C}}(\mathscr{A})\leq r$.

The induction runs as follows, where the start of induction (dimension one) is trivial. Here, of course, “An” means “A for $X$ of dimension at most $n$”, and ditto for Bn.

• $\circ$

An implies Bn, and

• $\circ$

Bn implies An+1.

While the proofs of both directions do draw on some of the more elementary arguments in [GKKP11], we stress that the technical core of that paper is not used. Hence it still seems fair to say that our proof is “new”.

9.A. Background on $\mathcal{C}$-pairs

For a more thorough treatment, see [Gra15, Sec. 5] and the references therein. A $\mathcal{C}$-pair is a pair $(X,D)$ in the usual sense, where $D=\sum_{i}(1-\frac{1}{n_{i}})D_{i}$ with $n_{i}\in\mathbb{N}\cup\left\{\infty\right\}$. One then defines adapted morphisms $\gamma\colon Y\to X$, essentially by requiring that the ramification order over $D_{i}$ is equal to $n_{i}$. The sheaves of $\mathcal{C}$-differentials are subsheaves

[TABLE]

defined by the condition that the pullback of a local section under an adapted morphism has at worst logarithmic poles along $\operatorname{supp}\gamma^{*}\lfloor{D}\rfloor$. We have

[TABLE]

and for $m=1$, this is an equality. Let \mathscr{A}\subset\operatorname{Sym}_{\mathcal{C}}^{[1]}\Omega_{X}^{r}\big{(}\!\log D\big{)} be a Weil divisorial subsheaf. The $\mathcal{C}$-Kodaira dimension $\kappa_{\mathcal{C}}(\mathscr{A})$ is defined to be the maximum of the dimensions of $\overline{\varphi_{m}(X)}$, where $\varphi_{m}$ is the rational map given by the global sections of $\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{A}$. Here $\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{A}$ is the saturation of $\mathscr{A}^{[m]}$ inside \operatorname{Sym}_{\mathcal{C}}^{[m]}\Omega_{X}^{r}\big{(}\!\log D\big{)}.

9.B. Residue and restriction sequences

We need to have the results of Section 5 at our disposal in this setting. These are, to a large extent, already contained in [GKKP11, Sec. 11] and [Gra15, Sec. 6]. Hence we only give a sketch of the proof.

Theorem 9.1** (Residue sequence).**

Let $(X,D)$ be a dlt $\mathcal{C}$-pair and $P\subset\lfloor{D}\rfloor$ an irreducible component. Setting $P^{c}\coloneqq\mathrm{Diff}_{P}(D-P)$, the pair $(P,P^{c})$ is again a dlt $\mathcal{C}$-pair, and the following holds: For any integer $r\geq 1$, there is a sequence

[TABLE]

which is exact on $X$ off a codimension three subset and on ${(X,D)}_{\mathrm{snc}}$ agrees with the usual residue sequence. Its restriction to $P$ induces a sequence

[TABLE]

which is exact on $P$ off a codimension two subset. More generally, for every $m\in\mathbb{N}$ there is a map

[TABLE]

surjective off a codimension three subset of $X$, which generically coincides with the $m$-th symmetric power of the residue map. ∎

Theorem 9.2** (Restriction sequence).**

Notation as above. Then there is a sequence

[TABLE]

exact off a codimension three subset, which on ${(X,D)}_{\mathrm{snc}}$ agrees with the usual restriction sequence. More generally, for every $m\in\mathbb{N}$ there is a map

[TABLE]

which is surjective in codimension two and generically coincides with the $m$-th symmetric power of the restriction map. ∎

Proof sketch of Theorems 9.1 and 9.2.

Sequences 9.1.1 and 9.1.2 along with the respective properties are in [GKKP11, Thm. 11.7]. The existence of $\operatorname{res}_{P}^{m}$ is shown in [Gra15, Thm. 6.9(i)]. Likewise, the maps $\operatorname{restr}_{P}^{m}$ are constructed in [Gra15, Thm. 6.9(ii)]. What is missing is the following:

• $\circ$

the existence of sequence 9.2.1, and

• $\circ$

the surjectivity properties of $\operatorname{res}_{P}^{m}$ and $\operatorname{restr}_{P}^{m}$.

For the first item, the argument is similar to Step 3 in the proof of 1.4, i.e. we reduce to the snc case by a reflexivity argument. For the second item (which is in fact not used in this paper), we resort to Step 2 in the above proof, but instead of 4.5 we use [Gra15, Prop. 6.12]. ∎

9.C. Lifting along a non-positive map

The analog of 6.1 is as follows.

Theorem 9.3** (Lifting forms).**

Assume Bn. Let $g\colon Y\to X$ be a proper birational map of normal varieties of dimension at most $n+1$, with $E=\operatorname{Exc}(g)$ the reduced divisorial part of the exceptional locus. Furthermore let $D$ be an effective divisor on $X$, and set $D_{Y}\coloneqq g^{-1}_{*}D+E$. Assume both of the following:

1. (9.3.1)

The pair $(Y,D_{Y})$ is dlt and $\mathbb{Q}$-factorial. 2. (9.3.2)

For any irreducible component $P\subset E$, setting $P^{c}\coloneqq\mathrm{Diff}_{P}(D_{Y}-P)$, we have that $K_{P}+P^{c}$ is not g\big{|}_{P}-big.

Then for any integer $r\geq 1$, the natural map

[TABLE]

is an isomorphism.

Recall that if $f$ is any map, a divisor on the source of $f$ is called $f$-big if its restriction to a general fibre of $f$ is big.

The proof relies crucially on the following Negativity Lemma, which should be compared to the usual one [BCHM10, Lemma 3.6.2(1)]. Indeed our version is somewhat stronger, as it does not merely make a numerical statement, but actually produces sections of a suitable line bundle.

Proposition 9.4** (Big Negativity Lemma, cf. [Gra15, Prop. 4.1]).**

Let $\pi\colon Y\to X$ be a proper birational map between normal quasi-projective varieties. Then for any nonzero effective $\pi$-exceptional $\mathbb{Q}$-Cartier divisor $E$, there is an irreducible component $P\subset E$ such that -E\big{|}_{P} is \pi\big{|}_{P}-big. ∎

Proof of 9.3.

We first contend that we may replace $D$ by $\lfloor{D}\rfloor$ and thus assume that $D$ is reduced. To this end, note that $\lfloor{D_{Y}}\rfloor=g^{-1}_{*}\lfloor{D}\rfloor+E$, so the conclusion we are aiming at only depends on $\lfloor{D}\rfloor$. Also, as $Y$ is $\mathbb{Q}$-factorial, the pair $(Y,\lfloor{D_{Y}}\rfloor)$ remains dlt. Finally, for any component $P\subset E$ we have

[TABLE]

where $\{D_{Y}\}\coloneqq D_{Y}-\lfloor{D_{Y}}\rfloor\geq 0$ is the fractional part of $D_{Y}$. So Condition (9.3.2) is likewise preserved.

Now let \sigma\in\mathrm{H}^{0}\!\left(X,\Omega_{X}^{[r]}\big{(}\!\log D\big{)}\right)\setminus\left\{0\right\} be an arbitrary nonzero logarithmic $r$-form, and pick an effective $g$-exceptional divisor $G$ such that

[TABLE]

Equivalently, there is an injective map i\colon\mathscr{O}_{Y}(-G)\to\Omega_{Y}^{[r]}\big{(}\!\log D_{Y}\big{)}. We may assume that $G\neq 0$, in which case by 9.4 there is a component $P\subset G$ such that -G\big{|}_{P} is g\big{|}_{P}-big. Set $P^{c}\coloneqq\mathrm{Diff}_{P}(D_{Y}-P)$. Let $F\subset P$ be a general fibre of g\big{|}_{P}, and set F^{c}\coloneqq P^{c}\big{|}_{F}. Then $F$ is normal and $(F,F^{c})$ is again a dlt $\mathcal{C}$-pair. Also, define \mathscr{A}\subset\operatorname{Sym}_{\mathcal{C}}^{[1]}\Omega_{Y}^{r}\big{(}\!\log D_{Y}\big{)} to be the image of $i$. Now consider the residue sequence 9.1.1 along $P$:

[TABLE]

Claim 9.5*.*

We have $\operatorname{res}_{P}(\mathscr{A})=0$, and hence $\mathscr{A}$ is contained in \Omega_{Y}^{[r]}\big{(}\!\log D_{Y}-P\big{)} as indicated by the dashed arrow in the above diagram.

In the following, note that the restriction of a reflexive sheaf on $P$ to the general fibre $F$ remains reflexive and hence in this case the double dual may be omitted.

Proof of 9.5.

Arguing by contradiction, let us assume that $\operatorname{res}_{P}(\mathscr{A})\neq 0$ and denote its saturation by \mathscr{B}\subset\operatorname{Sym}_{\mathcal{C}}^{[1]}\Omega_{P}^{r-1}\big{(}\!\log P^{c}\big{)}, a Weil divisorial sheaf. By [Gra15, Prop. 7.3], there are a number $0\leq q\leq r-1$ and embeddings

[TABLE]

for all $k$, satisfying the compatibility conditions that $\mathscr{C}_{k}$ and $\mathscr{C}_{1}^{[k]}$ generically agree as subsheaves of \operatorname{Sym}_{\mathcal{C}}^{[k]}\Omega_{F}^{q}\big{(}\!\log F^{c}\big{)}. We will show that $\mathscr{C}\coloneqq\mathscr{C}_{1}$ is “$\mathcal{C}$-big” in the sense that $\kappa_{\mathcal{C}}(\mathscr{C})=\dim F$. If $q<\dim F$, this contradicts Bn. If $q=\dim F$, it contradicts Assumption (9.3.2), which says that K_{F}+F^{c}=(K_{P}+P^{c})\big{|}_{F} is not big.

To this end, we claim that for any natural number $m$ there is an inclusion

[TABLE]

The first inclusion holds because $i$ does not vanish along $P$ nor $F$ (otherwise we would necessarily have $\operatorname{res}_{P}(\mathscr{A})=0$). For the second one, the map $\operatorname{res}_{P}^{m}$ from 9.1 gives an inclusion

[TABLE]

which by (2.1.3) factors via the saturated subsheaf $\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{B}$. Restricting to $F$, we see that the middle term of 9.5.1 is contained in $\mathscr{C}_{m}$. But $\mathscr{C}_{m}$, $\mathscr{C}^{[m]}$ and $\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{C}$ all generically agree. Hence $\mathscr{C}_{m}\subset\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{C}$ by another application of (2.1.3) and we obtain 9.5.1.

Now let $m$ be sufficiently divisible so that $mG$ is Cartier. In this case, on the left-hand side of 9.5.1 the double dual may be dropped and we simply get the big line bundle \mathscr{O}_{F}\big{(}{-mG}\big{|}_{F}\big{)}. As a consequence, also $\operatorname{Sym}_{\mathcal{C}}^{[m]}\mathscr{C}$ is big, establishing our claim that $\kappa_{\mathcal{C}}(\mathscr{C})=\dim F$. ∎

We next consider the restriction sequence 9.2.1:

[TABLE]

The same line of argument as in the proof of 9.5 then shows that we have $\operatorname{restr}_{P}(\mathscr{A})=0$ and so $\mathscr{A}$ is contained in \Omega_{Y}^{[r]}\big{(}\!\log D_{Y}\big{)}(-P)^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}} \rotatebox{90.0}{{\T1\guilsinglleft}}}}}. The details are omitted for their similarity. The proof of 9.3 is now finished in exactly the same fashion as 6.1: we have shown that $G$ can be replaced by $G-P$ in 9.4.1. Hence after finitely many repetitions we arrive at $G=0$. ∎

9.D. Proof of “An $\Rightarrow$ Bn”

This implication is by far the easier of the two. By [BCHM10], the dlt pair $(X,D)$ admits a $\mathbb{Q}$-factorialization. This is essentially a dlt modification in the special case of dlt pairs, cf. Section 3. For the proof, see [KK10, Thm. 3.1] and [Gra15, Thm. 9.2].

As the Kodaira dimension is invariant under small morphisms, we may assume that $\mathscr{A}$ is $\mathbb{Q}$-Cartier. Under this assumption, the proof of [GKKP11, Thm. 7.2] shows how to deduce that $\kappa(\mathscr{A})\leq r$ from An and the standard Bogomolov–Sommese vanishing theorem for snc pairs [EV92, Cor. 6.9]. The stronger statement that $\kappa_{\mathcal{C}}(\mathscr{A})\leq r$ can then be obtained from this by a branched covering trick as explained in [JK11, Sec. 7]. Compare also [GKKP11, Thm. 7.3] (which erroneously contains the extra assumption that $\dim X\leq 3$). ∎

9.E. Proof of “Bn $\Rightarrow$ An+1”

Let $\pi\colon Y\to X$ be a log resolution of $(X,D)$. Then $\pi$ can be factored as $f\circ\varphi_{r-1}\circ\cdots\circ\varphi_{0}$ just as in 3.1, whose notation we adopt here. The only difference is that some of the $\varphi_{i}$ might be flips. Also the proof is essentially the same, except that instead of [Tan18] we need to appeal to [BCHM10]. Also, because we cannot in general run the MMP on a dlt pair, we have to use the perturbation trick from the proof of [KK10, Thm. 3.1] to reduce the situation to the case of klt pairs.

We will lift forms along each step separately, just as in 7.3 but using 9.3 instead of 6.1. For this, we need to make sure Condition (9.3.2) is satisfied. As far as the map $f$ is concerned, this is quite clear: since $K_{Z}+\widetilde{D}_{r}=f^{*}(K_{X}+D)$, the restriction of $K_{Z}+\widetilde{D}_{r}$ to any fibre of $f$ is trivial and in particular not big. But for any component $P\subset E$, we have (K_{Z}+\widetilde{D}_{r})\big{|}_{P}=K_{P}+P^{c} by adjunction and so (9.3.2) is satisfied.

We now fix $0\leq i\leq r-1$ and turn to the map $\varphi_{i}\colon Y_{i}\to Y_{i+1}$. If $\varphi_{i}$ is a flip, then it is an isomorphism in codimension one and hence extension of forms from $Y_{i+1}$ to $Y_{i}$ is automatic by reflexivity. We may therefore assume that $\varphi_{i}$ is a divisorial contraction, with irreducible exceptional divisor $P$. By construction, $K_{Y_{i}}+\widetilde{D}_{i}$ is $\varphi_{i}$-anti-ample. Also, (K_{Y_{i}}+\widetilde{D}_{i})\big{|}_{P}=K_{P}+P^{c} by adjunction. Thus $K_{P}+P^{c}$ is \varphi_{i}\big{|}_{P}-anti-ample and in particular (9.3.2) is satisfied. ∎

10. Sharpness of results

In this section we discuss some examples that show to what extent our main results are optimal. First, we show that 1.2 fails for the rational double point $E_{8}^{0}$ in characteristic $p\leq 5$, using notation from [Art77].

Example 10.1* (No Logarithmic Extension Theorem in low characteristics).*

Fix any field $k$ of characteristic $p=2,3\text{ or }5$. Then for the (non-$F$-pure) $E_{8}^{0}$ singularity

[TABLE]

the Logarithmic Extension Theorem does not hold. More precisely, note that Kähler differentials on $X$ satisfy the relation

[TABLE]

and hence we may consider the $1$-form

[TABLE]

As any two coordinate functions on $X$ vanish simultaneously only at the origin, $\sigma\in\mathrm{H}^{0}\!\left({X}_{\mathrm{reg}},\Omega_{X}^{1}\right)=\mathrm{H}^{0}\!\left(X,\Omega_{X}^{[1]}\right)$ is a reflexive differential form on $X$. We blow up the origin of $\mathbb{A}^{3}_{k}$ (and points lying over it) four times in a row, yielding a map $\varphi\colon\widetilde{\mathbb{A}^{3}_{k}}\to\mathbb{A}^{3}_{k}$. In suitable coordinates on $\widetilde{\mathbb{A}^{3}_{k}}$, this map is given by

[TABLE]

We compute

[TABLE]

We see that $\widetilde{X}$ can be parametrized rationally by the $(u,w)$-plane, namely by setting $v=-\frac{w^{2}}{u(u+1)}$. In this parametrization, for $p=2$ the pullback of $\sigma$ is given by

[TABLE]

A similar calculation for the other characteristics gives

[TABLE]

This shows that the extension of $\sigma$ to $\widetilde{X}$ has worse than logarithmic poles along the exceptional divisor $\left\{w=0\right\}$.

Next, we show that in 1.3, the assumption on $\det(E_{i}\cdot E_{j})$ not being divisible by $p$ really is necessary.

Example 10.2* (No Regular Extension Theorem in spite of Logarithmic Extension Theorem).*

Let $k$ be a field of characteristic $p>0$, and consider the $p$-th Veronese subring $R$ of $k[x,y]$, that is, $R=k[x^{p},x^{p-1}y,\dots,y^{p}]$. Then $X=\operatorname{Spec}R$ is a strongly $F$-regular surface, since $R$ is a direct summand of the regular ring $k[x,y]$. In particular, $X$ is klt. If $\pi\colon Y\to X$ is the minimal resolution, then $E=\operatorname{Exc}(\pi)$ consists of a single smooth rational curve of self-intersection $-p$. In particular, the assumptions of 1.3 are not satisfied. For later use, let us record the discrepancy $a=a(E,X)$ along $E$: by adjunction,

[TABLE]

and hence $a=-1+\frac{2}{p}$.

One can see by direct calculation that $X$ satisfies the Logarithmic Extension Theorem, but not the Regular Extension Theorem. More precisely, $Y$ is covered by two open sets $U_{0},U_{1}\cong\mathbb{A}^{2}_{k}$, where $U_{i}$ has coordinates $x_{i},y_{i}$ and the coordinate change is given by $(x_{1},y_{1})=(x_{0}^{-1},x_{0}^{p}y_{0})$. The exceptional curve $E$ is given by the equation $y_{i}=0$ in the chart $U_{i}$. Consider the form $\sigma\in\mathrm{H}^{0}\!\left(Y\setminus E,\Omega_{Y}^{1}\right)$ given by $y_{0}^{-1}\mathrm{d}y_{0}$ on $U_{0}$ and by $y_{1}^{-1}\mathrm{d}y_{1}$ on $U_{1}$. It obviously does not extend regularly over $E$, showing that the Regular Extension Theorem fails for $X$. But $\sigma$ has only a logarithmic pole along $E$, and it generates the quotient \left.\raise 2.0pt\hbox{\Omega_{X}^{[1]}}\right/\hskip-2.0pt\raise-2.0pt\hbox{\pi_{*}\Omega_{Y}^{1}}. Thus the Logarithmic Extension Theorem does hold for $X$. (Of course, this last fact also follows from 1.2, at least if $p\geq 7$.)

An elaboration of the previous example shows that 1.5 fails for dlt pairs whose canonical divisor is not $\mathbb{Z}_{(p)}$-Cartier.

Example 10.3* (No restriction sequence for fiercely dlt pairs).*

Using notation from 10.2, let $D\subset X$ be the $\pi$-image of the curve $D_{Y}\subset U_{0}\subset Y$ given by the equation $\left\{x_{0}=\text{const.}\right\}$ (where the constant is arbitrary). Then $D$ is a smooth curve passing through the singular point $x\in X$, and it is isomorphic to its strict transform $D_{Y}$.

[TABLE]

We have $\pi^{*}D=D_{Y}+rE$, where $0=\pi^{*}D\cdot E=1-rp$ and hence $r=1/p$. It follows that the discrepancy of $(X,D)$ along $E$ is

[TABLE]

This shows that the pair $(X,D)$ is plt. On the other hand, $K_{X}+D$ cannot be $\mathbb{Z}_{(p)}$-Cartier, since otherwise $p$ would not appear in the denominator of $a(E,X,D)$ (written in lowest terms). Hence $(X,D)$ is fiercely dlt. In particular, we cannot apply 4.3 to compute $\mathrm{Diff}_{D}(0)$. But [Kol13, Thm. 3.36] tells us that we still have D^{c}\coloneqq\mathrm{Diff}_{D}(0)=\big{(}1-\frac{1}{p}\big{)}\cdot[x]. So, if 1.5 held, we would have a restriction map as in 1.5.1

[TABLE]

Consider however the form $\sigma$ from the previous example, viewed as a section of $\Omega_{X}^{[1]}$. As $D_{Y}\cong D$, we can compute $\operatorname{restr}_{D}(\sigma)$ on $Y$. We have already seen that $\sigma$ acquires a logarithmic pole along $E$. So \sigma\big{|}_{D_{Y}} has a logarithmic (i.e. simple) pole at the unique point in the intersection $D_{Y}\cap E$, which under $\pi$ maps to $x$. Summing up, this means that we do have a restriction map

[TABLE]

but it does not factor via $\Omega_{D}^{1}$. Looking at higher powers $\sigma^{[m]}$, we see that also the other maps $\operatorname{restr}_{D}^{m}$ from 1.5 do not exist in this example.

Example 10.4* (No Logarithmic Extension Theorem for singularities reduced from characteristic zero).*

Finally, we would like to remark that if we start with a log canonical singularity in characteristic zero and then reduce it modulo some small prime $p$, the resulting singularity may not satisfy the Logarithmic Extension Theorem even if it remains log canonical. Indeed, 10.1 furnishes a counterexample since $z^{2}+x^{3}+y^{5}=0$ defines an $E_{8}$ rational double point also in characteristic zero.

11. Counterexamples in higher dimensions

In this final section, we will prove 1.6. As a starting point, in [Kol95] Kollár has given a fairly explicit method for constructing counterexamples to Bogomolov–Sommese vanishing over fields of positive characteristic. We will recall Kollár’s construction in Section 11.A below, both for the benefit of the reader and in order to bring the result in the precise form we need. It turns out that in the examples, the line bundle in question is not just big, but even ample. Thus also Nakano vanishing is violated:

Proposition 11.1** (Failure of Nakano vanishing).**

Fix an algebraically closed field $k$ of characteristic $p>0$, and an integer $n\geq 2$.

1. (11.1.1)

If $n\geq 2p-2$, then there exists an $n$-dimensional Fano variety $Y/k$ with only isolated canonical hypersurface singularities such that

[TABLE] 2. (11.1.2)

If $n\geq 3p-2$, then there exists $Y$ as above, but such that $\omega_{Y}^{-1}$ admits a square root $L$ (i.e. an ample line bundle with $L^{2}\cong\omega_{Y}^{-1}$) and

[TABLE] 3. (11.1.3)

If $n\geq p-2$, then there exists an $n$-dimensional variety $Y/k$ with only isolated canonical hypersurface singularities, satisfying $\omega_{Y}\cong\mathscr{O}_{Y}$ and

[TABLE]

for some ample line bundle $L$ on $Y$.

In all cases, $Y$ actually has $F$-pure singularities. If $n\geq 3$, then $Y$ is even terminal and strongly $F$-regular.

In Section 11.B, we will turn our attention to cones over projective varieties and study when the Logarithmic Extension Theorem holds for such spaces. The conclusion is that cones over the examples from 11.1 are sufficient to prove 1.6, which is accomplished in Section 11.C.

11.A. Kollár’s construction

Kollár’s method is quite flexible in the sense that it does not rely on resolution of singularities and gives very good control on the canonical divisor of the resulting example. On the other hand, it only works in dimensions that satisfy a certain lower bound depending on the characteristic, and the spaces obtained are virtually never smooth. Also, the violation of Nakano vanishing is only guaranteed in degree $n-1$, where $n$ is the dimension.

Let $X$ be an $n$-dimensional smooth projective variety over an algebraically closed field of characteristic $p>0$ and $L$ a line bundle on $X$. Assume that $L^{p}$ is “globally generated to second order” in the sense that the restriction map

[TABLE]

is surjective for every (closed) point $x\in X$ with ideal sheaf $\mathfrak{m}_{x}\subset\mathscr{O}_{X}$. Choose a general section $s\in\mathrm{H}^{0}\!\left(X,L^{p}\right)$ and consider the cover

[TABLE]

as before. By [Kol95, (14.2)] there is a short exact sequence

[TABLE]

Taking determinants, we see that K_{Y}=\pi^{*}\big{(}K_{X}+(p-1)L\big{)}. On the other hand, the $(n-1)$-th reflexive wedge power of the first map in 11.1.1 shows that

[TABLE]

Thus we obtain interesting examples if $K_{X}+pL$ is ample, but $K_{X}+(p-1)L$ is not.

Proof of 11.1.

Let $X\subset\mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d=n-2p+3\geq 1$, and take $L=\mathscr{O}_{X}(2)$. The global generation hypothesis on $L^{p}$ is automatically satisfied, hence we may construct $\pi\colon Y\to X$ as above. Then $Y$ is Fano since

[TABLE]

is anti-ample. On the other hand, K_{X}+pL=\mathscr{O}_{X}(1)=\big{(}K_{X}+(p-1)L\big{)}^{-1} is ample and so by 11.1.2, the variety $Y$ violates Nakano vanishing in the required form. By [Kol95, (20.3), (22.1)], the singularities of $Y$ are locally of the form

[TABLE]

Using this description, it can be checked that $Y$ has only isolated canonical hypersurface singularities, which are terminal for $n\geq 3$. This proves (11.1.1).

The argument for (11.1.2) is very similar. We start with $X$ a smooth hypersurface of degree $d=n-3p+3\geq 1$ and $L=\mathscr{O}_{X}(3)$. Then

[TABLE]

and $K_{X}+pL=\mathscr{O}_{X}(1)$. Again we conclude by 11.1.2.

For (11.1.3), we tweak the numbers once more. Let $X$ be of degree $d=n-p+3\geq 1$ and $L=\mathscr{O}_{X}(1)$. Then

[TABLE]

so $\omega_{Y}\cong\mathscr{O}_{Y}$, and $K_{X}+pL=\mathscr{O}_{X}(1)$.

The claim about $F$-purity can likewise be checked using 11.1.3 and Fedder’s criterion [Fed83]. If $n\geq 3$, then even 11.1.3 multiplied by the non-unit $x_{1}$ is $F$-pure and so $Y$ is strongly $F$-regular. Note also that a strongly $F$-regular Gorenstein singularity is automatically canonical, therefore this provides an alternative proof of $Y$ being canonical. ∎

Remark 11.2*.*

One might be tempted to try and construct lower-dimensional examples by starting with a more interesting $X$ than just a hypersurface in $\mathbb{P}^{n+1}$. This, however, is not possible because the Fano index of $X$ is always $\leq\dim X+1$ by [Kol96, Ch. V, Thm. 1.6].

11.B. Extension properties on cones

Fix an integer $n\geq 2$, a smooth projective variety $Y$ with $\dim Y=n-1$, and an ample line bundle $L$ on $Y$. Following [Kol13, Ch. 3.1], let

[TABLE]

be the affine cone over $(Y,L)$. Blowing up the vertex gives a log resolution $\pi\colon\widetilde{X}\to X$, where $\widetilde{X}$ is the total space of the line bundle $L^{-1}$ and the exceptional locus $E$ is the zero section of $L^{-1}$. In particular, there is an affine map $r\colon\widetilde{X}\to Y$, which maps $E$ isomorphically onto $Y$.

For any integer $q\geq 0$, we will say that Condition $*$ ‣ 11.Bq holds if

[TABLE]

Note that $*$ ‣ 11.Bq always holds in any of the following cases: $q=0$, $q\geq n$, or if $L$ is sufficiently ample. In characteristic zero, $*$ ‣ 11.Bq holds for any $q\neq n-1$ by Nakano vanishing.

With this notation in place, we have the following result. It should be compared to the non-logarithmic, characteristic zero version in [KS21, Prop. B.2].

Proposition 11.3** (Logarithmic Extension Theorem on cones).**

Notation as above. Then the following equivalences hold:

1. (11.3.1)

$X$* satisfies the Logarithmic Extension Theorem for $1$-forms $\Leftrightarrow$ *$*$ ‣ 11.B1 holds. 2. (11.3.2)

$X$* satisfies the Logarithmic Extension Theorem for $n$-forms $\Leftrightarrow$ $*$ ‣ 11.Bn-1** holds.*

More generally, for arbitrary values of $q$ we have the following:

1. (11.3.1)

If $*$ ‣ 11.Bq* and $*$ ‣ 11.Bq-1** hold, then $X$ satisfies the Logarithmic Extension Theorem for $q$-forms.* 2. (11.3.2)

Conversely, if $X$ satisfies the Logarithmic Extension Theorem for $q$-forms, then $*$ ‣ 11.Bq* holds. If in addition $n\geq 3$ and $L$ is sufficiently ample, then also $*$ ‣ 11.Bq-1** holds.*

Proof.

The sequence of relative differentials for $r$ reads

[TABLE]

and its logarithmic version is

[TABLE]

For 11.3.2, choose a system of local parameters $y_{1},\dots,y_{n-1}$ of $Y$ and let $t$ be a nowhere vanishing local section of $L$, considered as a fibrewise linear coordinate on $\widetilde{X}$. Then the middle term of 11.3.2 is locally freely generated by

[TABLE]

and the first map is the inclusion of the subsheaf generated by the $r^{*}\mathrm{d}y_{i}$. Consequently, the quotient sheaf is invertible, locally generated by $\mathrm{d}t/t$. Since $E=\left\{t=0\right\}$, this shows that the quotient is isomorphic to $r^{*}L(E)$.

Now 11.3.1 and 11.3.2 sit inside the diagram shown in Figure 8 8.

Also, for forms of higher degree, from 11.3.2 we get [Har77, Ch. II, Ex. 5.16]

[TABLE]

Recalling that both $r$ and its restriction r^{\prime}\coloneqq r\big{|}_{\widetilde{X}\setminus E} are affine, with

[TABLE]

from 11.3.3 we obtain the following diagram with exact rows and injective vertical arrows:

[TABLE]

It is clear that $\alpha$ is an isomorphism $\Leftrightarrow$ $*$ ‣ 11.Bq holds, $\beta$ is an isomorphism $\Leftrightarrow$ the Logarithmic Extension Theorem for $q$-forms holds on $X$, and $\gamma$ is an isomorphism $\Leftrightarrow$ $*$ ‣ 11.Bq-1 holds. Furthermore, if $n\geq 3$ and $L$ is sufficiently ample then $\delta$ is an isomorphism by Serre vanishing and Serre duality. All claims thus follow from straightforward diagram chases (cf. [GKKP11, Lemma B.2]). ∎

11.C. Proof of 1.6

With all preliminaries in place, the construction of counterexamples to the Logarithmic Extension Theorem becomes very easy. Take $(Y,L)$ as in (11.1.3), and let $X$ be the affine cone over $(Y,L)$. Blowing up the vertex gives an exceptional divisor of discrepancy $-1$ because $\omega_{Y}\cong\mathscr{O}_{Y}$. The result is the total space of $L^{-1}$, which has canonical singularities just as $Y$. We conclude that $X$ is log canonical. By (11.3.2), the Logarithmic Extension Theorem for $(n-2)$-forms does not hold on $X$. This proves (1.6.1).

For (1.6.2), we use the Fano variety $Y$ from (11.1.1) instead. In this case, $X$ is the cone over $(Y,\omega_{Y}^{-1})$. A calculation shows that the first discrepancy is zero. Hence, since $Y$ has canonical singularities, so does $X$. The Logarithmic Extension Theorem fails for the same reason as above.

For (1.6.3), we appeal to (11.1.2), i.e. the cone $X$ is taken with respect to a square root of $\omega_{Y}^{-1}$. In this case the first discrepancy is equal to one. Since $\dim Y\geq 3p-2\geq 4$, we know that $Y$ has only terminal singularities and then the same is true of $X$.

In each case, a log resolution of $X$ can be obtained by first blowing up the vertex of the cone and then pulling back everything along a resolution of $Y$, which exists by [Kol95, §21]. ∎