arXiv:0707.2054v1 [math.AG] 13 Jul 2007

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THE STRUCTURE OF SURFACES MAPPING TO THE MODULI STACK OFCANONICALLY POLARIZED VARIETIES

STEFAN KEBEKUS AND SÁNDOR J. KOVÁCS

ABSTRACT. Generalizing the well-known Shafarevich hyperbolicity conjecture, it hasbeen conjectured by Viehweg that a quasi-projective manifold that admits a genericallyfinite morphism to the moduli stack of canonically polarizedvarieties is necessarily of loggeneral type. Given a quasi-projective surface that maps tothe moduli stack, we employextension properties of logarithmic pluri-forms to establish a strong relationship betweenthe moduli map and the minimal model program of the surface. As a result, we can describethe fibration induced by the moduli map quite explicitly. A refined affirmative answer toViehweg’s conjecture for families over surfaces follows asa corollary.

CONTENTS

1. Introduction and main results 1

PART I. TECHNIQUES 32. Extending pluri-forms over subvarieties of codimensionone 33. Viehweg-Zuo sheaves on log minimal models 84. Global index-one covers for varieties of logarithmic Kodaira dimension 0 95. Unwinding families 11

PART II. PROOF OF THEOREM 1.1 146. Setup and Notation 147. Proof in caseκ(S) = −∞ 158. Proof thatκ(S) 6= 0 159. Proof in caseκ(S) = 1 18References 18

1. INTRODUCTION AND MAIN RESULTS

1.A. Introduction. Let S be a quasi-projective manifold that admits a morphismµ :S → M to the moduli stack of canonically polarized varieties. Generalizing the classicalShafarevich hyperbolicity conjecture, [Sha63], Viehweg conjectured in [Vie01, 6.3] thatS

is necessarily of log general type ifµ is generically finite. Equivalently, iff : X → S

is a smooth family of canonically polarized varieties, thenS is of log general type as soonas the variation off is maximal, i.e.,Var(f) = dimS. We refer to [KK05], for therelevant notions, for detailed references, and for a brief history of the problem.

Viehweg’s conjecture was confirmed for2-dimensional manifoldsS in [KK05]; seealso [KS06]. Here, we complete the picture. The cornerstoneof the proof is an extensiontheorem for logarithmic pluri-forms, Theorem 2.10. This theorem and its consequences

Date: August 25, 2021.Stefan Kebekus was supported in part by the DFG-Forschergruppe “Classification of Algebraic Surfaces and

Compact Complex Manifolds”. Sándor Kovács was supported inpart by NSF Grant DMS-0554697 and the CraigMcKibben and Sarah Merner Endowed Professorship in Mathematics.

1

http://arxiv.org/abs/0707.2054v1

2 STEFAN KEBEKUS AND SÁNDOR J. KOVÁCS

are used to establish a strong relationship between the moduli mapµ and the logarithmicminimal model program of the surfaceS. This allows us to give a complete descriptionof the moduli map in those cases where the variation cannot bemaximal: the logarithmicminimal model program always ends with a fiber space, and the family comes from thebase of this fibration, at least birationally and after suitable étale cover. Previous resultsand a refined affirmative answer to Viehweg’s conjecture for families over surfaces followas a corollary.

The proof of our main result is rather conceptual and completely independent of theargumentation of [KK05] which essentially relied on combinatorial arguments for curvearrangements on surfaces and on Keel-McKernan’s solution to the Miyanishi conjecture indimension 2, [KMc99]. The present proof, besides giving a more complete picture, doesnot depend on the Keel-McKernan result at all. Many of the techniques introduced heregeneralize well to higher dimensions; most others at least conjecturally.

1.B. Main results. The following is the main result of this paper.

Theorem 1.1. Let f : X → S be a smooth projective family of canonically polarizedvarieties over a quasi-projective surfaceS and S a compactification ofS such thatD := S \ S is a divisor with simple normal crossings. Assume thatVar(f) > 0.

Thenκ(S) 6= 0. Furthermore, ifκ(S) < 2, then any log minimal model programof the pair(S,D) will terminate at a fiber space, and the moduli map factors through theinduced fibration ofS. More precisely, we have the following:

(1.1.1) If κ(S) = −∞, then there exists an open setU ⊂ S of the formU = V ×A1

such thatX∣∣U

is the pull-back of a family overV . In particular,Var(f) = 1.(1.1.2) If κ(S) = 1, then there exists an open setU ⊂ S and a Cartesian diagram

of one of the following two types,

Uγ

étale//

eπ

ellipticfibration

Uπ

ellipticfibration

V étale// V

or

Uγ

étale//

eπ

smoothalgebraicC

∗-bundle

Uπ

smooth,algebraicC

∗-bundle

V V

such thatX ×U U is the pull-back of a family overV . In particular,Var(f) = 1.

Remark1.2. Neither the compactificationS nor the minimal model program discussed inTheorem 1.1 is unique. When running the minimal model program, one often needs tochoose the extremal ray that is to be contracted.

A somewhat more precise version of Viehweg’s conjecture forsurfaces also follows asan immediate consequence of Theorem 1.1, cf. [KK05, Conjecture 1.6].

Corollary 1.3 (Viehweg’s conjecture for surfaces, [KK05, Thm. 1.4]). Letf : X → S

be a smooth projective family of canonically polarized varieties over a quasi-projectivesurfaceS. Then eitherκ(S) = −∞ andVar(f) < dimS, or Var(f) ≤ κ(S).

1.C. Conventions and notation. Throughout the present paper we work over the com-plex number field. When dealing with sheaves that are not necessarily locally free, wefrequently use square brackets to indicate taking the reflexive hull.

Notation1.4. Let Y be a normal variety andA a coherent sheaf ofOY -modules. Letn ∈ N and setA [n] := (A ⊗n)∗∗, Sym[n]

A := (SymnA )∗∗, etc. Likewise, for a

morphismf : X → Y of normal varieties, setf [∗]A := (f∗A )∗∗.

We will later discuss the Kodaira dimension of singular pairs and the Kodaira-Iitakadimension of reflexive sheaves on normal spaces. Since this is perhaps not quite standard,we recall the definition here.

SURFACES MAPPING TO THE MODULI STACK OF CANONICALLY POLARIZED VARIETIES 3

Notation1.5. Let Y be a normal projective variety andA a reflexive sheaf of rank one onY . If h0

(Y, A [n]

)= 0 for all n ∈ N, then we say thatA has Kodaira-Iitaka dimension

κ(A ) := −∞. Otherwise, recall that the restriction ofA to the smooth locus ofY islocally free and consider the rational mapping

φn : Y 99K P(H0

(Y, A

[n])∗)

.

The Kodaira-Iitaka dimension ofA is then defined as

κ(A ) := maxn∈N

(dimφn(Y )

).

If D ⊂ Y is an effective Weil divisor, define the Kodaira dimension ofthe pair(Y,D) asκ(Y,D) := κ

(OY (KY +D)

). If Y is smooth andD is a simple normal crossing divisor,

define the Kodaira dimension of the complementY = Y \ D asκ(Y ) := κ(Y,D).Recall, thatκ(Y ) is independent of the choice of the compactificationY .

1.D. Outline of proof, outline of paper. The technical core of this paper is the extensionresult for pluri-log forms, formulated in Theorems 2.10 and2.15 of Section 2. In essence,it states the following: If(S,D) is a pair of a smooth surface and a reduced divisor withsimple normal crossings,(Sλ, Dλ) a log-minimal model, andAλ ⊂ Sym[n] Ω1

Sλ(logDλ)

any rank-one reflexive sheaf of pluri-log forms, thenAλ pulls back to a reflexive sheaf ofpluri-log forms inSymnΩ1

S(logD′), whereD′ is a divisor that is only slightly larger than

D. Under the conditions of Theorem 1.1, a fundamental result of Viehweg and Zuo assertsthat a rank-one reflexive subsheafAλ ⊂ Sym[n] Ω1

Sλ(logDλ) of positive Kodaira-Iitaka

dimension always exists.The extension theorem is applied, e.g., in Section 3, in order to give a criterion that

is later used to show the fiber space structure of certain minimal models. For an idea ofthe statement and its proof, consider the setup of Theorem 1.1 in the simplest case whereκ(S) = −∞. The log-minimal model(Sλ, Dλ) will then either be log-Fano of Picard-number one, or a Mori-Fano fiber space. To show that(Sλ, Dλ) is a Mori-Fano fiberspace, we argue by contradiction and assume thatρ(Sλ) = 1. Using this assumption and

the existence ofAλ, an analysis of the stability ofΩ[1]Sλ

(logDλ) yields the existence of a

Q-ample rank-one subsheafBλ ⊂ Ω[1]Sλ

(logDλ). The Extension Theorem will then showthe existence of a big invertible subsheafB ⊂ Ω1

S(logD′). This, however, contradicts

the well-known Bogomolov-Sommese vanishing result, and the existence of a fiber spacestructure is shown.

The argumentation in caseκ(S) = 0 follows a similar outline, but is technically muchmore involved. Section 4 gathers results that are particular to the caseκ = 0, work in anydimension and may be of independent interest. The detailed description of the moduli mapfor fiber spaces is done in a unified framework in Section 5.

1.E. Acknowledgments. The work on which this article is based was finished while bothauthors participated in the workshop “Rational curves on algebraic varieties” at the Amer-ican Institute of Mathematics in May, 2007. We would like to thank the AIM for thestimulating atmosphere. We would also like to thank János Kollár for valuable suggestionsthat undoubtedly made the article better, and Duco van Straten for a number of discussionson the extension problem.

PART I. TECHNIQUES

2. EXTENDING PLURI-FORMS OVER SUBVARIETIES OF CODIMENSION ONE

If X is a surface,E ⊂ X a (−1)-curve andω ∈ H0(X, ΩpX(∗E)

)a p-form that is

allowed to have arbitrary poles alongE, then an elementary computation shows thatω isin fact everywhere regular onX , i.e.,ω ∈ H0

(X, ΩpX

).


Much of the argumentation in this paper is based on the observation that a slightlyweaker result also holds for pluri-log forms, and for somewhat larger classes of divisors.We refer to [vSS85, Fle88] for more general extension results that apply to holomorphicp-forms.

2.A. Notation and standard facts about logarithmic differentials. We introduce nota-tion and recall two standard facts before stating and proving the extension result in Sec-tion 2.C below. These make sense and will be used both in the algebraic and in the analyticcategory. We refer to [Iit82, Chapt. 11c] and [Del70, Chap. 3] for details and proofs.

Definition 2.1. A reduced pair(Z,∆) consists of a normal varietyZ and a reduced, but notnecessarily irreducible Weil divisor∆ ⊂ Z. A morphism of reduced pairsγ : (Z, ∆) →

(Z,∆) is a morphismγ : Z → Z such thatγ−1(∆) = ∆ set-theoretically.A reduced pair is calledlog smoothif Z is smooth and∆ has simple normal cross-

ings. Given a reduced pair(Z,∆), let (Z,∆)reg be the maximal open set ofZ where(Z,∆) is log-smooth, and let(Z,∆)sing be its complement, with the structure of a reducedsubscheme. By alog-resolutionof (Z,∆) we will mean a birational morphism of pairsγ : (Z, ∆) → (Z,∆) where(Z, ∆) is log-smooth, andγ is isomorphic along(Z,∆)reg.

Fact 2.2([Hir62]). Let(Z,∆) be a reduced pair. Then a log-resolution exists. If(Z,∆) islog-smooth, then the sheaf of log-differentialsΩ1

Z(log∆) is locally free.

Fact 2.3. Let γ : (Z, ∆) → (Z,∆) be a morphism of log-smooth reduced pairs,U ⊆ Z

an open set andU = γ−1(U). Then there exists a natural pull-back map of log-forms

γ∗ : H0(U, Ω1

Z(log∆))→ H0

(U , Ω1

eZ(log ∆)

).

and an associated sheaf morphism

dγ : γ∗(Ω1Z(log∆)) → Ω1

eZ(log ∆).

If γ is finite and unramified overZ \∆, thendγ is isomorphic.

Remark2.3.1. The pull-back morphism also gives a pull-back of pluri-log forms,

γ∗ : H0(Z, SymnΩ1

Z(log∆))→ H0

(Z, SymnΩ1

eZ(log ∆)

),

that obviously extends to a pull-back of rational forms.

We state one immediate consequence of Fact 2.3 for future reference.

Corollary 2.4. Under the conditions of Fact 2.3, assume thatγ is a finite morphismwhich is unramified overZ \ ∆. Let E ⊂ Z be an effective divisor andσ ∈H0

(Z, Symn

(Ω1Z(log∆)

)(∗E)

)a pluri-log form that might have poles alongE.

Thenσ has no poles alongE, i.e.,σ ∈ H0(Z, SymnΩ1

Z(log∆))

if and only ifγ∗(σ)

has no poles alongγ−1E, i.e.,γ∗(σ) ∈ H0(Z, SymnΩ1eZ(log ∆)).

Notation2.5. In the setup of Corollary 2.4, we say that “σ has poles as a pluri-log form ifand only ifγ∗(σ) has poles as a pluri-log form”.

2.B. Finitely dominated pairs. The formulation of the main extension result in Theo-rem 2.10 uses the following notion, which slightly generalizes quotient singularities.

Definition 2.6. A reduced pair(Z,∆) is said to befinitely dominated by smooth analyticpairs if for any pointz ∈ Z, there exists an analytic neighborhoodU of z and a finite,surjective morphism of reduced pairs(U , ∆) → (U,∆ ∩ U) where(U , ∆) is log-smooth.

Surface singularities that appear in certain variants of the minimal model program areoften finitely dominated by smooth analytic pairs. In the rest of this subsection we discussa class of examples that will become important later.

Definition 2.7. A reduced pair(Z,∆) is calleddlc if (Z,∆) is lc andZ \∆ is lt.


Example2.8. It follows immediately from the definition thatdlt pairs aredlc. For a lessobvious example, letZ be the cone over a conic and∆ the union of two rays through thevertex. Then(Z,∆) is dlc, but notdlt.

Lemma 2.9. Let(Z,∆) be a dlc pair of dimension2. Then(Z,∆) is finitely dominated bysmooth analytic pairs. In particular, if(Z,∆) is dlt, then it is finitely dominated by smoothanalytic pairs.

Proof. Let z ∈ (Z,∆)sing be an arbitrary singular point. Ifz 6∈ ∆, then the statement fol-lows from [KM98, 4.18]. We can thus assume without loss of generality for the remainderof the proof thatz ∈ ∆.

To continue, observe that for any rational number0 < ε < 1, the non-reduced pair(Z, (1− ε)∆) is numerically dlt; see [KM98, 4.1] for the definition and use [KM98, 3.41]for an explicit discrepancy computation. By [KM98, 4.11],Z is thenQ-factorial. UsingQ-factoriality, we can then choose a sufficiently small Zariski neighborhoodU of z andconsider the index-one cover for∆∩U . This gives a finite morphism of pairsγ : (U , ∆) →(U,∆∩U), where the morphismγ is branched only over the singularities ofU , and where∆ = γ∗(∆ ∩ U) is Cartier—see [KM98, 5.19] for the construction. Choose any point z ∈γ−1(z). Since discrepancies only increase under taking finite covers, [KM98, 5.20], thepair(U , ∆) will again be dlc. In particular, it suffices to prove the claim for a neighborhoodof z in (U , ∆). We can thus assume without loss of generality thatz ∈ ∆ and that∆ isCartier in our original setup.

Next, we claim that(Z, ∅) is canonical atz. In fact, letE be any divisor centered abovez, as in [KM98, 2.24]. Sincez ∈ ∆, and since∆ is Cartier, the pull-back of∆ to anyresolution whereE appears will containE with multiplicity at least1. In particular, wehave the following inequality for the log discrepancies:0 ≤ a(E,Z,∆)+1 ≤ a(E,Z, ∅).This shows that(Z, ∅) is canonical atz as claimed.

By [KM98, 4.20-21],Z has a Du Val quotient singularity atz. Again replacingZ by afinite cover of a suitable neighborhood ofz, and replacingz by its preimage in the coveringspace, we can henceforth assume without loss of generality thatZ is smooth. But then theclaim follows from [KM98, 4.15].

2.C. The extension theorem for finitely dominated pairs.The following is the mainresult of the present section. It asserts that any log form defined outside of a divisorE canbe extended to the whole space ifE contracts to a singularity which is finitely dominatedby a smooth analytic pair. Theorem 2.10 holds in arbitrary dimension.

Theorem 2.10(Extension Theorem for finitely dominated singularities). Let (Z,∆) bea reduced pair in the sense of Definition 2.1, and assume that(Z,∆) is finitely domi-nated by smooth analytic pairs. Letψ : (Y,Γ) → (Z,∆) be a log-resolution,EΓ ⊂ Y

the union of theψ-exceptional divisors that are not contained inΓ, andn ∈ N. Thenψ∗ Sym

nΩ1Y (log(Γ + EΓ)) is reflexive.

Remark2.10.1. LetE ⊂ Y be the exceptional set ofψ. Then Theorem 2.10 is equivalentto the statement that for any open setU ⊂ Z with preimageV := ψ−1(U) and any formσ ∈ H0

(V \ E, Symn Ω1

V \E(log Γ))

defined outside theψ-exceptional setE ∩ V , the

form σ extends to a formσ ∈ H0(V, SymnΩ1

V (log(Γ + EΓ)))

on all of V . Hence thename “extension theorem”.

Remark2.10.2. For an example in the simple case where∆ = ∅, let Y be the total spaceof OP1(−2), and letE be the zero-section. It is not very difficult to write down a pluri-logform

σ ∈ H0(Y, Sym2 Ω1

Y (logE))\H0

(Y, Sym2 Ω1

Y

).

BecauseE contracts to a quotient singularity, this example shows that Theorem 2.10 holdsonly for log-differentials, and that the boundary given in Theorem 2.10 is the smallestpossible.


In order to constructσ, consider the standard coordinate cover ofY with open setsU1,2 ≃ A2, whereUi carries coordinatesxi, yi and coordinate change is given as

φ1,2 : (x1, y1) 7→ (x2, y2) = (x−11 , x21y1).

In these coordinates the bundle mapUi → P1 is given as(xi, yi) → xi, and the zero-sectionE is given asE ∩ Ui = yi = 0. Now take

σ2 := y−12 (dy2)

2 ∈(Sym2(Ω1

Y (logE)))(U2)

and observe thatφ∗1,2(σ) extends to a form in(Sym2(Ω1

Y (logE)))(U1).

Proof of Theorem 2.10.Assume that we are given an open setU and a formσ as in Re-mark 2.10.1. Since the extension problem is local onZ in the analytic topology, we canshrinkZ and assume without loss of generality that there exists a finite, surjective mor-phismγ : (Z, ∆) → (Z,∆) from a smooth pair(Z, ∆).

Let Y be the normalization ofY × eZZ andΓ ⊂ Y the reduced preimage ofΓ. Then we

obtain a commutative diagram of surjective morphisms of pairs as follows,

(Y , Γ)eγ, finite

//

eψ

contractseE

(Y,Γ)ψ

log-resolution,contractsE

(Z, ∆)γ, finite

// (Z,∆)

whereE := (γ−1(E))red =((γ ψ

)−1(Z,∆)sing

)red

is the exceptional set of the mor-

phism ψ. Let B ⊂ Z be the branchdivisor of γ, i.e., the minimal codimension-1 setsuch thatγ

∣∣eZ\γ−1(B)

is étale in codimension one. Letψ−1∗ (B) ⊂ Y be its strict transform.

Finally, setY 0 := Y \

(ψ−1∗ (B) ∪ γ((Y , Γ)sing)

).

The setY 0 is then the maximal open subset ofY \ ψ−1∗ (B) such thatY 0 := γ−1(Y 0)

is contained in the log-smooth locus of(Y , Γ). We will use two of its main propertiesexplicitly. These are contained in the following Claims.

Claim 2.10.3. The complementY \ Y 0 intersects theψ-exceptional setE only in a set ofcodimensioncodimY (E \ Y 0) ≥ 2.

Proof. We need to show that

2 ≤ codimY

((ψ−1∗ (B) ∪ γ((Y , Γ)sing)

)∩ E

)

= mincodimY (ψ−1∗ (B) ∩E), codimY (γ((Y , Γ)sing) ∩E).

SinceY is normal, the log-singular locus(Y , Γ)sing has codimension at least2. Sinceγ isfinite, this givescodimY γ

((Y , Γ)sing

)≥ 2. It is also clear thatψ−1

∗ (B) andE have nocommon component, socodimY ψ

−1∗ (B) ∩ E ≥ 2.

Claim 2.10.4. The morphismγ∣∣

eY 0 is étale outside ofE ∩ Y 0.

Proof. By construction,γ is étale outside ofE ∪ ψ−1∗ (B).

To prove Theorem 2.10, we need to show thatσ extends to all ofY as a pluri-log form,i.e., that the associated section

σ ∈ H0(Y,

(SymnΩ1

Y (log(Γ + EΓ)))(∗E)

)

has no poles alongE as a pluri-log form. Sinceσ certainly has no poles outside ofE,and sinceSymnΩ1

Y (log(Γ + EΓ)) is locally free, Claim 2.10.3 implies that it suffices toshow that the restrictionσ

∣∣Y 0 has no poles alongE ∩Y 0 as a pluri-log form. In particular,

σ ∈ H0(Y, SymnΩ1

Y (log(Γ + EΓ))).


By Corollary 2.4 and Claim 2.10.4, it suffices to show that thepull-back γ∗(σ)∣∣

eY 0

does not have any poles alongY 0 ∩ E as a pluri-log form. For that, recall thatψ is anisomorphism over(Z,∆)reg. Hence the formσ gives rise to a form

τ ∈ H0(Z, Sym[n] Ω1

Z(log∆)).

Since(Z, ∆) is log-smooth, Fact 2.3 asserts that the pull-back ofτ extends to a pluri-logform τ ∈ H0

(Z, SymnΩ1

eZ(log ∆)

)on all of Z. The pull-back toY 0,

(2.10.5) τ := ψ∗(τ ) ∈ H0(Y 0, SymnΩ1

eY(log Γ)

∣∣eY 0

),

is then a pluri-log form onY 0 without poles that agrees withγ∗(σ) outsideE. This formnecessarily equals(γ

∣∣eY 0 )

∗(σ), which then does not have any poles alongY 0 ∩ E, asasserted. Theorem 2.10 follows.

Corollary 2.11. Under the conditions of Theorem 2.10 we obtain an embedding

ψ[∗] Sym[n] Ω1Z(log∆) → SymnΩ1

Y (log(Γ + EΓ)).

Proof. Asψ induces an isomorphismY \ E ≃ Z \ ψ(E), Theorem 2.10 implies that

Sym[m] Ω1Z(log∆) ≃ ψ∗ Sym

m Ω1Y (log(Γ + EΓ))

and hence we obtain that there exists a morphism

ψ∗ Sym[m] Ω1Z(log∆) ≃ ψ∗ψ∗ Sym

m Ω1Y (log(Γ + EΓ)) → Symm Ω1

Y (log(Γ + EΓ)),

which is an isomorphism, in particular an embedding, onY \ E. This remainstrue after taking the double dual of these sheaves. Therefore the kernel of the mapψ[∗] Sym[m]Ω1

Z(log∆) → Symm Ω1Y (log(Γ + EΓ)) is a torsion sheaf and the fact that

ψ[∗] Sym[m]Ω1Z(log∆) is torsion-free implies the statement.

2.D. Extensions of Viehweg-Zuo sheaves.We believe that the conclusion of Theo-rem 2.10 holds for a larger class of singularities than thosethat we need to discuss here.Thus it makes sense to introduce the following notation.

Definition 2.12. Let (Z,∆) be a reduced pair in the sense of Definition 2.1. Then we willsay thatthe extension theorem holds for(Z,∆) if for any log-resolutionψ : (Y,Γ) →(Z,∆), the sheafψ∗ Sym

nΩ1Y (log(Γ + EΓ)) is reflexive, whereEΓ denotes the union of

theψ-exceptional divisors that are not contained inΓ, andn ∈ N is arbitrary.

Example2.13. Example 2.9 and Theorem 2.10 imply that the extension theorem holds fordlc surface pairs.

We will later consider log-smooth reduced pairs(Z,∆) and morphismsf : Y → Z

whose restriction toZ \∆ is a smooth family of canonically polarized varieties. Iff haspositive variation,Var(f) > 0, then Viehweg and Zuo have shown in [VZ02, Thm. 1.4]that there exists a positive numbern and an invertible subsheafA ⊂ SymnΩ1

Z(log∆) ofKodaira-Iitaka dimensionκ(A ) ≥ Var(f). We call this aViehweg-Zuo sheafon (Z,∆).More generally and more precisely, we use the following definition.

Definition 2.14. Let (Z,∆) be a reduced pair. A reflexive sheafA of rank1 is called aViehweg-Zuo sheafif for somen ∈ N there exists an embeddingA ⊂ Sym[n] Ω1

Z(log∆).

The extension theorem will be used later to pull-back Viehweg-Zuo sheaves to log res-olutions. The following Theorem shows how this is done.

Theorem 2.15(Extension of Viehweg-Zuo sheaves). Let (Z,∆) be a reduced pair forwhich the extension theorem holds. Using the setup of Definition 2.12, assume that thereexists a Viehweg-Zuo sheafA with inclusionι : A → Sym[n]Ω1

Z(log∆). Then there


exists an invertible Viehweg-Zuo sheafC ⊂ SymnΩ1Y (log(Γ + EΓ)) with the following

property: Letm ∈ N and

ι[m] : A[m] → Sym[m·n] Ω1

Z(log∆)

the associated morphism of reflexive powers. Thenι[m] pulls back to give a sheaf morphismthat factors throughC ⊗m,

ι[m] : ψ[∗]A

[m] → C⊗m ⊂ Symm·nΩ1

Y (log(Γ + EΓ)).

Proof. By Corollary 2.11,ψ[∗]A embeds intoSymnΩ1Y (log(Γ + EΓ)). Let C ⊂

SymnΩ1Y (log(Γ + EΓ)) be the saturation of the image, which is automatically reflex-

ive by [OSS80, Lem. 1.1.16 on p. 158]. By [OSS80, Lem. 1.1.15 on p. 154], C isthen invertible as desired. Further observe that for anym ∈ N, the subsheafC ⊗m ⊂Symm·nΩ1

Y (log(Γ +EΓ)) is likewise saturated. Again, by Corollary 2.11, there exists anembedding,

ι[m] : ψ[∗]A

[m] → Symm·nΩ1Y (log(Γ + EΓ)).

It is easy to see thatι[m] factors throughC⊗m as it does so on the open set whereψ isisomorphic, and becauseC ⊗m is saturated.

Remark2.16. Under the conditions of Theorem 2.15, observe that the Kodaira-Iitaka di-mension ofC is at least the Kodaira-Iitaka dimension ofA , i.e.,κ(C ) ≥ κ(A ).

3. VIEHWEG-ZUO SHEAVES ON LOG MINIMAL MODELS

The existence of a Viehweg-Zuo sheaf of positive Kodaira-Iitaka dimension clearly hasconsequences for the geometry of the underlying space. The following theorem will laterbe used to show that a given pair is a Mori-Fano fiber space. This will turn out to be akey step in the proof of our main results. We refer to Definition 2.14 for the notion of aViehweg-Zuo sheaf.

Theorem 3.1. Let (Z,∆) be a reduced pair such thatZ is a normal andQ-factorialsurface. Assume that the following holds:

(3.1.1) there exists a Viehweg-Zuo sheafA ⊂ Sym[n] Ω1Z(log∆) of positive Kodaira-

Iitaka dimension,(3.1.2) the extension theorem holds for(Z,∆), and(3.1.3) the anti-log-canonical divisor−(KZ +∆) is nef.

Thenρ(Z) > 1.

Proof. We argue by contradiction and assume thatρ(Z) = 1. Let C ⊂ Z be a generalcomplete intersection curve. SinceC is general, it avoids the singular locus(Z,∆)sing.By (3.1.3), the restrictionΩ1

Z(log∆)∣∣C

is a vector bundle of non-positive degree,

(3.2.1) degΩ1Z(log∆)

∣∣C= (KZ +∆).C ≤ 0.

We claim that the restrictionΩ1Z(log∆)

∣∣C

is not anti-nef, i.e., that the dual vector bundleΩ1Z(log∆)∗

∣∣C

is not nef. In particular, we claim thatΩ1Z(log∆)

∣∣C

admits a subsheafof positive degree. Indeed, ifΩ1

Z(log∆)∣∣C

were anti-nef, then none of its symmetricproductsSymnΩ1

Z(log∆)∣∣C

could contain a subsheaf of positive degree. However, sinceC is general, the restriction of the Viehweg-Zuo sheaf toC is a locally free subsheafA

∣∣C

⊂ SymnΩ1Z(log∆)

∣∣C

of positive Kodaira-Iitaka dimension, and hence of positivedegree. This proves the claim.

As a consequence of the claim and of Equation (3.2.1), we obtain thatΩ[1]Z (log∆) is not

semi-stable and ifB ⊂ Ω[1]Z (log∆) denotes the maximal destabilizing subsheaf, its slope

µ(B) is positive. The assumption thatρ(Z) = 1 andQ-factoriality then guarantees thatB is Q-ample. In particular, its Kodaira-Iitaka dimension is maximal,κ(B) = 2.


Now consider a log-resolutionψ : (Y,Γ) → (Z,∆) as in Definition 2.12. The Exten-sion Theorem for Viehweg-Zuo sheaves, Theorem 2.15, Remark2.16, and the assumptionthat ρ(Z) = 1 guarantee the existence of a Viehweg-Zuo sheafC ⊂ Ω1

Y (log Γ + EΓ)of Kodaira-Iitaka dimensionκ(C ) = 2. As there are no symmetric tensors involved, thiscontradicts the Bogomolov-Sommese vanishing theorem, [EV92, Cor. 6.9].

4. GLOBAL INDEX -ONE COVERS FOR VARIETIES OF LOGARITHMICKODAIRA

DIMENSION 0

In this section, we consider a smooth pair(Y,D) of Kodaira dimension0, go to aminimal model and take the global index-one cover. If(Y,D) carries a Viehweg-ZuosheafA ⊂ SymnΩ1

Y (logD) of positive Kodaira-Iitaka dimension, then we show that thecover is uniruled and that its boundary is not empty. All results of this section hold inarbitrary dimension.

4.A. Construction of the cover. First we briefly recall the main properties of the index-one cover, as described in [KM98, 2.52] or [Rei87, Sect. 3.6f].

Proposition 4.1. Let (Y,D) be a reduced, log-smooth pair of dimensiondim Y ≥ 2 andKodaira dimensionκ(Y,D) = 0. Assume that there exists a birational mapλ : Y 99K Yλto a normal varietyYλ, such that the following holds.

(4.1.1) The inverseλ−1 does not contract any divisor.(4.1.2) (Yλ, Dλ) is a log minimal model of(Y,D), whereDλ denotes the cycle-

theoretic image ofD.(4.1.3) The log abundance conjecture holds for(Yλ, Dλ).

Then there exists a diagram

Y

λ

Y

eλ

log resolution

Yλ Yλγ, index-one cover

finite, étale whereYλ is smoothoo

with the following properties.

(4.1.4) If Dλ := γ∗(Dλ), thenKeYλ+ Dλ is Cartier withOeYλ

(KeYλ+ Dλ) ≃ OeYλ

(4.1.5) The pair(Yλ, Dλ) is dlt. If y ∈ Yλ is a point where(Yλ, Dλ) is not log-smooth,then(Yλ, Dλ) is canonical aty.

(4.1.6) If D = λ∗(Dλ)red, thenκ(Y , D) = 0.

For the reader’s convenience, we recall a few notions of higher dimensional geometryused in the formulation of Proposition 4.1.

Notation4.2. A log minimal model is a dlt pair(Yλ, Dλ) whereYλ is Q-factorial andwhereKYλ

+ Dλ is nef, cf. [KM98, 3.29–31]. If(Yλ, Dλ) is a log minimal model andhas Kodaira dimensionκ(Yλ, Dλ) = 0, we say thatthe log abundance conjecture holdsfor (Yλ, Dλ) if there exists a numberk ∈ N+ such thatk · (KYλ

+ Dλ) is Cartier andOYλ

(k · (KYλ

+Dλ))≃ OYλ

, cf. [KM98, 3.12].

Remark4.3. The existence of log minimal models and log abundance for minimal modelsis currently known fordimY ≤ 3, see [KM98, 3.13] for references concerning abundance.Both are expected to hold in any dimension—see [BCHM06, Siu06] for the latest progress.

Proof of Proposition 4.1.Let k ∈ N+ be the index ofKYλ+Dλ, i.e., the smallest number

such thatOYλ

(k · (KYλ

+Dλ))≃ OYλ

and letγ : Yλ → Yλ be the associated index-one

cover. We obtain thatKeYλ+ Dλ is a Cartier divisor for the trivial bundle, as claimed

in (4.1.4).


The assertion thatYλ is dlt follows from the definition and from the fact that discrepan-cies increase under finite morphisms, [KM98, 5.20]. Ify ∈ Yλ is any point where(Yλ, Dλ)is not log-smooth, then by the definition of dlt, the discrepancy of any divisorE that liesovery is a(E, Yλ, Dλ) > −1. But sinceKeYλ

+ Dλ is Cartier, this number must be an

integer, soa(E, Yλ, Dλ) ≥ 0. It follows that the pair(Yλ, Dλ) is canonical aty, hence(4.1.5) is shown.

It remains to prove thatκ(Y , D) = 0, as claimed in (4.1.6). Since(Yλ, Dλ) is canonicalwherever it is not log-smooth, the definition of canonical, [KM98, 2.26, 2.34], implies thatKeY

+ D is represented by an effective,λ-exceptional divisor, hence (4.1.6) follows.

Corollary 4.4. Under the conditions of Proposition 4.1 further assume thatdimY = 2.Then(Yλ, Dλ) is log-smooth alongDλ andYλ isQ-factorial.

Proof. TheQ-factoriality follows from (4.1.5) and [KM98, 4.11]. Log-smoothness followsfrom the classification of canonical surface singularities, [KM98, 4.5].

4.B. The index-one cover in the presence of a Viehweg-Zuo sheaf.We will later con-sider the index-one cover in the presence of a Viehweg-Zuo sheafA . If κ(A ) > 0, wewill show that Y is uniruled, and that the boundary cannot be empty. A similarline ofargumentation was used in [KK05, KK07].

Proposition 4.5. Under the conditions of Proposition 4.1 further assume thatthere exists aViehweg-Zuo sheafA ⊂ SymnΩ1

Y (logD) of positive Kodaira-Iitaka dimension,κ(A ) >

0. ThenY andY are uniruled.

The following—rather elementary—statements are used in the proof of Proposition 4.5.We formulate two separate lemmas for later reference.

Lemma 4.6. Let (Y,D) be a log-smooth pair and assume that there exists a Viehweg-ZuosheafA ⊂ Symn Ω1

Y (logD). If λ : Y 99K Yλ is a birational map whose inverse doesnot contract any divisor,Yλ is normal andDλ is the cycle-theoretic image ofD, thenthere exists a Viehweg-Zuo sheafAλ ⊂ Sym[n]Ω1

Yλ(logDλ) of Kodaira-Iitaka dimension

κ(Aλ) ≥ κ(A ).

Proof. The assumption thatλ−1 does not contract any divisors and the normality ofYλguarantee thatλ−1 : Yλ 99K Y is well-defined and injective over an open subsetY

λ ⊂ Yλwhose complement has codimensioncodimYλ

(Yλ \ Y λ ) ≥ 2. In particular,Dλ

∣∣Y λ

=(λ−1

∣∣Y λ

)−1D. Let ι : Y

λ → Yλ denote the embedding and setAλ := ι∗((λ−1

∣∣Y λ

)[∗]A).

Fact 2.3 gives an inclusionAλ ⊂ Sym[n]Ω1Yλ(logDλ). By constructionh0

(Yλ, A

[m]λ

)≥

h0(Y, A ⊗m) for all m > 0, henceκ(Aλ) ≥ κ(A ).

Lemma 4.7. Under the conditions of Proposition 4.1 further assume thatthere exists aViehweg-Zuo sheafA ⊂ SymnΩ1

Y (logD). Then there exists a Viehweg-Zuo sheafAλ ⊂

Sym[n] Ω1eYλ

(log Dλ) of Kodaira-Iitaka dimensionκ(Aλ) ≥ κ(A ).

Proof. Let Aλ be defined as in Lemma 4.6, and setAλ := γ[∗]Aλ. The factsthat Aλ is reflexive and thatγ is étale imply that there exists an embeddingAλ →

Sym[n] Ω1eYλ

(log Dλ), as claimed.

Proof of Proposition 4.5.Since uniruledness is a birational property, and since images ofuniruled varieties are again uniruled, it suffices to show the claim forYλ. We argue by con-tradiction and assume thatYλ (and then alsoY ) is not uniruled —by [BDPP04, Cor. 0.3]this is equivalent to assuming thatKeY

is pseudo-effective. Again by [BDPP04, Thm. 0.2],

this is in turn equivalent to the assumption thatKeY· C ≥ 0 for all moving curvesC ⊂ Y .


As a first step, we will show that the assumption implies that the (Weil) divisorDλ iszero. To this end, choose a polarization ofYλ and consider a general complete intersectioncurveCλ ⊂ Yλ. BecauseCλ is a complete intersection curve, it intersects the supportof the effective divisorDλ if the support is not empty. By general choice, the curveCλ

is contained in the smooth locus ofYλ and avoids the indeterminacy locus ofλ−1. ItspreimageC := λ−1(Cλ) is then a moving curve inY which intersectsD positively if andonly if the Weil divisorDλ is not zero. But

0 = (KeYλ+ Dλ) · Cλ = (KeY

+ D) · C = KeY· C

︸︷︷︸≥0, as eC is moving

+ D · C︸︷︷︸

≥0, as eC not in eD

,

so D · C = 0. In particular,Dλ is the zero divisor. This, combined with the fact thatOeYλ

(KeYλ+Dλ) ≃ OeYλ

implies that the canonical divisorKeYλis Cartier and its associated

sheaf is trivial. In particular, the restrictionsΩ1eYλ

∣∣eCλ

andTeYλ

∣∣eCλ

are vector bundles of

degree zero and so is the symmetric productSymnΩ1eYλ

∣∣eCλ

.Recall from Lemma 4.7 that there exists a Viehweg-Zuo sheaf of positive Kodaira-

Iitaka dimension, sayAλ ⊂ Sym[n] Ω1eYλ

(log Dλ). As Cλ is a general curve, the re-

striction Aλ

∣∣eCλ

⊂ SymnΩ1eYλ

∣∣eCλ

has positive degree. In particular,SymnΩ1eYλ

∣∣eCλ

isnot semi-stable. Since symmetric products of semi-stable vector bundles are again semi-stable [HL97, Cor. 3.2.10], this implies thatΩ1

eYλ

∣∣eCλ

is likewise not semi-stable. Since

degΩ1eYλ

∣∣eCλ

= degTeYλ

∣∣eCλ

= 0, this also implies thatTeYλ

∣∣eCλ

is not semi-stable.

In particular, the maximal destabilizing subsheaf ofTeYλ

∣∣eCλ

is of positive degree, henceample. In this setup, a variant [KST07, Cor. 5] of Miyaoka’s uniruledness criterion [Miy87,Cor. 8.6] applies to give the uniruledness ofYλ. For more details on this criterion see thesurvey [KS06]. This ends the proof of Proposition 4.5.

Corollary 4.8. Under the conditions of Proposition 4.5 the boundary divisor Dλ is notempty. In particular,D, Dλ andD are not empty.

Proof. Again, we assume to the contrary thatDλ is empty. Proposition 4.1 then impliesthatκ(Y ) = 0, while Proposition 4.5 asserts thatY is uniruled, a contradiction.

5. UNWINDING FAMILIES

We will consider projective familiesg : Y → T where the baseT itself admits afibrationρ : T → B such thatg is isotrivial on allρ-fibers. It is of course generally falsethatg would be the pull-back of a family defined overB. We will, however, show in thissection that in some situations the familyg does become a pull-back after a suitable basechange.

We use the following notation for fibered products that appear in our setup.

Notation5.1. LetT be a scheme,Y andZ schemes overT andh : Y → Z aT -morphism.If t ∈ T is any point, letYt andZt denote the fibers ofY andZ overt. Furthermore, letht denote the restriction ofh to Yt. More generally, for anyT -schemeT , let

heT: Y ×T T︸︷︷︸

=:Y eT

→ Z ×T T︸︷︷︸=:Z eT


denote the pull-back ofh to T . The situation is summarized in the following commutativediagram.

YeT

??

????

??

((

h eT

// Z eT

((Y

==

====

== h// Z

T// T

The setup of the current section is then formulated as follows.

Assumption 5.2. Throughout the present section, consider a sequence of morphisms be-tween algebraic varieties,

Yg

smooth, projective// T

ρ

smooth, rel. dim.=1// B,

whereg is a smooth projective family andρ is smooth of relative dimension 1, but notnecessarily projective. Assume further that for allb ∈ B, there exists a smooth varietyFbsuch that for allt ∈ Tb, there exists an isomorphismYt ≃ Fb.

5.A. Relative isomorphisms of families over the same base.To start, recall the well-known fact that an isotrivial family of varieties of generaltype over a curve becomes trivialafter passing to an étale cover of the base. As we are not awareof an adequate reference,we include a proof here.

Lemma 5.3. Let b ∈ B and assume thatAut(Fb) is finite. Then the natural morphismι : I = IsomTb

(Yb, Tb × Fb) → Tb is finite and étale. Furthermore, pull-back toI yieldsan isomorphism ofI-schemesYI ≃ I × Fb.

Proof. Consider theTb-scheme

H := HilbTb

(Yb ×Tb

(Tb × Fb))≃ HilbTb

(Yb × Fb

).

By Assumption 5.2,Ht ≃ Hilb(Fb × Fb) for all t ∈ Tb. Similarly, It ≃ Aut(Fb), henceI is one-dimensional andlength(It) is constant onTb. SinceI is open inH , the union ofcomponents ofH that containI, denoted byHI , is also one-dimensional.

Recall thatH → Tb is projective, soHI → Tb is also projective, hence finite. SinceH → Tb is flat, length(Ht) is constant. Furthermore,I ⊆ HI is open, soHI

t = It andhencelength(Ht) = length(It) for a generalt ∈ Tb. However, we observed above thatlength(It) is also constant, so we must have thatlength(Ht) = length(It) for all t ∈ Tb,and sinceI ⊆ HI , this means thatI = HI andι : I → Tb is finite and unramified, henceétale.

In order to prove the global triviality ofYI , considerIsomI(YI , I × Fb). Recall thattakingHilb andIsom commutes with base change, and so we obtain an isomorphism

IsomI(YI , I × Fb) ≃ I ×TbIsomTb

(Yb, Tb × Fb) ≃ I ×TbI.

This scheme admits a natural section overTb, namely its diagonal, which induces anI-isomorphism betweenYI andI × Fb.

The preceding Lemma 5.3 can be used to compare two families whose associated mod-uli maps agree. We show that in our setup any two such familiesbecome globally isomor-phic after changing base.

Lemma 5.4. In addition to Assumption 5.2, assume that there exists another projectivemorphism,Z → T , with the following property: for anyb ∈ B and anyt ∈ Tb, we haveYt ≃ Zt ≃ Fb. Then


(5.4.1) there exists a surjective morphismτ : T → T such that the pull-back familiesof Y andZ to T are isomorphic asT -schemes, i.e., we have a commutativediagram as follows:

YeT

??

????

??

((ooeT−isom.

// Z eT

((Y

??

????

??Z

Tτ // T

ρ

B.

Furthermore, if for allb ∈ B, the groupAut(Fb) is finite, thenT can be chosen such thatthe following holds. LetT ′ ⊂ T be any irreducible component. Then

(5.4.2) τ is quasi-finite,(5.4.3) the image setτ(T ′) is a union ofρ-fibers, and(5.4.4) if T ′ dominatesB, then there exists an open subsetB ⊂ (ρτ)(T ′) such that

τ∣∣

eT ′ is finite and étale overB. More precisely, if we setT := ρ−1(B) and

T := τ−1(T ) ∩ T ′, then the restrictionτ∣∣

eT : T → T is finite and étale.

Remark5.4.5. In Lemma 5.4 we do not claim thatT is irreducible or connected.

Proof of Lemma 5.4.SetT := IsomT (Y, Z) and letτ : T → T be the natural morphism.Again, takingIsom commutes with base change, and we have an isomorphismT ×T T ≃IsomeT

(YeT, Z eT

). Similarly, for all b ∈ B, and for allt ∈ Tb, there is a natural one-to-one

correspondence betweenTt andAut(Fb). In particular, we obtain thatτ is surjective. Asbefore, observe thatT ×T T admits a natural section, the diagonal. This shows (5.4.1).

If for all b ∈ B, Aut(Fb) is finite, then the restriction ofτ to anyρ-fiber,τb : Tb → Tbis finite étale by Lemma 5.3. This shows (5.4.2) and (5.4.3). Furthermore, it implies thatif T ′ ⊂ T is a component that dominatesB, neither the ramification locus ofτ

∣∣eT ′ nor the

locus whereτ∣∣

eT ′ is not finite dominatesB. In fact, if we letB denote the open set ofBwhere#Aut(Fb) is constant, then (5.4.4) holds forB.

5.B. Families whereρ has a section.Now consider Assumption 5.2 in case the morphismρ admits a sectionσ : B → T such thatZ = YB ×B T . As a corollary to Lemma 5.4, wewill show that in this situationT always contains a componentT ′ such that the pull-backfamily YeT ′ comes fromB. Better still, the restrictionτ

∣∣eT ′

is “relatively étale” in the sensethatτ

∣∣eT ′ is étale and thatρ τ

∣∣eT ′ has connected fibers.

Corollary 5.5. Under the conditions of Lemma 5.4 assume thatρ admits a sectionσ :

B → T , and thatZ = YB ×B T . Then there exists an irreducible componentT ′ ⊂ T suchthat

(5.5.1) T ′ surjects ontoB, and(5.5.2) the restricted morphismρ τ

∣∣eT ′ : T

′ → B has connected fibers.

Proof. It is clear from the construction thatYB ≃ ZB. This isomorphism corresponds toa morphismσ : B → IsomT (Y, Z) = T . Let T ′ ⊂ T be an irreducible component thatcontains the image ofσ. The existence of a section guarantees thatρ τ

∣∣eT ′ : T ′ → B is

surjective and its fibers are connected.

One particular setup where a section is known to exist is whenT is a birationally ruledsurface overB. The following will become important later.


Corollary 5.6. In addition to Assumption 5.2, suppose thatB is a smooth curve and thatthe generalρ-fiber is isomorphic toP1, A1 or (A1)∗ = A1 \ 0. Then there exist non-empty Zariski open setsB ⊂ B, T := ρ−1(B) and a commutative diagram

T τ

étale//

conn. fibers $$

T

ρ

B

such that

(5.6.1) the fibers ofρ τ are again isomorphic toP1, A1 or (A1)∗, respectively, and(5.6.2) the pull-back familyYeT comes fromB, i.e., there exists a projective family

Z → B and aT -isomorphism

YeT ≃ Z eT .

Remark5.6.3. If the generalρ-fiber is isomorphic toP1 or A1, the morphismτ is neces-sarily an isomorphism. ShrinkingB further, if necessary,ρ : T → B will then even bea trivialP1– orA1–bundle, respectively.

Proof. ShrinkingB, if necessary, we may assume that allρ-fibers are isomorphic toP1,A1

or (A1)∗, and hence thatT is smooth. Then it is always possible to find a relative smoothcompactification ofT , i.e. a smoothB-varietyT → B and a smooth divisorD ⊂ T suchthatT \D andT are isomorphicB-schemes.

By Tsen’s theorem, [Sha94, p. 73], there exists a sectionσ : B → T . In fact, thereexists a positive dimensional family of sections, so that wemay assume without loss ofgenerality thatσ(B) is not contained inD.

LetB ⊂ B be the open subset such that for allb ∈ B, T b ≃ P1, Tb is isomorphic toP1,A1 or (A1)∗, respectively, andσ(b) 6∈ D. Using that any connected finite étale cover ofTb is again isomorphic toTb, and shrinkingB further, Corollary 5.5 yields the claim.

Remark5.7. Throughout the article we work over the field of complex numbersC, thuswe kept that assumption here as well. However, we would like to note that the results ofthis section work over an arbitrary algebraically closed base fieldk.

PART II. PROOF OF THEOREM 1.1

6. SETUP AND NOTATION

The casesκ(S) = −∞, 0 and1 are considered separately in Sections 7–9 below.The following setup and notation will be used throughout therest of the article: As in

Theorem 1.1, we fix a smooth compactificationS ⊂ S such thatD := S \ S is a divisorwith simple normal crossings. The log minimal model programthen yields a birationalmorphismλ : S → Sλ, with the following properties.

(6.0.1) The surfaceSλ is normal andQ-factorial.(6.0.2) IfDλ is the cycle-theoretic image ofD, then(Sλ, Dλ) is a reduced dlt pair.(6.0.3) By Lemma 2.9 and Theorem 2.10, the extension theoremholds for(Sλ, Dλ).

Again, recall from [VZ02, Thm. 1.4] that there exists a Viehweg-Zuo sheafA ⊂SymnΩ1

S(logD) of positive Kodaira-Iitaka dimensionκ(A ) ≥ Var(f) > 0. ByLemma 4.6, there exists a Viehweg-Zuo sheafAλ ⊂ Sym[n] Ω1

Sλ(logDλ) of Kodaira-

Iitaka dimensionκ(Aλ) ≥ κ(A ).


7. PROOF IN CASEκ(S) = −∞

In this case, the log canonical bundleKS +D has negative Kodaira-Iitaka dimension,and(Sλ, Dλ) is a pair that either has the structure of a Mori-Fano fiber space or is a log-Fano pair with Picard numberρ(Sλ) = 1. However, since the extension theorem holds,Theorem 3.1 rules out the case thatρ(Sλ) = 1. The pair(Sλ, Dλ) thus always admitsa fibration, independently of the choices made in its construction. In particular, thereexists a smooth curveC and a fibrationπλ : Sλ → C with connected fibers, such that−(KSλ

+Dλ) intersects the general fiber positively.Settingπ := πλ λ, the general fiberF of π is then a rational curve that intersects the

boundary in one point, if at all. In particular, the restriction of the familyf to F ∩ S

is necessarily isotrivial by [Kov00]. The detailed description of the moduli map in caseκ(S) = −∞ then follows from Corollary 5.6 and Remark 5.6.3.

8. PROOF THATκ(S) 6= 0

8.A. Setup. To prove Theorem 1.1 in this case, we argue by contradiction and assumethat κ(S) = 0. Let (Sλ, Dλ) be the index-one cover of a log-minimal model, as inProposition 4.1. The main properties of(Sλ, Dλ) are summarized as follows.

(8.1.1) The pair(Sλ, Dλ) isQ-factorial and dlt (Proposition 4.1 and Corollary 4.4).(8.1.2) There exists a Viehweg-Zuo sheafAλ ⊂ Sym[n] Ω1

eSλ

(log Dλ) of positiveKodaira-Iitaka dimension (Lemma 4.7).

(8.1.3) If Sλ,reg ⊂ Sλ is the maximal smooth open subset, then the restriction

Aλ

∣∣eSλ,reg

is invertible ([OSS80, 1.1.15 on p. 154]).

(8.1.4) Sλ is uniruled, and the boundaryDλ is not empty (Proposition 4.5 and Corol-lary 4.8).

(8.1.5) The divisorKeSλ+ Dλ is Cartier and trivial (Proposition 4.1).

(8.1.6) If p ∈ Dλ is any point, then(Sλ, Dλ) is log-smooth atp. In particular,Ω1

eSλ

(log Dλ) is locally free alongDλ (Corollary 4.4).

8.B. Outline of the proof. As a first step in the proof of Theorem 1.1, we aim to applyTheorem 3.1, in order to show thatSλ is fibered over a curve, with rational fibers thatintersect the boundary twice. Since Theorem 3.1 works best in the caseκ = −∞ weneed to decrease the boundary coefficients slightly and perform extra contractions beforeTheorem 3.1 can be applied to prove the existence of a fibration.

The fiber space structure ofSλ is then used to analyze the restriction of the Viehweg-Zuo sheafAλ to a suitable boundary componentD′

λ ⊂ Dλ. Even though there is no

smooth family overD′λ, it will turn out that the restrictionAλ

∣∣fD′

λ

can be interpreted as a

Viehweg-Zuo sheaf onD′λ, which again has positive Kodaira-Iitaka dimension. This leads

to contradiction and thus finishes the proof.

8.C. Minimal models of (Sλ, Dλ). SinceDλ is not empty andKeSλ≡ −Dλ, it follows

that for any rational number0 < ε < 1,

κ(Sλ, (1− ε)Dλ

)= κ

(KeSλ

+ (1− ε)Dλ

)= κ

(ε ·KeSλ

)= κ

(Sλ

)= −∞.

Choose a rational number0 < ε < 1 and perform a minimal model program for the pair(Sλ, (1 − ε)Dλ

). This will produce a birational morphismµ : Sλ → Sµ. Let Dµ be

the cycle-theoretic image ofDλ. Since(Sλ, (1 − ε)Dλ

)has dlt singularities, the pair(

Sµ, (1 − ε)Dµ

)will also be dlt, in fact, it will be klt.

Remark8.2. Sinceκ(Sλ, (1 − ε)Dλ) = −∞, eitherρ(Sµ) > 1 and the pair(Sµ, (1 −

ε)Dµ

)is a Mori-Fano fiber space, orρ(Sµ) = 1.


Remark8.3. It follows from the equationKeSλ≡ −Dλ that for any0 < ε′, ε′′ < 1, the

divisorsKeSλ+(1−ε′)Dλ andKeSλ

+(1−ε′′)Dλ are multiples of one another. In particular,

the birational morphismµ is a minimal model program for the pair(Sλ, (1 − ε)Dλ

),

independently of the chosen0 < ε < 1. It follows that(Sµ, (1−ε)Dµ

)has dlt singularities

for all ε. In particular, it follows directly from the definition of discrepancy [KM98, 2.26]that the reduced pair(Sµ, Dµ) is dlc in the sense of Definition 2.7.

8.D. The fiber space structure ofSλ. We apply Theorem 3.1 in order to show thatSµ isa fiber space.

Proposition 8.4. One has thatρ(Sµ) > 1. In particular, Sµ has the structure of a non-trivial Mori-Fano fiber space.

Proof. If −(KSµ+Dµ) is not ample, thenρ(Sµ) > 1 and the statement follows from Re-

mark 8.2. If−(KSµ+Dµ) is ample, then Remark 8.3 and Example 2.9 imply that(Sµ, Dµ)

is finitely dominated by smooth analytic pairs. Then by Theorem 2.10, the extension the-orem holds for(Sµ, Dµ). According to Lemma 4.6 there exists a Viehweg-Zuo subsheafAµ ⊂ Sym[n] Ω1

Sµ(logDµ) of positive Kodaira-Iitaka dimension and then Theorem 3.1

and Remark 8.2 imply the desired statement.

Corollary 8.5. There exists a morphismπ : Sλ → C to a smooth curve, and an opensetC ⊂ C such that for anyc ∈ C, the associated fiberFc := π−1(c) is a smoothrational curve which is entirely contained in the log-smooth locus(Sλ, Dλ)reg and whichintersects the boundaryDλ transversally in exactly two points. In particular, the sheafΩ1

eSλ

(log Dλ)∣∣Fc

is trivial.

Proof. The existence ofπ and the rationality of the general fiber follows from Proposi-tion 8.4. The number of intersection points follows fromKeSλ

+ Dλ ≡ 0. The triviality of

Ω1eSλ

(log Dλ)∣∣Fc

follows from standard sequences, see [KK05, 2.14] and (8.9.1) below.

Corollary 8.6. If c ∈ C is a general point, then the restrictionAλ

∣∣Fc

is trivial.

Proof. SinceFc is a general fiber,A [r]λ

∣∣Fc

is an invertible sheaf for anyr ∈ Z by (8.1.3).

In particular,A [r]λ

∣∣Fc

≃(Aλ

∣∣Fc

)⊗r. Fix anr ∈ N such thath0

(Sλ, A

[r]λ

)> 0. Then

there exists a non-trivial and hence injective morphism

A[r]λ

∣∣Fc

→(Sym[r·n]Ω1

eSλ(log Dλ)

)∣∣Fc

≃ Symr·n(Ω1

eSλ(log Dλ)

∣∣Fc

).

The triviality of the sheafΩ1eSλ

(log Dλ)∣∣Fc

implies thatdeg(A

[r]λ

∣∣Fc

)≤ 0. SinceFc

passes through a general point ofDλ, a general section of the sheafA[r]λ does not vanish

along all ofFc. ThereforeA [r]λ

∣∣Fc

is a line bundle of non-positive degree that has a globalsection. Consequently it is trivial. SinceFc ≃ P1, this implies the statement.

8.E. Non-triviality of Aλ

∣∣D

. Now consider a sectionσ ∈ H0(Sλ, A

[r]λ

), let Fc be a

generalπ-fiber andy ∈ Fc a general point ofFc. The triviality of A [r]λ onFc can now be

used to compare the value ofσ at ay with its value at a point whereFc hits the boundaryDλ. It will follow that σ is completely determined by the values it takes on the boundary.

Lemma 8.7. There exists an irreducible componentD′λ ⊂ Dλ such that for anyr ∈ N,

the natural restriction morphism

H0(Sλ, A

[r]λ

)→ H0

(D′λ, A

[r]λ

∣∣fD′

λ

)


is injective. In particular, the restrictionAλ

∣∣fD′

λ

is a non-trivial invertible sheaf and

its Kodaira-Iitaka dimension equals the Kodaira-Iitaka dimension ofAλ, i.e.,κ(Aλ) =

κ(Aλ

∣∣fD′

λ

).

Proof. Corollary 8.5 implies that there exists a componentD′λ ⊂ Dλ that is hit by all the

curves(Fc)c∈C . Now let r be any given number. Ifh0(Sλ, A

[r]λ

)= 0, there is nothing

to show. Otherwise, using the notation of Corollary 8.5, set

D′λ, := D′

λ ∩ π−1(C).

Since a section in the trivial bundle is determined by its value at any given point, a sectionσ ∈ H0

(Sλ, A

[r]λ

)is uniquely determined by its restriction toD′

λ, ⊂ D′λ by Corol-

lary 8.6. Finally, note thatAλ

∣∣fD′

λ

is invertible by (8.1.3) and (8.1.6). Thus the claim is

shown.

Remark8.8. Let ι : Aλ → Sym[n]Ω1eSλ

(log Dλ) denote the injection of our Viehweg-Zuosheaf into the sheaf of pluri-log differentials. Then its restrictionι|fD′

λ

is injective.

8.F. Existence of pluri-forms on D′

λ. As a last ingredient in the proof, we show that

sections in tensor powers of the invertible sheafAλ

∣∣fD′

λ

can again be interpreted as pluri-

forms on the boundary.

Lemma 8.9. LetD′′λ = (Dλ−D

′λ)∣∣

fD′λ

. Then there exists a numberm ≤ n and an injective

sheaf morphismAλ

∣∣fD′

λ

→ Symm Ω1eD′λ

(log D′′λ).

Proof. SinceAλ

∣∣fD′

λ

is invertible by (8.1.3) and (8.1.6), it is enough to show that there

exists a non-zero morphismAλ

∣∣fD′

λ

→ Symm Ω1eD′λ

(log D′′λ), for somem ∈ N. We will

use the following sequence that relates restrictions of log-forms with log-forms on therestriction—the sequence is discussed in [KK05, 2.13].

(8.9.1) 0 // Ω1eD′λ

(log D′′λ)

α // Ω1eSλ

(log Dλ)∣∣

fD′λ

β// O eD′

λ

// 0.

Along with this sequence comes the standard filtration of thesymmetric product,

SymnΩ1eSλ(log Dλ)

∣∣fD′

λ

= F0 ⊇ F

1 ⊇ · · · ⊇ Fn ⊇ F

n+1 = 0,

with quotients

(8.9.2) 0 //F p+1

αp// F p

βp// SympΩ1

eD′λ

(log D′′λ) // 0.

See [Har77, ex. II.5.16] for details. As in Remark 8.8, letι be the injection of the Viehweg-Zuo sheafAλ into the sheaf of pluri-log differentialsSym[n] Ω1

eSλ

(log Dλ). Recall from

Lemma 8.7 thatAλ

∣∣fD′

λ

has positive Kodaira-Iitaka dimension and from Remark 8.8 that it

embeds intoSym[n] Ω1eSλ

(log Dλ)∣∣

fD′λ

≃ SymnΩ1eSλ

(log Dλ)∣∣

fD′λ

.

First consider the sequence in (8.9.2) forp = 0. SinceSym0 Ω1eD′λ

(log D′′λ) = O eD′

λ

,

and since any morphism from an invertible sheaf of positive Kodaira-Iitaka dimension tothe structure sheaf is necessarily zero, the compositionβ0 ι|fD′

λ

is zero, and the restriction

ι|fD′λ

factors via an injectionι1 : A∣∣

fD′→ F 1.

Next consider (8.9.2) forp = 1. If β1 ι1 is non-zero, the proof is finished. Otherwise,ι1 factors via an injectionι2 : Aλ

∣∣fD′

λ

→ F 2, and we consider (8.9.2) forp = 2, etc. This

process must stop after no more thann steps. Thus the claim is shown.


8.G. End of the proof. Using the notation introduced in Lemma 8.9, the adjunction for-mula shows thatK eD′

λ

+ D′′λ = (KeSλ

+ Dλ)∣∣

fD′λ

≡ 0. In particular,degΩ1eD′λ

(log D′′λ) = 0.

On the other hand, Lemma 8.9 asserts the existence of an injective morphism of sheavesAλ

∣∣fD′

λ

→ Symm Ω1eD′λ

(log D′′λ). By Lemma 8.7,Aλ

∣∣fD′

λ

has positive Kodaira-Iitaka di-

mensionκ(Aλ

∣∣fD′

λ

)= κ(Aλ) > 0. This is clearly absurd, and the proof of Theorem 1.1 is

thus finished in the caseκ(S) = 0.

9. PROOF IN CASEκ(S) = 1

In this case the statements of Theorem 1.1 follow from the results of Section 5 when oneapplies the logarithmic minimal model program. The following proposition summarizesthe standard description of surfaces with logarithmic Kodaira dimension 1.

Proposition 9.1. If κ(S) = 1, then there exists a smooth curveC and a fibrationπ : S →C with connected fibers, such thatKS+D is trivial on the general fiber. In particular, oneof the following holds:

(9.1.1) The general fiber is an elliptic curve and no component ofD dominatesC, or(9.1.2) The general fiber is isomorphic toP1 andD intersects the general fiber in

exactly two points.

Proof. The logarithmic abundance theorem in dimension 2, see e.g. [KM98, 3.3], assertsthat forn≫ 0 the linear system|n(KSλ

+Dλ)| yields a morphism to a curveπλ : Sλ → C,such thatKSλ

+Dλ is trivial on the general fiberFλ of πλ. Likewise, ifπ := πλ φ andF ⊂ S is a general fiber ofπ, thenKS +D is trivial onF . Statements (9.1.1) and (9.1.2)describe the only two ways this can happen.

To finish the proof of Theorem 1.1, consider the morphismπ : S → C provided byProposition 9.1. LetV ⊆ C be the locus over whichπ is smooth and eitherD∩π−1(V ) =∅ or π

∣∣D

is étale. Consider the restriction ofπ toU := π−1(V ) ∩ S. By Proposition 9.1,the general fiber ofπ

∣∣U

is either an elliptic curve, or it is isomorphic toC∗. In both cases,it follows from [Kov96] and [Kov00] thatf is isotrivial on the fibers ofπ : U → V . Thefactorization of the moduli map follows.

It remains to give the detailed description of the moduli map. If the general fibers ofπ are isomorphic toC∗, Corollary 5.6 yields the claim. Otherwise, take an irreduciblemultisectionV ⊂ S, restrictV further if necessary soV is étale overV and take a basechange toV . We end up with a sectionσ : V → U := U×V V . Finally, setX := X×U U ,andZ := V ×σ X. ShrinkingV further, if necessary, an application of Lemma 5.4completes the proof of Theorem 1.1.

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STEFAN KEBEKUS, MATHEMATISCHES INSTITUT, UNIVERSITÄT ZU KÖLN, WEYERTAL 86–90, 50931KÖLN, GERMANY

E-mail address: [email protected]: http://www.mi.uni-koeln.de/∼kebekus

SÁNDOR KOVÁCS, UNIVERSITY OF WASHINGTON, DEPARTMENT OF MATHEMATICS, BOX 354350,SEATTLE, WA 98195, U.S.A.

E-mail address: [email protected]: http://www.math.washington.edu/∼kovacs



http://www.ictp.trieste.it/~pub_off/services

mailto:[email protected]

http://www.mi.uni-koeln.de/~kebekus

mailto:[email protected]

http://www.math.washington.edu/~kovacs

arXiv:0707.2054v1 [math.AG] 13 Jul 2007

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