A note on Flenner's extension theorem

TL;DR

This paper presents a simplified proof of Flenner's extension theorem for differential forms on complex spaces, showing extension with possible logarithmic poles under certain codimension conditions, and discusses limitations in positive characteristic.

Contribution

Provides a shorter, simpler proof of Flenner's extension theorem for p-forms on complex spaces, extending the known results to include forms with logarithmic poles.

Findings

Extension of p-forms with logarithmic poles under codimension constraints

Shorter, more accessible proof of Flenner's theorem

Counterexamples in positive characteristic

Abstract

We show that any $p$-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as $p \le \mathrm{codim}_X (X_{\mathrm{sg}}) - 2$, where $c$ is the codimension of the singular locus. A stronger version of this result, allowing no poles at all, is originally due to Flenner. Our proof, however, is not only completely different, but also shorter and technically simpler. We furthermore give examples to show that the statement fails in positive characteristic.

Equations

$$\mathrm{H}^{0}\big{(}Y,\Omega_{Y}^{p}\big{)}\longrightarrow\mathrm{H}^{0}\big{(}{X}_{\mathrm{reg}},\Omega_{X}^{p}\big{)}$$

$$\mathrm{H}^{0}\big{(}Y,\Omega_{Y}^{p}(\log E)\big{)}\longrightarrow\mathrm{H}^{0}\big{(}{X}_{\mathrm{reg}},\Omega_{X}^{p}\big{)}$$

$$\mathrm{H}^{0}\big{(}Y,\Omega_{Y}^{q}(\log E)\big{)}\longrightarrow\mathrm{H}^{0}\big{(}{X}_{\mathrm{reg}},\Omega_{X}^{q}\big{)}$$

$$\pi^{*}\sigma\in\mathrm{H}^{0}\!\left(Y,\Omega_{Y}^{p}\big{(}\!\log E\big{)}(G)\right).$$

$$\textstyle{\mathscr{O}_{P}\big{(}{-G\big{|}_{P}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$

$$\rho\coloneqq\pi\big{|}_{P}\colon P\longrightarrow B\coloneqq\pi(P),$$

$$\dim F=\dim P-\dim B\ge n-1-(n-c)=c-1$$

$$0\longrightarrow\rho^{*}\Omega_{B}^{1}\longrightarrow\Omega_{P}^{1}\big{(}\!\log P^{c}\big{)}\longrightarrow\Omega_{P/B}^{1}\big{(}\!\log P^{c}\big{)}\longrightarrow 0$$

$$\hskip 67.50009pt\mathscr{G}_{i}\coloneqq\rho^{*}\Omega_{B}^{i}\otimes\Omega_{P/B}^{r-i}\big{(}\!\log P^{c}\big{)},\qquad i=0,\dots,r.$$

$$\kappa (\mathcal{L})\le r-i\le r\le p.$$

$$\kappa (\mathcal{L})=\dim F\ge c-1$$

$${\mathrm{H}}^0\left(X,{\Omega }_X^q\otimes L^{-1}\right)\ne 0\text{for all 1≤q≤n−1.}$$

$$\pi :X[\sqrt[d]{s}]⟶X,$$

$$X:=\mathrm{Spec}\underset{m\ge 0}{⨁}{\mathrm{H}}^0\left(Y,L^m\right)$$

Full text

The author was supported in full by a DFG Research Fellowship. This research was carried out at the University of Utah.

Let $X$ be a normal complex space and $\pi\colon Y\to X$ a resolution of singularities. It is an important question whether reflexive differentials, i.e. differential forms on the smooth locus ${X}_{\mathrm{reg}}$ of $X$, extend holomorphically to forms on $Y$. Two important papers proving such extension properties under additional assumptions on the singularities of $X$ are [GKKP11] and the recent [KS19]. In a slightly different direction, Ohsawa [Ohs82] and independently Steenbrink–van Straten [SvS85] showed that extension always holds if $X$ has isolated singularities and $p\leq\dim X-2$. Later, Flenner [Fle88] generalized this to not necessarily isolated singularities as follows:

[TABLE]

is bijective for all $p\leq\mathrm{codim}_{X}({X}_{\mathrm{sg}})-2$.

Both [Ohs82, SvS85] and [Fle88] heavily rely in their proofs on results about the Hodge theory of isolated singularities. The purpose of this short note is twofold: Firstly, to give a novel and technically significantly simpler proof of the following logarithmic version of 1. While this variant is obviously somewhat weaker, we point out that by combining it with [GK14, Thm. 3.1], at least for $1$-forms we do obtain a new (and still simpler) proof of 1 in full strength.

[TABLE]

is bijective for all $p\leq\mathrm{codim}_{X}({X}_{\mathrm{sg}})-2$.

Secondly, since we can prove it in characteristic zero slowly, we can find a counterexample in positive characteristic quickly. Of course, nobody ever claimed that Flenner holds in characteristic $p$, but this means in particular that no counterexamples have been given so far.

[TABLE]

is not surjective for any $1\leq q\leq n-2$.

Similar examples also exist for higher values of $n$, assuming resolution of singularities in dimension $n-1$.

Extension with logarithmic poles

The only substantial ingredients to our proof of 2 (besides certain standard results) are 4 below and the Bogomolov–Sommese vanishing theorem, asserting that on a projective snc pair $(X,D)$, any line bundle \mathscr{L}\subset\Omega_{X}^{p}\big{(}\!\log D\big{)} has Kodaira dimension $\kappa(\mathscr{L})\leq p$. The latter statement follows from the closedness of logarithmic forms on $(X,D)$ and the branched covering trick. While closedness is usually viewed as a consequence of Deligne’s results on degeneracy of some spectral sequence [Del71], a very short proof based on the classical theory of harmonic integrals has been given by Noguchi [Nog95]. In the case $D=\emptyset$ (which, however, is not sufficient for our purposes), all one needs for closedness is actually Stokes’ theorem.

Proposition 4** (Big Negativity Lemma).**

Let $\pi\colon Y\to X$ be a projective bimeromorphic morphism of normal complex spaces. Then for any nonzero effective $\pi$-exceptional $\mathbb{Q}$-Cartier divisor $E$ on $Y$, there is an irreducible component $P\subset E$ such that -E\big{|}_{P} is \pi\big{|}_{P}-big (i.e. the restriction to a general fibre of \pi\big{|}_{P} is big).

The proof is the same as for the algebraic version in [Gra15, Prop. 4.1], except that in the beginning, $H\subset Y$ should be chosen to be a relatively ample divisor (this exists because $\pi$ is assumed to be projective).

Proof of 2

Since any two log resolutions can be dominated by a third one, it suffices to prove the theorem for one particular choice of $\pi$. By general results on resolution [Kol07, Thm. 3.45], we may therefore assume that $\pi$ is projective and an isomorphism over ${X}_{\mathrm{reg}}$. Let $\sigma\in\mathrm{H}^{0}\!\left({X}_{\mathrm{reg}},\Omega_{X}^{p}\right)$ be a nonzero $p$-form, where $p\leq c-2$ with $c\coloneqq\mathrm{codim}_{X}({X}_{\mathrm{sg}})$. By Grauert’s Direct Image Theorem [GR84, Ch. 10, §4], the sheaf \pi_{*}\Omega_{Y}^{p}\big{(}\!\log E\big{)} is coherent. The Rückert Nullstellensatz [GR84, Ch. 3, §2] therefore applies to show that the pullback $\pi^{*}\sigma$ is a meromorphic form on $Y$. Set $G$ to be the pole divisor of $\pi^{*}\sigma$ as a logarithmic form, i.e. the minimal effective $\pi$-exceptional divisor such that

[TABLE]

We want to show that $G=0$. Otherwise, there is a component $P\subset G$ such that -G\big{|}_{P} is \pi\big{|}_{P}-big, by 4. Set P^{c}\coloneqq(E-P)\big{|}_{P}. Note that 1 may be equivalently regarded as a map \mathscr{O}_{Y}(-G)\to\Omega_{Y}^{p}\big{(}\!\log E\big{)}, which by minimality of $G$ does not vanish along $P$. Denoting its restriction to $P$ by $i$, the residue sequence for $p$-forms along $P$ reads

[TABLE]

Hence \mathscr{O}_{P}\big{(}{-G\big{|}_{P}}\big{)} injects into \Omega_{P}^{r}\big{(}\!\log P^{c}\big{)} either for $r=p-1$ (if ${\operatorname{res}_{P}}\circ i\neq 0$) or for $r=p$ (if ${\operatorname{res}_{P}}\circ i=0$). In both cases, $r\leq p$.

Now let $F\subset P$ be a general fibre of

[TABLE]

and set F^{c}\coloneqq P^{c}\big{|}_{F}. Since $(P,P^{c})$ is an snc pair, so is $(F,F^{c})$. Furthermore, with $n=\dim X$, we have

[TABLE]

since $B\subset{X}_{\mathrm{sg}}$. By generic smoothness, we may shrink $B$ and assume that $\rho$ is an snc morphism of the pair $(P,P^{c})$. This means that $B$ is smooth and all components of $P^{c}$, as well as all their intersections (including $P$), are smooth over $B$. In this case, there is a log differential sequence [EV90, Sec. 4.1]

[TABLE]

inducing a filtration of \Omega_{P}^{r}\big{(}\!\log P^{c}\big{)} with quotients [Har77, Ch. II, Ex. 5.16]

[TABLE]

Consequently, \mathscr{O}_{P}\big{(}{-G\big{|}_{P}}\big{)} injects into $\mathscr{G}_{i}$ for some $i$. Restricting to $F$, we get \mathscr{L}\coloneqq\mathscr{O}_{F}\big{(}{-G\big{|}_{F}}\big{)}\hookrightarrow\Omega_{F}^{r-i}\big{(}\!\log F^{c}\big{)}^{\oplus\operatorname{rk}\Omega_{B}^{i}} and then even \mathscr{L}\hookrightarrow\Omega_{F}^{r-i}\big{(}\!\log F^{c}\big{)} by projecting onto a suitable summand. In view of the Bogomolov–Sommese vanishing theorem [EV92, Cor. 6.9], this implies

[TABLE]

On the other hand, as -G\big{|}_{P} is $\rho$-big, $\mathscr{L}$ is a big line bundle on $F$ and so

[TABLE]

thanks to 2. Put together, 3 and 4 yield $p\geq c-1$, contradicting our assumption that $p\leq c-2$. We conclude that $G=0$, as desired. ∎

Counterexamples in positive characteristic

A reformulation of Bogomolov–Sommese vanishing is that on an $n$-dimensional smooth projective variety $X$, we have $\mathrm{H}^{0}\!\left(X,\Omega_{X}^{q}\otimes L^{-1}\right)=0$ for $q<n$ and any big line bundle $L$. If $L$ is even assumed to be ample, the statement becomes a special case of (Kodaira–Akizuki–)Nakano vanishing [AN54, Thm. 1*′′*]. Utilizing a construction due to Mumford [Mum61], we show that over fields of positive characteristic, Nakano vanishing fails in a strong sense. Taking cones over these examples then allows us to prove 3.

Proposition 5** (Characteristic $p$ failure of Nakano vanishing).**

Assume resolution of singularities in some fixed dimension $n\geq 2$ and characteristic $p>0$. Then over any algebraically closed field of characteristic $p$, there exists an $n$-dimensional smooth projective variety $X$ together with an ample line bundle $L$ on $X$ such that

[TABLE]

In particular, there exist surfaces and threefolds with this property.

Mumford’s construction

By a twisted differential $q$-form on a variety $X$, we will mean a section of $\Omega_{X}^{q}\otimes M$ for some line bundle $M$ on $X$. The following lemma is valid only in positive characteristic.

Lemma 6** (Regularizing forms).**

Let $X$ be a smooth projective variety, $U\subset X$ a dense open subset and \alpha\in\mathrm{H}^{0}\!\left(U,\Omega_{U}^{q}\otimes M\big{|}_{U}\right)\setminus\left\{0\right\} a nonzero rational twisted $q$-form. Then there exists a finite separable map $f\colon Y\to X$ such that $Y$ is normal and $f^{*}\alpha\in\mathrm{H}^{0}\!\left(Y,\Omega_{Y}^{q}\otimes f^{*}M\right)$ is a nonzero regular twisted $q$-form.

Proof.

Let $X=\bigcup U_{i}$ be a finite open covering of $X$ which trivializes $M$, i.e. M\big{|}_{U_{i}}\cong\mathscr{O}_{U_{i}}. Using these trivializations, the \alpha_{i}\coloneqq\alpha\big{|}_{U_{i}} become “ordinary” rational $q$-forms. As in [Mum61, p. 340], we may find finite separable maps $f_{i}\colon V_{i}\to U_{i}$ with $V_{i}$ smooth and $f_{i}^{*}(\alpha_{i})$ regular.111Mumford only considers $1$-forms on surfaces, but an inspection of his argument reveals that it also applies to forms of higher degree. Consider the compositum $L$ of all the function fields $k(V_{i})$, as a subfield of say the algebraic closure of $k(X)=k(U_{i})$. Then the extension $L/k(X)$ is finite and separable. Take $Y$ to be the normalization of $X$ in $L$ and $f\colon Y\to X$ the natural map. It is clear that $f^{*}\alpha$ is a regular twisted form. Also, $f^{*}\alpha\neq 0$ because $\alpha\neq 0$ and $f$ is separable. ∎

Proof of 5.

Pick an arbitrary smooth projective variety $Z$ of dimension $n$, an ample line bundle $L_{Z}$ on $Z$ and nonzero rational $q$-forms $\alpha_{q}$ twisted by $M=L_{Z}^{-1}$, for each $1\leq q\leq n-1$. (This obviously exists.) By 6, there is a finite cover $f\colon W\to Z$ such that the $f^{*}\alpha_{q}$ are regular. Note that $L_{W}\coloneqq f^{*}L_{Z}$ is still ample. Let $g\colon X\to W$ be a resolution of $W$. Then $L_{X}\coloneqq g^{*}L_{W}$ is big and nef (but not ample unless $W$ is already smooth). The forms $(f\circ g)^{*}(\alpha_{q})$ are nonzero global sections of $\Omega_{X}^{q}\otimes L_{X}^{-1}$.

We may assume that $g$ is a composition of blow-ups of codimension two centers, in which case $L_{X}-E$ is ample for some suitable ($g$-anti-ample) effective $g$-exceptional $\mathbb{Q}$-divisor $E\geq 0$. The trouble, of course, is that $E$ will almost never have integral coefficients and thus $L_{X}-E$ does not correspond to a line bundle. We will overcome this difficulty by performing suitable (ramified) finite covers.

After a perturbation, we may assume that $E=\sum_{P}m_{P}P$ is a $\mathbb{Z}_{(p)}$-divisor, i.e. if the coefficients $m_{P}=c_{P}/d_{P}$ are written in lowest terms, then $p$ does not divide $d_{P}$. Let $P\subset\operatorname{supp}E$ be an arbitrary irreducible component, and set $d=d_{P}$. By [BG71, Lemma 2.1], after a finite cover we may assume that the line bundle $\mathscr{O}_{X}(P)$ admits a $d$-th root. The divisor $P$ may not be irreducible anymore, but it is still smooth. If $s\in\mathrm{H}^{0}\!\left(X,\mathscr{O}_{X}(P)\right)$ is a section corresponding to $P$, consider

[TABLE]

the cover obtained by extracting a $d$-th root of that section [Kol13, (2.44)]. Since $X$ and $P$ are smooth, so is $X[{\sqrt[d]{s}}]$. Furthermore, $\pi$ is totally ramified along $P$ by construction and hence $\pi^{*}(m_{P}P)$ becomes integral. Also note that since $p$ does not divide $d$, all covers performed are in fact separable, so the twisted forms $\alpha_{q}$ stay nonzero.

Repeating this procedure finitely often, we arrive at a situation where $L\coloneqq L_{X}-E$ becomes a $\mathbb{Z}$-divisor (and it stays ample). The forms $\alpha_{q}\in\mathrm{H}^{0}\!\left(X,\Omega_{X}^{q}\otimes L_{X}^{-1}\right)\subset\mathrm{H}^{0}\!\left(X,\Omega_{X}^{q}\otimes L^{-1}\right)$ show that we have found the desired example, finishing the proof of 5. ∎

Proof of 3

Fix an integer $n\in\left\{3,4\right\}$, and pick an $(n-1)$-dimensional smooth projective variety $Y$ together with an ample line bundle $L$ violating Nakano vanishing as in 5. 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 $L^{-1}$ and the exceptional locus $E$ is the zero section. As we have carefully proven in [Gra19, Prop. 11.3], the non-vanishing of $\mathrm{H}^{0}\!\left(Y,\Omega_{Y}^{q}\otimes L^{-1}\right)$ implies non-extension of $q$-forms on $X$. Thus $X$ is an example proving 3. ∎