An Optimal Extension Theorem for

1-Forms and the Lipman-Zariski

Conjecture

Patrick Graf, Sándor J Kovács

Received: January 23, 2014 Revised: May 25, 2014

Communicated by Thomas Peternell

Abstract. Let X be a normal variety. Assume that for some reduced divisor D ⊂ X, logarithmic 1-forms defined on the snc locus of (X, D) extend to a log resolution X̃ → X as logarithmic differential forms. We prove that then the Lipman-Zariski conjecture holds for X. This result applies in particular if X has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair (X, ∅) which acquires a logarithmic pole along an exceptional divisor of discrepancy zero, thereby improving on a similar example of Greb, Kebekus, Kovács and Peternell.

2010 Mathematics Subject Classification: 14B05, 32S05 Keywords and Phrases: Singularities of the minimal model program, differential forms, Lipman-Zariski conjecture

1

Contents

1. Introduction  816
2. Dropping non-exceptional divisors  819
3. Dropping certain exceptional divisors  820
4. Proof of Theorem 1.2  824
5. Proof of Theorem 1.4  825
6. Optimality of Theorem 1.4  826

1The first named author was partially supported by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”. The second named author was supported in part by NSF Grants DMS-0856185, DMS-1301888 and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Washington.

7. Non-Extension without poles over klt places  827
References  829

1. Introduction

1.A. Main results. The Lipman-Zariski conjecture [Lip65] asserts the following.

Conjecture 1.1. Let X be a complex variety such that the tangent sheaf 𝒯_{X} := ℋ om_{𝒪_{X}} (Ω_{X}^{1}, 𝒪_{X}) is locally free. Then X is smooth.

Despite being almost 50 years old, this conjecture remains open in general. It is therefore natural to consider special cases. The conjecture is known to hold in the case that the singular locus of X has codimension at least 3 by the work of Flenner [Fle88] and for complete intersections by the work of Källström [Käl11]. Recently, Jörder [Jör13] proved the conjecture under the assumption that 𝒯_{X} locally has a basis consisting of commuting vector fields. Earlier work on Conjecture 1.1 includes [SS72], [Hoc75], and [SvS85]. In a slightly different direction, one may consider varieties with only singularities arising in the minimal model program. The minimal model program aims at the birational classification of varieties, and it is well-known that even if one is interested only in smooth varieties, running the mmp requires dealing with singular models. For us the most important classes will be klt (Kawamata log terminal) and log canonical. Klt singularities form the largest class of singularities where most of the mmp is known to work, while the class of log canonical singularities is the largest class where the relevant notions of the mmp make sense. It is also a more stable class than that of klt singularities; log canonical singularities and their non-normal versions play an important role in compactifications of moduli spaces of canonically polarized varieties. For the precise definitions, we refer to [KM98, Sec. 2.3]. Greb-Kebekus-Kovács-Peternell [GKKP11, Theorem 6.1] showed Conjecture 1.1 for klt spaces. In this paper, we prove the following new special case of Conjecture 1.1.

Theorem 1.2 (Lipman-Zariski conjecture given an extension theorem). Let X be a normal complex variety, and let π: X̃ → X be a log resolution. Assume that the sheaf

π_{∗}Ω1 ; X̃(log D̃)

is reflexive for some snc divisor D̃ ⊂ X̃. Then the Lipman-Zariski conjecture holds for X, i.e., 𝒯_{X} being locally free implies that X is smooth.

Note that Lipman [Lip65, Thm. 3] proved that if 𝒯_{X} is locally free, then X is normal. Hence the normality assumption in our theorem is not a real restriction. Also note that the reflexivity assumption in the theorem is equivalent to saying that logarithmic 1-forms defined on the snc locus of (X, π_{∗}D̃) extend to logarithmic 1-forms on (X̃, D̃) – cf. [GKKP11, Rem. 1.5.2].

By [GKKP11, Thm. 1.5], we have the following immediate corollary.

Corollary 1.3 (Lipman-Zariski conjecture for log canonical pairs). Let (X, D) be a complex log canonical pair. Then the Lipman-Zariski conjecture holds for X.

By the same method of proof as in Theorem 1.2, we obtain the following result.

Theorem 1.4 (Extension theorem for 1-forms on log canonical pairs). Let (X, D) be a complex log canonical pair, and let π: X̃ → X be a log resolution of (X, D). Then the sheaf

π_{∗}Ω1 ; X̃(log D̃)

is reflexive, where D̃ is any reduced divisor such that

Exc(π) ∧ π^{−1}(⌊D⌋) ⊆ supp D̃ ⊆ π^{−1}(⌊D⌋).

Here ⌊D⌋ denotes the coefficient-wise round-down of D. In our case, ⌊D⌋ is simply the union of all components of D that have coefficient one. The expression Exc(π) ∧ π^{−1}(⌊D⌋) denotes the largest divisor contained in both Exc(π) and π^{−1}(⌊D⌋). In a similar fashion as above, the reflexivity assertion is equivalent to saying that any logarithmic 1-form defined on the snc locus of (X, π_{∗}D̃) can be extended to X̃, possibly acquiring logarithmic poles along D̃. Theorem 1.4 should be compared to the extension theorem [GKKP11, Thm. 1.5]. There the conclusion is similar, with D̃ replaced by D̂, the largest reduced divisor contained in π^{−1}(non-klt locus). (The non-klt locus is the smallest closed subset W ⊂ X such that (X, D) is klt away from W. Note that this contains ⌊D⌋.) Theorem 1.4 says that [GKKP11, Thm. 1.5] is not optimal: we allow logarithmic poles only along a smaller divisor. For example, if D = ∅ but X is not klt, e.g. if X is a cone over an abelian variety, then D̃ = 0 while D̂ is nonzero.

1.B. Further results. We show that Theorem 1.4 in turn is optimal, both with respect to the pole divisor D̃ and with respect to the degree of the forms considered. To be more precise, concerning the first point we prove the following.

Theorem 1.5 (Optimality of Theorem 1.4). Let (X, D) and π be as in Theorem 1.4, and let D̃ = π^{−1}(⌊D⌋). Assume that one of the following holds: (1.5.1) X is ℚ-factorial, or (1.5.2) dim X = 2. Then for any divisor B such that π_{∗}^{−1}⌊D⌋ ≤ B ⪇ D̃, the sheaf π_{∗}Ω1 ; X̃(log B) is not reflexive.

As to the second point, we show that for resolutions of log canonical pairs, forms of higher degree may acquire logarithmic poles along exceptional divisors of discrepancy strictly greater than −1, even if the boundary divisor of the pair is empty. This improves upon an example given in [GKKP11, Ex. 3.2]. The precise statement is as follows.

Theorem 1.6 (Non-extension over klt places, cf. Theorem 7.4). There exists a three-dimensional complex log canonical pair (X, ∅) with empty boundary such that there is a divisor E_{0} ⊂ X̃ of discrepancy 0 in some log resolution X̃ → X, and a 2-form on the smooth locus of X that acquires a logarithmic pole along E_{0} when pulled back to X̃.

1.C. Overview of proofs. The proofs of our main results are based on the two auxiliary Theorems 2.1 and 3.1. The purpose of these theorems is to shrink the divisor along which we allow pulled-back 1-forms to acquire logarithmic poles. Theorem 2.1 deals with non-exceptional components, while Theorem 3.1 handles the exceptional ones. To prove Theorem 1.2, we first apply Theorem 2.1 and then Theorem 3.1 in order to shrink that pole divisor to zero. Then an argument going back to [SvS85, (1.6)] completes the proof. To prove Theorem 1.4, we take [GKKP11, Thm. 16.1] as our starting point and then apply Theorems 2.1 and 3.1.

1.D. Recent work by Druel. In [Dru13, Thm. 1.1], Druel has recently obtained Corollary 1.3 by an independent proof. He employs a cutting-down procedure to reduce to the surface case, where the main work is done. Note however that this case is essentially already contained in [SvS85]. To be more precise, if x ∈ X is a normal surface singularity with smooth locus U and π: X̃ → X is a log resolution with exceptional divisor E, then [SvS85, Cor. 1.4] says that the map

H^{0}(U, Ω_{U}^{1})/H^{0}(X̃, Ω1 ; X̃) ⟶ H^{0}(U, Ω_{U}^{2})/H^{0}(X̃, Ω2 ; X̃(log E))

induced by differentiation is injective. If x ∈ X is log canonical, then by definition the right-hand side is zero, hence so is the left-hand side. This means that all 1-forms defined on U extend to X̃ without poles. Now the argument given in [SvS85, (1.6)] shows that if 𝒯_{X} is free, then x ∈ X is in fact smooth.

1.E. Acknowledgements. The authors would like to thank Daniel Greb, Clemens Jörder and Stefan Kebekus for interesting discussions on the subject of this paper.

1.F. Notation, definitions, and conventions. Throughout this paper, we work over the field of complex numbers ℂ. A pair (X, D) consists of a normal variety X over ℂ and an effective ℝ-Weil divisor D on X. Let (X, D) be a pair and x ∈ X a point. We say that (X, D) is snc at x if there exists a Zariski-open neighbourhood U ⊆ X of x such that U is smooth and supp D ∩ U is either empty, or a divisor with simple normal crossings. The pair (X, D) is called an snc pair or simply snc if it is snc at every point of X. For the definitions of klt and log canonical pairs, we refer to [KM98, Sec. 2.3].

Given a pair (X, D), let (X, D)_{reg} denote the maximal open subset of X where (X, D) is snc, and (X, D)_{sing} its complement, with the induced reduced subscheme structure. Let (X, D) be a pair. A log resolution of (X, D) is a proper birational morphism π: X̃ → X such that X̃ is smooth, both the pre-image π^{−1}(supp D) of supp D and the exceptional set E = Exc(π) are of pure codimension one in X̃, and (X̃, D̃ + E) is an snc pair where D̃ = π^{−1}(supp D)_{red} is the reduced divisor supported on π^{−1}(supp D). Let D be a divisor on a normal variety, and let D = ∑ a_{i}D_{i} be its decomposition into irreducible components. The round-down ⌊D⌋ of D is defined to be ∑⌊a_{i}⌋D_{i}, where ⌊a_{i}⌋ is the largest integer less than or equal to a_{i}. Let D_{1}, D_{2} be divisors on a normal variety. Then D_{1} ∨ D_{2} denotes the smallest divisor that contains both D_{1} and D_{2}, while D_{1} ∧D_{2} denotes the largest divisor that is contained in both D_{1} and D_{2}.

2. Dropping non-exceptional divisors

In this section we prove that if the extension theorem holds for a pair (X, Γ+Δ) where Δ is a reduced effective divisor, then it also holds for (X, Γ). More precisely we prove the following.

Theorem 2.1 (Dropping non-exceptional divisors). Let X be a normal variety and π: X̃ → X a log resolution of X. Assume that the sheaf

π_{∗}Ω1 ; X̃(log D̃)

is reflexive for some snc divisor D̃. Let Δ be a reduced effective divisor on X such that supp Δ ⊂ π_{∗}D̃. Then the sheaf

π_{∗}Ω1 ; X̃(log B̃)

is also reflexive, where B̃ = D̃ − π_{∗}^{−1}Δ.

Proof. Notice that one may assume that Δ is irreducible and conclude the general case via replacing D̃ by B̃ and iterating the process for all irreducible components of Δ. For simplicity let us denote π_{∗}^{−1}Δ by Δ̃. Consider the following short exact sequence given by the residue map, cf. [EV92, 2.3(b)]:

0 → Ω1 ; X̃(log B̃) → Ω1 ; X̃(log D̃) → 𝒪Δ̃ → 0.

Pushing this forward via π gives

0 → π_{∗}Ω1 ; X̃(log B̃) → π_{∗}Ω1 ; X̃(log D̃) → 𝒬 → 0,

where 𝒬 ⊂ π_{∗}𝒪Δ̃. In particular, 𝒬 is supported on Δ and it is torsion-free as an 𝒪_{Δ}-module. It follows that the only associated prime of 𝒬 has height 1. Then the statement follows from [Har80, Cor. 1.5]. □

3. Dropping certain exceptional divisors

The aim of the present section is to show that if ω is a logarithmic 1-form on a smooth variety whose poles are contained in an exceptional divisor, then ω in fact does not have any poles.

Theorem 3.1 (Dropping exceptional divisors). Let X be a normal variety and π: X̃ → X a log resolution. Let E be a reduced π-exceptional divisor. Then the natural inclusion map

H^{0}(X̃, Ω1 ; X̃) ↪ H^{0}(X̃, Ω1 ; X̃(log E))

is an isomorphism. Equivalently, the inclusion π_{∗}Ω1 ; X̃ ⊂ π_{∗}Ω1 ; X̃(log E) is an isomorphism of sheaves.

Theorem 3.1 is a consequence of the following two propositions.

Proposition 3.2 (Theorem 3.1 for isolated singularities). Let X be a normal variety and π: X̃ → X a log resolution. Let E be a reduced divisor which is mapped to a single point by π. Then the natural inclusion map

H^{0}(X̃, Ω1 ; X̃) ↪ H^{0}(X̃, Ω1 ; X̃(log E))

is an isomorphism.

Proposition 3.3. Proposition 3.2 implies Theorem 3.1.

Proposition 3.2 was first observed by Wahl in the case of surfaces, cf. [Wah85, Lemma 1.3]. Our proof of Proposition 3.3 follows the lines of [GKK10, Section 7.D].

3.A. The first Chern class. We collect some well-known facts about the first Chern class of a line bundle.

Notation 3.4 (First Chern class). Let X be a smooth variety and ℒ ∈ Pic(X) a line bundle. The first Chern class c_{1}(ℒ) ∈ H^{1}(X, Ω_{X}^{1}) is the image of ℒ under the map Pic(X) = H^{1}(X, 𝒪_{X}^{∗}) → H^{1}(X, Ω_{X}^{1}) induced by the map d log: 𝒪_{X}^{∗} → Ω_{X}^{1} that sends f ↦ f^{−1}df.

Lemma 3.5 (Connecting homomorphism of the residue sequence). Let X be a smooth variety and E ⊂ X an snc divisor, consisting of irreducible components E_{1}, …, E_{k}. Consider the short exact sequence

(3.6)  0 → Ω_{X}^{1} → Ω_{X}^{1}(log E) → ⨁_{i=1}^{k} 𝒪_{E_{i}} → 0

given by the residue map (cf. [EV92, 2.3(a)]). The associated connecting homomorphism

δ: ⨁_{i=1}^{k} H^{0}(E_{i}, 𝒪_{E_{i}}) → H^{1}(X, Ω_{X}^{1})

sends

1_{E_{i}} ↦ c_{1}(𝒪_{X}(E_{i})), 1 ≤ i ≤ k.

Here 1_{E_{i}} denotes the function that is constant with value 1 on E_{i} and vanishes on the other components.

Proof. This is well-known, and easy to prove by a Čech cohomology computation. □

For a proof of the following fact see [For81, Paragraph 17].

Fact 3.7 (Residue map on curves). Let C be a smooth projective curve. Then there is a canonically defined linear map Res : H^{1}(C, Ω_{C}^{1}) → ℂ, which is an isomorphism. □

Lemma 3.8 (Residue and degree). If ℒ ∈ Pic(C) is a line bundle on a smooth projective curve, then Res(c_{1}(ℒ)) = deg ℒ.

Proof. For P ∈ C a point and ℒ = 𝒪_{C}(P), the claim is easily seen to be true from the description of Res given in [For81, Thm. 17.3]. By linearity, this is enough. □

3.B. Proof of Proposition 3.2. We may assume X to be affine of dimension ≥ 2. Let E_{1}, …, E_{k} be the irreducible components of E. Consider the short exact sequence (3.6),

0 → Ω1 ; X̃ → Ω1 ; X̃(log E) → ⨁_{i=1}^{k} 𝒪_{E_{i}} → 0.

By the corresponding long exact sequence, it suffices to show injectivity of the induced map

δ: ⨁_{i=1}^{k} H^{0}(E_{i}, 𝒪_{E_{i}}) → H^{1}(X̃, Ω1 ; X̃).

Note that we may assume π to be a projective morphism and then X̃ is quasiprojective. After choosing a (locally closed) embedding of X̃ into a projective space, let H ⊂ X̃ be the intersection of general hyperplanes H_{1}, …, H_{dim X−2} ⊂ X̃. (If X is a surface, then H = X̃.) We formulate the properties of H in a separate lemma.

Lemma 3.9. Using the notation introduced above we have that (H, E|_{H}) is an snc surface pair. Furthermore, for any i, C_{i} := E_{i}|_{H} is irreducible (in particular, nonempty), and π|_{H} is proper and birational onto its image.

Proof. If dim X = 2, then there is nothing to prove, so we may assume that dim X ≥ 3. In particular the intersection of two irreducible divisors on X̃ is still positive dimensional and hence if one of them is ample, then the intersection is connected. We will use this fact below. We proceed inductively, cutting by one hyperplane at a time. First we cut by H_{1}. By Bertini’s theorem E + H_{1} is snc and H_{1} is connected and hence irreducible by the above discussion. Hence (H_{1}, E|_{H_{1}}) is an snc pair, and the E_{i}|_{H_{1}} are smooth and irreducible for all i by Bertini again. It is clear that π|_{H_{1}} is proper and since H_{1} is general, π|_{H_{1}} is birational.

Now we are in the same situation as before cutting by H_{1}, so we may apply the same argument again and obtain the statement for H_{1} ∩ H_{2}. After finitely many steps, we arrive at H. □

The image π(H) need not be normal, but we may normalize it and get a birational morphism H → π(H)^{ν}, which contracts all the C_{i}. Then negative definiteness ([KM98, Lemma 3.40]) asserts that the intersection matrix A := (C_{i} · C_{j}) is invertible. Recall that we need to show the injectivity of δ. To this end, think of the C_{i} as smooth projective curves in X̃, consider the restriction morphism

r: H^{1}(X̃, Ω1 ; X̃) → ⨁_{i=1}^{k} H^{1}(C_{i}, Ω_{C_{i}}^{1}),

and observe that the composition

r ∘ δ: ⨁_{i=1}^{k} H^{0}(E_{i}, 𝒪_{E_{i}}) → ⨁_{i=1}^{k} H^{1}(C_{i}, Ω_{C_{i}}^{1})

is an isomorphism: On the left-hand side, choose the basis consisting of the

functions 1_{E_{i}}, and on each summand of ⨁_{i=1}^{k} H^{1}(C_{i}, Ω_{C_{i}}^{1}), choose the basis

canonically determined by the residue map of Fact 3.7. By Lemmas 3.5 and 3.8 the map r ∘δ with respect to these bases is given by the matrix A. We have already noted that this matrix is invertible. □

3.C. Proof of Proposition 3.3. We will make essential use of the following proposition.

Proposition 3.10 (Negativity lemma, see [GKK10, Proposition 7.5]). Let φ: Ỹ → Y be a projective birational morphism between normal quasi-projective varieties of dimension ≥ 2, where Ỹ is smooth. Let y ∈ Y be a point whose preimage φ^{−1}(y) has (not necessarily pure) codimension one and let F_{0}, …, F_{k} ⊂ φ^{−1}(y) be the reduced divisorial components. If all the F_{i} are smooth and ∑ e_{i}F_{i} is a nonzero effective divisor, then there is a 0 ≤ j ≤ k such that e_{j} ≠ 0 and h^{0}(F_{j}, 𝒪Ỹ (∑ e_{i}F_{i})|_{F_{j}}) = 0. □

Proof of Proposition 3.3. We may assume X to be affine of dimension ≥ 2. Let

(3.11)  σ ∈ H^{0}(X̃, Ω1 ; X̃(log E))

be a logarithmic 1-form. Assuming Proposition 3.2, we will show that

(3.12)  σ ∈ H^{0}(X̃, Ω1 ; X̃).

To this end, we will consider an irreducible component of E′ ⊂ E for which dim π(E′) is maximal among the irreducible components of E, and for any such E′ we will show that σ ∈ H^{0}(X̃, Ω1 ; X̃(log(E − E′))). Then replace E by E − E′ and repeat the argument until E disappears.

No.  1 2 3 4 5 6 7 8 9 10 ⋯
dim X  2 3 3 4 4 4 5 5 5 5 ⋯
codim_{X} π(E′) 2 2 3 2 3 4 2 3 4 5 ⋯

We proceed by induction on pairs of numbers (dim X, codim_{X} π(E′)) ordered lexicographically as indicated in the following table: In order to simplify notation, we number the irreducible components E_{i} of E such that E′ = E_{0} and π(E_{i}) = π(E_{0}) if and only if 0 ≤ i ≤ k, for some k. Let e_{i} be the pole orders of σ along the E_{i}. These are the minimal non-negative numbers such that

σ ∈ H^{0}(X̃, Ω1 ; X̃ ⊗ 𝒪X̃(∑ e_{i}E_{i})).

By (3.11), we already know all the e_{i} are either 0 or 1, and our aim is to show that e_{0} = 0. Start of induction. This is the case dim X = codim_{X} π(E_{0}) = 2. For surfaces, every exceptional divisor is contracted to a point, so Proposition 3.2 applies. Inductive step. We distinguish two possibilities: the divisor E_{0} may be mapped to a point by π, or it may be mapped to a positive-dimensional variety. If dim π(E_{0}) = 0, then by the choice of E_{0}, every exceptional divisor contained in E is contracted to a point, so Proposition 3.2 applies again. If dim π(E_{0}) > 0, choose general hyperplanes H_{1}, …, H_{dim π(E_{0})} ⊂ X, let H be the intersection H_{1} ∩ ⋯ ∩ H_{dim π(E_{0})} and H̃ the preimage π^{−1}(H). Applying [GKKP11, Lemmas 2.23 and 2.24], we obtain that H is normal and π|H̃ is a log resolution. The intersection H ∩ π(E_{0}) is finite, but nonempty. Shrinking X, we may assume that H ∩ π(E_{0}) consists of a single point, say x. Now set F_{x} = (π|H̃)^{−1}(x) and

F_{x,i} = F_{x} ∩ E_{i} = (π|_{E_{i}})^{−1}(x).

Then F_{x} is the union of the F_{x,i}.

Claim 3.13. The subsets F_{x,0}, …, F_{x,k} are smooth, irreducible, and have codimension one in H̃, while the other F_{x,i} are empty.

Proof. If 0 ≤ i ≤ k, then being a general fibre of π|_{E_{i}}, F_{x,i} is smooth of dimension dim E_{i} −dim π(E_{0}) = dim H̃ −1. Since F_{x,i} = H̃ ∩E_{i}, it is an ample divisor on E_{i}, hence connected and by being smooth it is also irreducible. On the other hand, if i > k, then by the choice of E′ = E_{0}, we have that π(E_{0}) ⊄ π(E_{i}), and hence x ∉ π(E_{i}) and so F_{x,i} = ∅. □

Claim 3.13 implies that it is possible to apply Proposition 3.10 to π|H̃: H̃ → H, x ∈ H, and F_{x,0}, …, F_{x,k}, which we will do later. Now consider the dual of the normal bundle sequence for H̃ ⊂ X̃,

\xymatrix{0 \ar[r] & N_{H̃/X̃}^{∗} \ar[r] & Ω1 ; X̃|H̃ \ar[r]^{ρ} & Ω1 ; H̃ \ar[r] & 0,}

twist it with ℱ := 𝒪H̃(∑ e_{i}E_{i}|H̃), and restrict to F_{x,j}, for 0 ≤ j ≤ k:

\xymatrix{N_{H̃/X̃}^{∗} ⊗ ℱ \ar[r]^{α} \ar[d] & Ω1 ; X̃|H̃ ⊗ ℱ \ar[r]^{β} \ar[d]^{r_{j}} & Ω1 ; H̃ ⊗ ℱ \ar[d] \\ N_{H̃/X̃}^{∗} ⊗ ℱ |_{F_{x,j}} \ar[r]^{α_{j}} & Ω1 ; X̃|H̃ ⊗ ℱ |_{F_{x,j}} \ar[r]^{β_{j}} & Ω1 ; H̃ ⊗ ℱ |_{F_{x,j}}.}

Since H has smaller dimension than X, the induction hypothesis gives us that β(σ|H̃) has no poles, that is

(3.14)  β(σ|H̃) ∈ H^{0}(H̃, Ω1 ; H̃) ⊂ H^{0}(H̃, Ω1 ; H̃ ⊗ ℱ).

Recall that we want to show that e_{0} = 0. We will show more generally that e_{j} = 0 for all 0 ≤ j ≤ k. So, assume to the contrary that there is an index j with e_{j} = 1. By the definition of the e_{i}, σ|H̃ as a section in Ω1 ; X̃|H̃ ⊗ℱ does not vanish along F_{x,j}. But by (3.14), β(σ|H̃) does vanish along F_{x,j}. So r_{j}(σ|H̃) is a nonzero global section in ker β_{j}, which means that H^{0}(F_{x,j}, N_{H̃/X̃}^{∗} ⊗ ℱ |_{F_{x,j}}) ≠ 0. Now note that N_{H̃/X̃}|_{F_{x,j}} is trivial, because N_{H̃/X̃} is the pullback of N_{H/X}. Hence from H^{0}(F_{x,j}, N_{H̃/X̃}^{∗} ⊗ ℱ |_{F_{x,j}}) ≠ 0 it follows that H^{0}(F_{x,j}, ℱ |_{F_{x,j}}) ≠ 0. Since this holds for all j with e_{j} = 1, we have a contradiction to Proposition 3.10, showing in particular that e_{0} = 0 and thus completing the proof of Proposition 3.3. □

4. Proof of Theorem 1.2

The aim of the present section is to prove Theorem 1.2. First, for the reader’s convenience we recall some facts about resolutions of singularities.

Lemma 4.1 (Reflexivity is independent of the choice of resolution). Let X be a normal variety such that π_{∗}Ω1 ; X̃ is reflexive for some resolution π: X̃ → X. Then φ_{∗}Ω_{X′}^{1} is reflexive for any resolution φ: X′ → X.

Proof. Let ψ: X̂ → X be a resolution of X that dominates both X̃ and X′, i.e. we have the following commutative diagram.

\xymatrix{ & X̂ \ar[dl]_{π̂} \ar[dr]^{φ̂} \ar[dd]^{ψ} \\ X̃ \ar[dr]_{π} &  & X′ \ar[dl]^{φ} \\  & X}

Since X̃ and X′ are smooth, we have π̂_{∗}Ω1 ; X̂ = Ω1 ; X̃ and φ̂_{∗}Ω1 ; X̂ = Ω_{X′}^{1}. We obtain

φ_{∗}Ω_{X′}^{1} = φ_{∗}φ̂_{∗}Ω1 ; X̂ = ψ_{∗}Ω1 ; X̂ = π_{∗}π̂_{∗}Ω1 ; X̂ = π_{∗}Ω1 ; X̃.

Since π_{∗}Ω1 ; X̃ is reflexive by assumption, so is φ_{∗}Ω_{X′}^{1}. □

The next theorem is a special case of [GKK10, Cor. 4.7].

Theorem 4.2 (Functorial resolutions). Let X be a normal variety. Then there exists a resolution φ : X′ → X with the property that φ_{∗}𝒯_{X′} is reflexive, i.e. for any vector field ξ on some open subset U ⊂ X, there is a vector field ξ̃ on φ^{−1}(U) that agrees with ξ wherever φ is an isomorphism.

Sketch of proof. It is a classical fact [Kau65, Satz 3] that vector fields on X are in one-to-one correspondence with local ℂ-actions on X. Loosely speaking, a local ℂ-action is a ℂ-action such that t • z is only defined for sufficiently small values of |t|, dependent on z. For any local ℂ-action, the action map ℂ × X ⊃ U → X is a smooth morphism. By [Kol07, Thm. 3.45], there exists a resolution functor ℛ which to any variety assigns a resolution in such a way that smooth morphisms between varieties can be lifted to the resolutions. The resolutions output by ℛ are called functorial resolutions. Let φ: X′ → X be the functorial resolution of X. Applying the functor ℛ to the action map associated to a vector field ξ on X, we obtain a diagram

\xymatrix{ℂ × X′ \ar[d]_{id×φ} & U′ \ar@{^{(}->}[l] \ar[d] \ar[r] & X′ \ar[d]^{φ} \\ ℂ × X & U \ar@{^{(}->}[l] \ar[r] & X.}

One then checks that the map U′ → X′ is a local ℂ-action, giving rise to a vector field ξ̃ on X′ which extends ξ as desired. For a rigorous proof of Theorem 4.2, the reader may consult [GKK10, Sec. 4].

□

Proof of Theorem 1.2. By assumption, we have a log resolution π : X̃ → X such that π_{∗}Ω1 ; X̃(log D̃) is reflexive. We may uniquely write D̃ = D̃_{bir} +E, where E is exceptional and no component of D̃_{bir} is exceptional. By Theorem 2.1, π_{∗}Ω1 ; X̃(log E) is reflexive. Then by Theorem 3.1, also π_{∗}Ω1 ; X̃ is reflexive. Let φ: X′ → X be the functorial resolution from Theorem 4.2, so that vector fields on X can be lifted to X′. By Lemma 4.1, φ_{∗}Ω_{X′}^{1} is reflexive. Now the proof of [GKKP11, Theorem 6.1] applies verbatim to show that if 𝒯_{X} is locally free, then X is smooth. □

5. Proof of Theorem 1.4

By Theorem 2.1 we may assume that D̃ = π^{−1}(⌊D⌋)_{red}. Let E denote the exceptional locus of π. Let D̂ = π_{∗}^{−1}⌊D⌋ + E. Note that D̂ is obtained from D̃ by adding finitely many irreducible π-exceptional divisors whose image via π is not contained in ⌊D⌋. By [GKKP11, Thm. 16.1], we know that π_{∗}Ω1 ; X̃(log D̂) is reflexive. Let E_{1} be an irreducible component of D̂ that is not contained in D̃. As we observed above, this means that π(E_{1}) ⊄ ⌊D⌋, so by localizing near the general point of π(E_{1}), that is, by further shrinking X, we may assume that ⌊D⌋ = ∅. In this case, D̂ is π-exceptional, hence Theorem 3.1 implies that π_{∗}Ω1 ; X̃(log D̂ − E_{1}) is reflexive. We may iterate this process as long as D̂ is larger than D̃ and so the statement follows. □

6. Optimality of Theorem 1.4

In this section, we show that extension of differential forms as in Theorem 1.4 fails in many cases if one shrinks the pole divisor D̃ further, or if one considers forms of higher degree.

6.A. Shrinking D̃ further. The aim of this subsection is to prove Theorem 1.5. First we need two lemmas.

Lemma 6.1 (Strictly logarithmic poles). Let X be a smooth variety and f ∈ 𝒪_{X}(X) a regular function such that its reduced zero set, D = {f = 0}_{red} ⊂ X, is a divisor with simple normal crossings. Then d log f ∈ H^{0}(X, Ω_{X}^{1}(log D)), and d log f ∉ H^{0}(X, Ω_{X}^{1}(log B)) for any reduced divisor 0 ≤ B < D.

Proof. This follows directly from the definition of logarithmic differentials. □

Lemma 6.2 (Non-reflexivity). Let (X, Σ) be a pair, where Σ is a reduced divisor, π: X̃ → X a log resolution of (X, Σ), and D̃ the largest reduced divisor contained in π^{−1}(Σ). Let E_{0} be an irreducible π-exceptional divisor that is mapped into an effective divisor D whose support is contained in Σ and which is ℚ-Cartier at the general point of π(E). Then the sheaf π_{∗}Ω1 ; X̃(log B) is not reflexive for any reduced divisor π_{∗}^{−1}Σ ≤ B ≤ D̃ − E_{0}.

Proof. By the assumptions, there is an open set U ⊂ X with π(E_{0}) ∩ U ≠ ∅, and a function f ∈ 𝒪_{X}(U) cutting out some multiple of D. Set g = π^{∗}f ∈ 𝒪X̃(π^{−1}(U)). By Lemma 6.1,

d log g ∈ H^{0}(U \ π(Exc(π)), π_{∗}Ω1 ; X̃(log B))

but

d log g ∉ H^{0}(U, π_{∗}Ω1 ; X̃(log B)).

So d log g cannot be extended over the codimension ≥ 2 subset π(Exc(π)). This implies that π_{∗}Ω1 ; X̃(log B) is not reflexive. □

Proof of Theorem 1.5. Let E_{0} be a component of D̃ − B. Then E_{0} is πexceptional and π_{∗}^{−1}D ≤ B ≤ D̃ − E_{0}. If we are in case (1.5.1), Lemma 6.2 applies immediately. Hence we may assume we are in case (1.5.2). Then π(E_{0}) is a point p ∈ X, and we have p ∈ D ⊂ X, because E_{0} ⊂ D̃ ⊂ π^{−1}(⌊D⌋) set-theoretically. We will show that D is ℚ-Cartier at p, then the claim follows from Lemma 6.2. By shrinking X we may assume that (X, D) is snc away from p. For 0 < ε ≤ 1, the pair (X, (1 − ε)D) is numerically dlt [KM98, Ntn. 4.1, Lem. 3.41]. By [KM98, Prop. 4.11], X is ℚ-factorial. In particular, D is ℚ-Cartier. □

6.B. Other values of p. The analogue of Theorem 1.4 does not hold for pforms with p ≥ 2. Counterexamples may be obtained by taking a p-dimensional normal Gorenstein singularity z ∈ Z which is log canonical but not klt (notice that this only exists if p ≥ 2), and considering the product X = Z × ℂ^{n−p}, for n ≥ p arbitrary. Let σ be a local generator for ω_{Z} and replace Z with a neighbourhood of z where σ is everywhere defined. Then pr_{1}^{∗}σ ∈ H^{0}(X_{reg}, Ω_{X_{reg}}^{p}) will not be extendable without logarithmic poles on any resolution of singularities of X. This way one obtains counterexamples to the analogue of Theorem 1.4 for any p ≥ 2 in arbitrary dimension n ≥ p.

7. Non-Extension without poles over klt places

In this section, we consider a reduced log canonical pair (X, D) and a log resolution π: X̃ → X of (X, D). Deviating slightly from our previous notation, we let

E = π_{∗}^{−1}D + Exc(π),
E_{nklt} = sum of all divisors in E with discrepancy −1,
E_{nklt} ∨ π^{−1}(D) = sum of all divisors contained in E_{nklt} or in π^{−1}(D),
D̃ = largest reduced divisor contained in π^{−1}(non-klt locus).

pt Then we obviously have

E_{nklt} ≤ E_{nklt} ∨ π^{−1}(D) ≤ D̃ ≤ E.

The extension theorem [GKKP11, Thm. 1.5] states that the sheaves π_{∗}Ωp ; X̃(log D̃) are reflexive for all values of p. In [GKKP11, Section 3.B], it was observed that basically by the definition of discrepancy, even the sheaf π_{∗}Ωn ; X̃(log E_{nklt}) is reflexive, where n = dim X. This leads to the following natural question.

Question 7.1. Are the sheaves π_{∗}Ωp ; X̃(log E_{nklt}) also reflexive when p < n?

The answer turns out to be “no”. However, in the counterexample given in [GKKP11, Ex. 3.2], X is the quadric cone, D consists of two rulings, and the exceptional divisor where extension fails is contained in the preimage of D. This means that [GKKP11, Ex. 3.2] does not answer the following refined version of Question 7.1:

Question 7.2. Are the sheaves π_{∗}Ωp ; X̃(log(E_{nklt} ∨ π^{−1}(D))) reflexive when p < n?

In this section, we will give an example showing that even this question has to be answered negatively. First we need a lemma.

Lemma 7.3 (Cusp singularities). There exists a log canonical Gorenstein surface singularity 0 ∈ S that has a log resolution S̃ → S containing two distinct exceptional curves C_{1}, C_{2} which both have discrepancy −1 and whose intersection is non-empty.

Proof. This follows from the classification of log canonical Gorenstein surface singularities [Kaw88, Sec. 9]. It is also possible to explicitly write down a hypersurface singularity with the desired property. Namely, consider the following polynomial in three variables:

f(x, y, z) = x^{2}(x + z) − y^{2}z + z^{4}.

A tedious but routine calculation shows that the origin in ℂ^{3} is an isolated singular point of S = {f = 0} ⊂ ℂ^{3}, so 0 ∈ S is a normal Gorenstein surface singularity. Furthermore, blowing up 0 ∈ X yields a resolution whose exceptional locus consists of a rational curve C_{1} with a single node, and which has discrepancy −1. Blowing up that node, we obtain a log resolution containing an additional exceptional rational curve C_{2}, also of discrepancy −1, such that C_{2} and the strict transform of C_{1} meet in two points. □

The next theorem tells us that the answer to Question 7.2 is “no”.

Theorem 7.4 (Non-extension over klt places, cf. Theorem 1.6). There exists a three-dimensional reduced log canonical pair (X, D) such that using the notation introduced at the beginning of this section, the sheaf π_{∗}Ω2 ; X̃(log(E_{nklt}∨π^{−1}(D))) is not reflexive.

Proof. Let 0 ∈ S and p ∈ C_{1} ∩ C_{2} ⊂ S̃ be as in Lemma 7.3. Take X := S × 𝔸_{ℂ}^{1} and D = ∅. Then X′ := S̃ × 𝔸_{ℂ}^{1} → X is a log resolution of (X, D). On X′, blow up a point of the form (p, t) with t ∈ 𝔸_{ℂ}^{1} arbitrary to obtain f: X̃ → X′. This gives a log resolution π : X̃ → X of (X, D). Denote by E_{0} ⊂ X̃ the exceptional divisor arising from blowing up the point (p, t), and note that its discrepancy a(E_{0}, X, D) = 0 by [KM98, Lemmata 2.29 and 2.30]. Hence we have the following diagram.

\xymatrix{X = S × 𝔸_{ℂ}^{1} \ar[d] & X′ = S̃ × 𝔸_{ℂ}^{1} \ar[l] & X̃ ⊃ E_{0} \ar[l]_{f} \ar[ll]^{π} \\ S}

In order to prove the claim, consider a local generator σ of the canonical sheaf ω_{S} in a neighborhood U of 0 ∈ S, and denote its pullback to X′ and X̃ by σ′ and σ̃, respectively. Choose local coordinates (u, v, w) on X′ and (x, y, z) on X̃ such that f: X̃ → X′ is given by

f(x, y, z) = (x, xy, xz)

in these coordinates. Then E_{0} = {x = 0}. Furthermore, we may assume that the exceptional divisor of X′ → X is given by the equation vw = 0. Then, up to a unit, we have

σ′ = d log v ∧ d log w

by construction, so

σ̃ = f^{∗}σ′ = (d log x + d log y) ∧ (d log x + d log z).

An Optimal Extension Theorem for 1-Forms 829

This shows that

σ̃ ∈ H^{0}(π^{−1}(U × 𝔸_{ℂ}^{1}), Ω2 ; X̃(log E))

has a pole along E_{0} = {x = 0}. However, since a(E_{0}, X, D) = 0 and D = ∅, we see that E_{0} is not contained in E_{nklt} ∨ π^{−1}(D). An argument similar to the proof of Lemma 6.2 now yields that π_{∗}Ω2 ; X̃(log(E_{nklt} ∨ π^{−1}(D))) is not reflexive. □

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Patrick Graf Lehrstuhl für Mathematik I Universität Bayreuth 95440 Bayreuth Germany patrick.graf@uni-bayreuth.de Sándor J Kovács University of Washington Department of Mathematics Box 354350 Seattle WA 98195-4350, USA skovacs@uw.edu