Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
DDifferential forms on singular spaces, the minimal modelprogram, and hyperbolicity of moduli stacks
Stefan Kebekus
In memory of Eckart Viehweg
Abstract.
The Shafarevich Hyperbolicity Conjecture, proven by Arakelovand Parshin, considers a smooth, projective family of algebraic curves over asmooth quasi-projective base curve Y . It asserts that if Y is of special type,then the family is necessarily isotrivial.This survey discusses hyperbolicity properties of moduli stacks and gener-alisations of the Shafarevich Hyperbolicity Conjecture to higher dimensions.It concentrates on methods and results that relate moduli theory with recentprogress in higher dimensional birational geometry. Contents
Received by the editors November 1, 2018.2000
Mathematics Subject Classification.
Primary 14D22; Secondary 14D05.
Key words and phrases. hyperbolicity properties of moduli spaces, minimal model program,differential forms on singular spaces. a r X i v : . [ m a t h . AG ] D ec Differential forms, MMP, and Hyperbolicity4.5 Existence of a pull-back morphism, idea of proof 314.6 Open problems 365 Viehweg’s conjecture for families over threefolds, sketch of proof 375.1 A special case of the Viehweg conjecture 375.2 Sketch of proof 37
1. Introduction
In his contribution to the 1962 International Congress ofMathematicians, Igor Shafarevich formulated an influential conjecture, consideringsmooth, projective families f ◦ : X ◦ → Y ◦ of curves of genus g >
1, over a fixedsmooth quasi-projective base curve Y ◦ . One part of the conjecture, known as the“hyperbolicity conjecture”, gives a sufficient criterion to guarantee that any suchfamily is isotrivial. The conjecture was shown in two seminal works by Parshinand Arakelov, including the following special case. Theorem 1.1 (Shavarevich Hyperbolicity Conjecture, [Sha63], [Par68, Ara71]) . Let f ◦ : X ◦ → Y ◦ be a smooth, complex, projective family of curves of genus g > , over a smooth quasi-projective base curve Y ◦ . If Y ◦ is isomorphic to oneof the following varieties, • the projective line P , • the affine line A , • the affine line minus one point C ∗ , or • an elliptic curve,then any two fibres of f ◦ are necessarily isomorphic. Notation-Assumption 1.2.
Throughout this paper, a family is a flat morphismof algebraic varieties with connected fibres. We always work over the complexnumber field.Remark . Following standard convention, we refer to Theorem 1.1 as “Sha-farevich hyperbolicity conjecture” rather than “Arakelov-Parshin theorem”. Thereader interested in a complete picture is referred to [Vie01, p. 253ff], where allparts of the Shafarevich conjecture are discussed in more detail.Formulated in modern terms, Theorem 1.1 asserts that any morphism froma smooth, quasi-projective curve Y ◦ to the moduli stack of algebraic curves isnecessarily constant if Y ◦ is one of the special curves mentioned in the theorem. If Y ◦ is a quasi-projective variety of arbitrary dimension, then any morphism from Y ◦ to the moduli stack contracts all rational and elliptic curves, as well as all affinelines and C ∗ s that are contained in Y ◦ .tefan Kebekus 3We refer to [HK10, Sect. 16.E.1] for a discussion of the relation between theShafarevich hyperbolicity conjecture and the notions of Brody– and Kobayashihyperbolicity. This survey is concerned with generalisations of theShafarevich hyperbolicity conjecture to higher dimensions, concentrating on meth-ods and results that relate moduli– and minimal model theory. We hope that themethods presented here will be applicable to a much wider ranges of problems, inmoduli theory and elsewhere. The list of problems that we would like to addressinclude the following.
Questions . Apart from the quasi-projective curves mentioned above, whatother varieties admit only constant maps to the moduli stack of curves? Whatabout moduli stacks of higher dimensional varieties? Given a variety Y ◦ , is therea good geometric description of the subvarieties that will always be contracted byany morphism to any reasonable moduli stack?Much progress has been achieved in the last years and several of the questionscan be answered today. It turns out that there is a close connection between theminimal model program of a given quasi-projective variety Y ◦ , and its possiblemorphisms to moduli stacks. Some of the answers obtained are in fact conse-quences of this connection.In the limited number of pages available, we say almost nothing about thehistory of higher dimensional moduli, or about the large body of important worksthat approach the problem from other points of view. Hardly any serious attemptis made to give a comprehensive list of references, and the author apologises toall those whose works are not adequately represented here, be it because of theauthor’s ignorance or simply because of lack of space.The reader who is interested in a broader overview of higher dimensionalmoduli, its history, complete references, and perhaps also in rigidity questions formorphisms to moduli stacks is strongly encouraged to consult the excellent surveysfound in this handbook and elsewhere, including [HK10, Kov06, Vie01]. A gentleand very readable introduction to moduli of higher dimensional varieties in alsofound in [Kov09], while Viehweg’s book [Vie95] serves as a standard technicalreference for the construction of moduli spaces.Most relevant notions and facts from minimal model theory can either befound in the introductory text [Mat02], or in the extremely clear and well-writtenreference book [KM98]. Recent progress in minimal model theory is surveyed in[HK10]. Section 2 introduces a number of conjectural generalisations of the Shafare-vich hyperbolicity conjecture and gives an overview of the results that have been Differential forms, MMP, and Hyperbolicityobtained in this direction. In particular, we mention results relating the modulimap and the minimal model program of the base of a family.Sections 3 and 4 introduce the reader to methods that have been developedto attack the conjectures mentioned in Section 2. While Section 3 concentrateson positivity results on moduli spaces and on Viehweg and Zuo’s constructionof (pluri-)differential forms on base manifolds of families, Section 4 summarisesresults concerning differential forms on singular spaces. Both sections containsketches of proofs which aim to give an idea of the methods that go into the proofs,and which might serve as a guideline to the original literature. The introduction toSection 4 motivates the results on differential forms by explaining a first strategyof proof for a special case of a (conjectural) generalisation of the Shafarevichhyperbolicity conjecture. Following this plan of attack, a more general case istreated in the concluding Section 5, illustrating the use of the methods introducedbefore.
This paper is dedicated to the memory of Eckart Viehweg. Like so many othermathematicians of his age group, the author benefited immensely from Eckart’spresence in the field, his enthusiasm, guidance and support. Eckart will be re-membered as an outstanding mathematician, and as a fine human being.The work on this paper was partially supported by the DFG Forschergruppe790 “Classification of algebraic surfaces and compact complex manifolds”. PatrickGraf kindly read earlier versions of this paper and helped to remove several errorsand misprints. Many of the results presented here have been obtained in jointwork of S´andor Kov´acs and the author. The author would like to thank S´andorfor innumerable discussions, and for a long lasting collaboration. He would also liketo thank the anonymous referee for careful reading and for numerous suggestionsthat helped to improve the quality of this survey.Not all the material presented here is new, and some parts of this surveyhave appeared in similar form elsewhere. The author would like to thank hiscoauthors for allowing him to use material from their joint research papers. Thefirst subsection of every chapter lists the sources that the author was aware of.
2. Generalisations of the Shafarevich hyperbolicity conjecture
Given its importance in algebraic and arithmetic geometry, much work hasbeen invested to generalise the Shafarevich hyperbolicity conjecture, Theorem 1.1.Historically, the first generalisations have been concerned with families f ◦ : X ◦ → Y ◦ where Y ◦ is still a quasi-projective curve, but where the fibres of f ◦ are allowedto have higher dimension. The following elementary example shows, however, thattefan Kebekus 5Theorem 1.1 cannot be generalised na¨ıvely, and that additional conditions mustbe posed. Example . Consider a smooth projective surface Y of general typewhich contains a rational or elliptic curve C ⊂ Y . Assume that the automorphismgroup of Y fixes the curve C pointwise. Examples can be obtained by choosing anysurface of general type and then blowing up sufficiently many points in sufficientlygeneral position —each blow-up will create a rational curve and lower the numberof automorphisms. Thus, if c and c ∈ C are any two distinct points, then thesurfaces Y c i obtained by blowing up the points c i are non-isomorphic.In order to construct a proper family, consider the product Y × C with itsprojection π : Y × C → C and with the natural section ∆ ⊂ Y × C . If X is theblow-up of Y × C in ∆, then we obtain a smooth, projective family f : X → C ofsurfaces of general type, with the property that no two fibres are isomorphic.It can well be argued that Counterexample 2.1 is not very natural, and thatthe fibres of the family f would trivially be isomorphic if they had not beenblown up artificially. This might suggest to consider only families that are “notthe blow-up of something else”. One way to make this condition is precise is toconsider only families of minimal surfaces , i.e., surfaces F whose canonical bundle K F is semiample. In higher dimensions, it is often advantageous to impose astronger condition and consider only families of canonically polarised manifolds ,i.e., manifolds F whose canonical bundle K F is ample.Hyperbolicity properties of families of minimal surfaces and families of min-imal varieties have been studied by a large number of people, including Miglior-ini [Mig95], Kov´acs [Kov96, Kov97] and Oguiso-Viehweg [OV01]. For families ofcanonically polarised manifolds, the analogue of Theorem 1.1 has been shown byKov´acs in the algebraic setup [Kov00]. Combining algebraic arguments with deepanalytic methods, Viehweg and Zuo prove a more general Brody hyperbolicitytheorem for moduli spaces of canonically polarised manifolds which also impliesan analogue of Theorem 1.1, [VZ03]. Theorem 2.2 (Hyperbolicity for families of canononically polarized varieties,[Kov00, VZ03]) . Let f ◦ : X ◦ → Y ◦ be a smooth, complex, projective familyof canonically polarised varieties of arbitrary dimension, over a smooth quasi-projective base curve Y ◦ . Then the conclusion of the Shafarevich hyperbolicityconjecture, Theorem 1.1, holds. (cid:3) This paper discusses generalisations of the Shafarevich hyperbolicity conjec-ture to families over higher dimensional base manifolds. To formulate any gener-alisation, two points need to be clarified. Differential forms, MMP, and Hyperbolicity(1) We need to define a higher dimensional analogue for the list of quasi-projective curves given in Theorem 1.1.(2) Given any family f ◦ : X ◦ → Y ◦ over a higher dimensional base, call twopoints y , y ∈ Y ◦ equivalent if the fibres ( f ◦ ) − ( y ) and ( f ◦ ) − ( y ) areisomorphic. If Y ◦ is a curve, then either there is only one equivalence class,or all equivalence classes are finite. For families over higher dimensionalbase manifolds, the equivalence classes will generally be subvarieties ofarbitrary dimension. We will need to have a quantitative measure for thenumber of equivalence classes and their dimensions.The problems outlined above justify the definition of the logarithmic Kodairadimension and of the variation of a family , respectively. Before coming to thegeneralisations of the Shafarevich hyperbolicity conjecture in Section 2.2.3 below,we recall the definitions for the reader’s convenience. The logarithmic Kodaira di-mension generalises the classical notion of Kodaira dimension to the category ofquasi-projective varieties.
Definition 2.3 (Logarithmic Kodaira dimension) . Let Y ◦ be a smooth quasi-projective variety and Y a smooth projective compactification of Y ◦ such that D := Y \ Y ◦ is a divisor with simple normal crossings. The logarithmic Kodairadimension of Y ◦ , denoted by κ ( Y ◦ ) , is defined to be the Kodaira-Iitaka dimensionof the line bundle O Y ( K Y + D ) ∈ Pic( Y ) . A quasi-projective variety Y ◦ is calledof log general type if κ ( Y ◦ ) = dim Y ◦ , i.e., the divisor K Y + D is big. It is a standard fact of logarithmic geometry that a compactification Y withthe described properties exists, and that the logarithmic Kodaira dimension κ ( Y ◦ )does not depend on the choice of the compactification. Observation . The quasi-projective curves listed in Theorem 1.1 are preciselythose curves Y ◦ with logarithmic Kodaira dimension κ ( Y ◦ ) ≤ The following definition provides a quanti-tative measure of the birational variation of a family. Note that the definition ismeaningful even in cases where no moduli space exists.
Definition 2.5 (Variation of a family, cf. [Vie83, Introduction]) . Let f ◦ : X ◦ → Y ◦ be a projective family over an irreducible base Y ◦ , and let C ( Y ◦ ) denote thealgebraic closure of the function field of Y ◦ . The variation of f ◦ , denoted by Var( f ◦ ) , is defined as the smallest integer ν for which there exists a subfield K of C ( Y ◦ ) , finitely generated of transcendence degree ν over C and a K -variety F such that X × Y ◦ Spec C ( Y ◦ ) is birationally equivalent to F × Spec K Spec C ( Y ◦ ) .Remark . In the setup of Definition 2.5, assume that all fibres if Y ◦ are canon-ically polarised complex manifolds. Then coarse moduli schemes are known totefan Kebekus 7exist, [Vie95, Thm. 1.11], and the variation is the same as either the dimension ofthe image of Y ◦ in moduli, or the rank of the Kodaira-Spencer map at the generalpoint of Y ◦ . Further, one obtains that Var( f ◦ ) = 0 if and only if all fibres of f ◦ are isomorphic. In this case, the family f ◦ is called “isotrivial”. Using the notion of “logarithmic Kodaira di-mension” and “variation”, the Shafarevich hyperbolicity conjecture can be refor-mulated as follows.
Theorem 2.7 (Reformulation of Theorem 1.1) . If f ◦ : X ◦ → Y ◦ is any smooth,complex, projective family of curves of genus g > , over a smooth quasi-projectivebase curve Y ◦ , and if Var( f ◦ ) = dim Y ◦ , then κ ( Y ◦ ) = dim Y ◦ . Aiming to generalise the Shafarevich hyperbolicity conjecture to families overhigher dimensional base manifolds, Viehweg has conjectured that this reformula-tion holds true in arbitrary dimension.
Conjecture 2.8 (Viehweg’s conjecture, [Vie01, 6.3]) . Let f ◦ : X ◦ → Y ◦ bea smooth projective family of varieties with semiample canonical bundle, over aquasi-projective manifold Y ◦ . If f ◦ has maximal variation, then Y ◦ is of log gen-eral type. In other words, Var( f ◦ ) = dim Y ◦ ⇒ κ ( Y ◦ ) = dim Y ◦ . Viehweg’s conjecture has been proven by S´andor Kov´acs and the author incase where Y ◦ is a surface, [KK08a, KK07], or a threefold, [KK08c]. The methodsdeveloped in these papers will be discussed, and an idea of the proof will be givenlater in this paper, cf. the outline of this paper given in Section 1.2 on page 3. Theorem 2.9 (Viehweg’s conjecture for families over threefolds, [KK08c,Thm. 1.1]) . Viehweg’s conjecture holds in case where dim Y ◦ ≤ . (cid:3) For families of canonically polarised varieties, much stronger results havebeen obtained, giving an explicit geometric explanation of Theorem 2.9.
Theorem 2.10 (Relationship between the moduli map and the MMP, [KK08c,Thm. 1.1]) . Let f ◦ : X ◦ → Y ◦ be a smooth projective family of canonically po-larised varieties, over a quasi-projective manifold Y ◦ of dimension dim Y ◦ ≤ .Let Y be a smooth compactification of Y ◦ such that D := Y \ Y ◦ is a divisor withsimple normal crossings.Then any run of the minimal model program of the pair ( Y, D ) will termi-nate in a Kodaira or Mori fibre space whose fibration factors the moduli mapbirationally.Remark . Neither the compactification Y nor the minimal model programdiscussed in Theorem 2.10 is unique. When running the minimal model program,one often needs to choose the extremal ray that is to be contracted. Differential forms, MMP, and HyperbolicityIn order to explain the statement of Theorem 2.10, let M be the appropriatecoarse moduli space whose existence is shown, e.g. in [Vie95, Thm. 1.11]. Further,let µ ◦ : Y ◦ → M be the moduli map associated with the family f ◦ , and let µ : Y (cid:57)(cid:57)(cid:75) M be the associated rational map from the compactification Y . If λ : Y (cid:57)(cid:57)(cid:75) Y λ is a rational map obtained by running the minimal model program, and if Y λ → Z λ is the associated Kodaira or Mori fibre space, then Theorem 2.10 assertsthe existence of a map Z λ (cid:57)(cid:57)(cid:75) M that makes the following diagram commutative, Y λ MMP of the pair (
Y,D ) (cid:47) (cid:47) moduli map induced by f ◦ (cid:15) (cid:15) Y λ Kodaira or Mori fibre space (cid:15) (cid:15) M Z λ . ∃ ! (cid:111) (cid:111) Now, if we assume in addition that κ ( Y ◦ ) ≥
0, then the minimal modelprogram terminates in a Kodaira fibre space whose base Z λ has dimensiondim Z λ = κ ( Y ◦ ), so that Var( f ◦ ) ≤ κ ( Y ◦ ). If we assume that κ ( Y ◦ ) = −∞ ,then the minimal model program terminates in proper Mori fibre space and weobtain that dim Z λ < dim Y and Var( f ◦ ) < dim Y ◦ . The following refined answerto Viehweg’s conjecture is therefore an immediate corollary of Theorem 2.10. Corollary 2.12 (Refined answer to Viehweg’s conjecture, [KK08c, Cor. 1.3]) . Let f ◦ : X ◦ → Y ◦ be a smooth projective family of canonically polarised varieties, overa quasi-projective manifold Y ◦ of dimension dim Y ◦ ≤ . Then either (1) κ ( Y ◦ ) = −∞ and Var( f ◦ ) < dim Y ◦ , or (2) κ ( Y ◦ ) ≥ and Var( f ◦ ) ≤ κ ( Y ◦ ) . (cid:3) Remark . Corollary 2.12 asserts that any family of canonically polarised vari-eties over a base manifold Y ◦ with κ ( Y ◦ ) = 0 is necessarily isotrivial. Remark . Corollary 2.12 has also been shown in case where Y ◦ is a projective manifold of arbitrary dimension, conditional to the standard conjectures of mini-mal model theory , cf. [KK08b, Thm. 1.4]. A very short proof that does not relyon minimal model theory has been announced by Patakfalvi as this paper goes toprint, [Pat11]. Example κ ( Y ◦ ) = −∞ ) . To see thatthe result of Corollary 2.12 is optimal in case κ ( Y ◦ ) = −∞ , let f ◦ : X ◦ → Y ◦ be any family of canonically polarised varieties with Var( f ◦ ) = 2, over a smoothsurface Y ◦ (which may or may not be compact). Setting X ◦ := X ◦ × P and Y ◦ := Y ◦ × P , we obtain a family f ◦ = f ◦ × Id P : X ◦ → Y ◦ with variationVar( f ◦ ) = 2, and with a base manifold Y ◦ of Kodaira dimension κ ( Y ◦ ) = −∞ . i.e., existence and termination of the minimal model program and abundance tefan Kebekus 9 Example κ ( Y ◦ ) = −∞ ) . In thesetup of Corollary 2.12, if Y ◦ is a projective Fano manifold, then a fundamentalresult of Campana and Koll´ar-Miyaoka-Mori asserts that Y ◦ is rationally con-nected, [Kol96, V. Thm. 2.13]. In other words, given any two points x , y in Y ◦ ,there exists a rational curve C ⊂ Y ◦ which contains both x and y . Recalling fromTheorem 2.2 that families over rational curves are isotrivial, it follows immediatelythat the family f ◦ is necessarily isotrivial itself.A much stronger version of this result has been shown by Lohmann, [Loh11].Given a projective variety Y and a Q -divisor D such that ( Y, D ) is a divisoriallylog terminal (=dlt) pair, consider the smooth quasi-projective variety Y ◦ := ( Y \ supp (cid:98) D (cid:99) ) reg . Lohmann shows that if (
Y, D ) is log-Fano, that is, if the Q -divisor − ( K Y + D )is ample, then any family of canonically polarized varieties over Y ◦ is necessarilyisotrivial. The proof relies on a generalization of Araujo’s result [Ara10] whichrelates extremal rays in the moving cone of a variety with fiber spaces that appearat the end of the minimal model program. Lohmann shows that the moduli mapfactorizes through any of the fibrations obtained in this way. In a series of papers, including [Cam04,Cam08], Campana introduced the notion of “geometric orbifolds” and “specialvarieties”. Campana’s language helps to formulate a very natural generalisationof Theorem 1.1, which includes the cases covered by the Viehweg Conjecture 2.8,and gives (at least conjecturally) a satisfactory geometric explanation of isotrivi-ality observed in some families over spaces that are not covered by Conjecture 2.8.Before formulating the conjecture, we briefly recall the precise definition ofa special logarithmic pair for the reader’s convenience. We take the classicalBogomolov-Sommese Vanishing Theorem as our starting point. We refer to [Iit82,EV92] or to the original reference [Del70] for an explanation of the sheaf Ω pY (log D )of logarithmic differentials. Theorem 2.17 (Bogomolov-Sommese Vanishing Theorem, cf. [EV92, Sect. 6]) . Let Y be a smooth projective variety and D ⊂ Y a reduced (possibly empty) divisorwith simple normal crossings. If p ≤ dim Y is any number and A ⊆ Ω pY (log D ) any invertible subsheaf, then the Kodaira-Iitaka dimension of A is at most p , i.e., κ ( A ) ≤ p . (cid:3) In a nutshell, we say that a pair (
Y, D ) is special if the inequality in theBogomolov-Sommese Vanishing Theorem is always strict.
Definition 2.18 (Special logarithmic pair) . In the setup of Theorem 2.17, a pair ( Y, D ) is called special if the strict inequality κ ( A ) < p holds for all p and all We refer to [KM98, Sect. 2.3] for the definition of a dlt pair, and for related notions concerningsingularities of pairs that are relevant in minimal model theory. invertible sheaves
A ⊆ Ω pY (log D ) . A smooth, quasi-projective variety Y ◦ is calledspecial if there exists a smooth compactification Y such that D := Y \ Y ◦ is adivisor with simple normal crossings and such that the pair ( Y, D ) is special.Remark . It is an elementary fact that if Y ◦ is a smooth, quasi-projective variety and Y , Y two smooth compactificationssuch that D i := Y i \ Y ◦ are divisors with simple normal crossings, then ( Y , D )is special if and only if ( Y , D ) is. The notion of special should thus be seen as aproperty of the quasi-projective variety Y ◦ . Fact . Ra-tionally connected manifolds and manifolds X with κ ( X ) = 0 are special. (cid:3) With this notation in place, Campana’s conjecture can be formulated asfollows.
Conjecture 2.21 (Campana’s conjecture, [Cam08, Conj. 12.19]) . Let f : X ◦ → Y ◦ be a smooth family of canonically polarised varieties over a smooth quasi-projective base. If Y ◦ is special, then the family f is isotrivial. In analogy with the construction of the maximally rationally connected quo-tient map of uniruled varieties, Campana constructs in [Cam04, Sect. 3] an almost-holomorphic “core map” whose fibres are special in the sense of Definition 2.18.Like the MRC quotient, the core map is uniquely characterised by certain maxi-mality properties, [Cam04, Thm. 3.3], which essentially say that the core map of X contracts almost all special subvarieties contained in X . If Campana’s Con-jecture 2.21 holds, this would imply that the core map always factors the modulimap, similar to what we have seen in Section 2.2.3 above, Y core mapalmost holomorphic (cid:47) (cid:47) moduli map induced by f ◦ (cid:15) (cid:15) Z ∃ ! (cid:111) (cid:111) M . As with Viehweg’s Conjecture 2.8, Campana’s Conjecture 2.21 has beenshown for surfaces [JK09b] and threefolds [JK09a].
Theorem 2.22 (Campana’s conjecture in dimension three, [JK09a, Thm. 1.5]) . Campana’s Conjecture 2.21 holds if dim Y ◦ ≤ . (cid:3) Viehweg’s Conjecture 2.8 and Campana’s Conjecture 2.21 have been shownfor families over base manifolds of dimension three or less. As we will see inSection 5, the restriction to three-dimensional base manifolds comes from thefact that minimal model theory is particularly well-developed for threefolds, andfrom our limited ability to handle differential forms on singular spaces of highertefan Kebekus 11dimension. We do not believe that there is a fundamental reason that restricts usto dimension three, and we do believe that the relationship between the modulimap and the MMP found in Theorem 2.10 will hold in arbitrary dimension.
Conjecture 2.23 (Relationship between the moduli map and the MMP) . Let f ◦ : X ◦ → Y ◦ be a smooth projective family of canonically polarised varieties,over a quasi-projective manifold Y ◦ . Let Y be a smooth compactification of Y ◦ such that D := Y \ Y ◦ is a divisor with simple normal crossings. Then any run ofthe minimal model program of the pair ( Y, D ) will terminate in a Kodaira or Morifibre space whose fibration factors the moduli map birationally. Conjecture 2.24 (Refined Viehweg conjecture, cf. [KK08a, Conj. 1.6]) . Corol-lary 2.12 holds without the assumption that dim Y ◦ ≤ . Given the current progress in minimal model theory, a proof of Conjec-tures 2.23 and 2.24 does no longer seem out of reach.
3. Techniques I: Existence of Pluri-differentials on the base of afamily
Throughout the present Section 3, we consider a smooth projective family f ◦ : X ◦ → Y ◦ of projective, canonically polarised complex manifolds, over asmooth complex quasi-projective base. We assume that the family is not isotrivial,and fix a smooth projective compactification Y of Y ◦ such that D := Y \ Y ◦ is adivisor with simple normal crossings. In this setup, Viehweg and Zuo have shownthe following fundamental result asserting the existence of many logarithmic pluri-differentials on Y . Theorem 3.1 (Existence of pluri-differentials on Y , [VZ02, Thm. 1.4(i)]) . Let f ◦ : X ◦ → Y ◦ be a smooth projective family of canonically polarised complexmanifolds, over a smooth complex quasi-projective base. Assume that the familyis not isotrivial and fix a smooth projective compactification Y of Y ◦ such that D := Y \ Y ◦ is a divisor with simple normal crossings.Then there exists a number m > and an invertible sheaf A ⊆
Sym m Ω Y (log D ) whose Kodaira-Iitaka dimension is at least the variation of thefamily, κ ( A ) ≥ Var( f ◦ ) . (cid:3) Remark . Observe that the Shafarevich hyperbolicity conjecture, Theorem 1.1,follows as an immediate corollary of Theorem 3.1.
Remark . A somewhat weaker version of Theorem 3.1 holds for families of pro-jective manifolds with only semiample canonical bundle if one assumes additionallythat the family is of maximal variation, i.e., that Var( f ◦ ) = dim Y ◦ , cf. [VZ02,Thm. 1.4(iv)].2 Differential forms, MMP, and HyperbolicityAs we will see in Section 5, the “Viehweg-Zuo” sheaf A is one of the crucialingredients in the proofs of Viehweg’s and Campana’s conjecture for families overthreefolds, Theorems 2.9, 2.10 and 2.22. A careful review of Viehweg and Zuo’sconstruction reveals that the “Viehweg-Zuo sheaf” A comes from the coarse modulispace M , at least generically. The precise statement, given in Theorem 3.6, usesthe following notion. Notation 3.4 (Differentials coming from moduli space generically) . Let µ : Y ◦ → M be the moduli map associated with the family f ◦ , and consider the subsheaf B ⊆ Ω Y (log D ) , defined on presheaf level as follows: if U ⊆ Y is any open setand σ ∈ H (cid:0) U, Ω Y (log D ) (cid:1) any section, then σ ∈ H (cid:0) U, B (cid:1) if and only if therestriction σ | U (cid:48) is in the image of the differential map dµ | U (cid:48) : µ ∗ (cid:0) Ω M (cid:1) | U (cid:48) −→ Ω U (cid:48) , where U (cid:48) ⊆ U ∩ Y ◦ is the open subset where the moduli map µ has maximal rank.Remark . By construction, it is clear that the sheaf B is a saturated subsheafof Ω Y (log D ), i.e., that the quotient sheaf Ω Y (log D ) / B is torsion free. We saythat B is the saturation of Image( dµ ) in Ω Y (log D ). Theorem 3.6 (Refinement of the Viehweg-Zuo Theorem 3.1, [JK09b, Thm. 1.4]) . In the setup of Theorem 3.1, there exists a number m > and an invertiblesubsheaf A ⊆
Sym m B whose Kodaira-Iitaka dimension is at least the variation ofthe family, κ ( A ) ≥ Var( f ◦ ) . Theorem 3.6 follows without too much work from Viehweg’s and Zuo’s orig-inal arguments and constructions, which are reviewed in Section 3.2 below. Com-pared with Theorem 3.1, the refined Viehweg-Zuo theorem relates more directlyto Campana’s Conjecture 2.21 and other generalizations of the Shafarevich con-jecture. To illustrate its use, we show in the surface case how Theorem 3.6 reducesCampana’s Conjecture 2.21 to the Viehweg Conjecture 2.8, for which a positiveanswer is known.
Corollary 3.7 (Campana’s conjecture in dimension two) . Conjecture 2.21 holdsif dim Y ◦ = 2 .Proof. We maintain the notation of Conjecture 2.21 and let f : X ◦ → Y ◦ be asmooth family of canonically polarised varieties over a smooth quasi-projectivebase, with Y ◦ a special surface. Since Y ◦ is special, it is not of log general type,and the solution to Viehweg’s conjecture in dimension two, [KK08c, Thm. 1.1],gives that Var( f ◦ ) < f ◦ ) = 1 and choose a compact-ification ( Y, D ) as in Definition 2.18. By Theorem 3.6 there exists a number m >
A ⊆
Sym m B such that κ ( A ) ≥
1. However, since B is saturated in the locally free sheaf Ω Y (log D ), it is reflexive, [OSS80, Claim ontefan Kebekus 13p. 158], and since Var( f ◦ ) = 1, the sheaf B is of rank 1. Thus B ⊆ Ω Y (log D ) isan invertible subsheaf, [OSS80, Lem. 1.1.15, on p. 154], and Definition 2.18 of aspecial pair gives that κ ( B ) <
1, contradicting the fact that κ ( A ) ≥
1. It followsthat Var( f ◦ ) = 0 and that the family is hence isotrivial. (cid:3) Outline of this section
Given its importance in the theory, we give a verybrief synopsis of Viehweg-Zuo’s proof of Theorem 3.1, showing how the theoremfollows from deep positivity results for push-forward sheaves of relative dualizingsheaves, and for kernels of Kodaira-Spencer maps, respectively. Even though noproof of the refined Theorem 3.6, is given, it is hoped that the reader who choosesto read Section 3.2 will believe that Theorem 3.6 follows with some extra work byessentially the same methods.The reader who is interested in a detailed understanding, including is referredto the papers [Kol86], [VZ02], and to the survey [Vie01]. The overview contained inthis section and the facts outlined in Section 3.2.5 can perhaps serve as a guidelineto [VZ02].Many of the technical difficulties encountered in the full proof of Theorem 3.1vanish if f ◦ is a family of curves. The proof becomes very transparent in this case.In particular, it is very easy to see how the Kodaira-Spencer map associated withthe family f ◦ transports the positivity found in push-forward sheaves into thesheaf of differentials Ω Y (log Y ). After setting up notation in Section 3.2.1, wehave therefore included a Section 3.2.2 which discusses the curve case in detail.Most of the material presented in the current Section 3, including the synopsisof Viehweg-Zuo’s construction, is taken without much modification from the paper[JK09b]. The presentation is inspired in part by [Vie01]. Throughout the present Section 3.2, we choose asmooth projective compactification X of X ◦ such that the following holds:(1) The difference ∆ := X \ X ◦ is a divisor with simple normal crossings.(2) The morphism f ◦ extends to a projective morphism f : X → Y .It is then clear that ∆ = f − ( D ) set-theoretically. Removing a suitable subset S ⊂ Y of codimension codim Y S ≥
2, the following will then hold automaticallyon Y (cid:48) := Y \ S and X (cid:48) := X \ f − ( S ), respectively.(3) The restricted morphism f (cid:48) := f | X (cid:48) is flat.(4) The divisor D (cid:48) := D ∩ Y (cid:48) is smooth.(5) The divisor ∆ (cid:48) := ∆ ∩ X (cid:48) is a relative normal crossing divisor, i.e. a normalcrossing divisor whose components and all their intersections are smoothover the components of D (cid:48) . The positivity results in question are formulated in Theorems 3.10 and Fact 3.22, respectively. f (cid:48) : X (cid:48) → Y (cid:48) is a “good partial compactification of f ◦ ”. Remark . Let G be a locally freesheaf on Y , and let F (cid:48) ⊆ G| Y (cid:48) be an invertible subsheaf. Since codim Y S ≥ F (cid:48) to an invertible subsheaf F ⊆ G on Y . Furthermore, the restriction map H (cid:0) Y, F (cid:1) → H (cid:0) Y (cid:48) , F (cid:48) (cid:1) is an isomorphism.In particular, the notion of Kodaira-Iitaka dimension makes sense for the sheaf F (cid:48) , and κ ( F (cid:48) ) = κ ( F ).We denote the relative dimension of X over Y by n := dim X − dim Y . Before sketch-ing the proof of Theorem 3.1 in full generality, we illustrate the main idea in aparticularly simple setting.
Simplifying Assumptions . Throughout the present introductory Section 3.2.2,we maintain the following simplifying assumptions in addition to the assumptionsmade in Theorem 3.1.(1) The quasi-projective variety Y ◦ is in fact projective. In particular, weassume that X = X ◦ , f = f ◦ , that D = ∅ and that ∆ = ∅ .(2) The family f : X → Y is a family of curves of genus g >
1. In particular,we have that ( T X/Y ) ∗ = Ω X/Y = ω X/Y , where T X/Y is the kernel of thederivative
T f : T X → f ∗ ( T Y ).(3) The variation of f ◦ is maximal, that is, Var( f ◦ ) = dim Y ◦ .The proof of Theorem 3.1 sketched here uses positivity of the push-forwardof relative dualizing sheaves as its main input. The positivity result required isdiscussed in Viehweg’s survey [Vie01, Sect. 1–3], where positivity is obtained asa consequence of generalised Kodaira vanishing theorems. The reader interestedin a broader overview might also want to look at the remarks and references in[Laz04, Sect. 6.3.E], as well as the papers [Kol86, Zuo00] Theorem 3.10 (Positivity of push-forward sheaves, cf. [VZ02, Prop. 3.4.(i)]) . Under the simplifying Assumptions 3.9, the push-forward sheaf f ∗ (cid:0) ω ⊗ X/Y (cid:1) is locallyfree of positive rank. If
A ∈
Pic( Y ) is any ample line bundle, then there existnumbers N, M (cid:29) and a sheaf morphism φ : A ⊕ M → Sym N f ∗ (cid:0) ω ⊗ X/Y (cid:1) which is surjective at the general point of Y . (cid:3) For the reader’s convenience, we recall two other facts used in the proof,namely the existence of a Kodaira-Spencer map, and Serre duality in the relativesetting.tefan Kebekus 15
Theorem 3.11 (Kodaira-Spencer map, cf. [Voi07, Sect. 9.1.2] or [Huy05,Sect. 6.2]) . Under the simplifying Assumptions 3.9, since
Var( f ) > , there existsa non-zero sheaf morphism κ : T Y → R f ∗ ( T X/Y ) which measures the variation ofthe isomorphism classes of fibres in moduli. (cid:3) Theorem 3.12 (Serre duality in the relative setting, cf. [Liu02, Sect. 6.4]) . Underthe simplifying Assumptions 3.9, if F is any coherent sheaf on X , then there existsa natural isomorphism f ∗ ( F ∗ ⊗ ω X/Y ) ∼ = (cid:0) R f ∗ ( F ) (cid:1) ∗ . (cid:3) Proof of Theorem 3.1 under the Simplifying Assumptions 3.9.
Consider the dualof the (non-trivial) Kodaira-Spencer map discussed in Theorem 3.11, say κ ∗ : (cid:0) R f ∗ ( T X/Y ) (cid:1) ∗ → ( T Y ) ∗ . Recalling that T ∗ X/Y equals the relative dualizing sheaf ω X/Y , and using the relative Serre Duality Theorem 3.12, the sheaf morphism κ ∗ is naturally identified with a non-zero morphism(3.13) κ ∗ : f ∗ (cid:0) ω ⊗ X/Y (cid:1) → Ω Y . Choosing an ample line bundle
A ∈
Pic( Y ) and sufficiently large and divisiblenumbers N, M (cid:29)
0, Theorem 3.10 yields a sequence of sheaf morphisms A ⊕ M φ gen. surjective (cid:47) (cid:47)
Sym N f ∗ (cid:0) ω ⊗ X/Y (cid:1)
Sym N ( κ ∗ )non-trivial (cid:47) (cid:47) Sym N Ω Y , whose composition A ⊕ M → Sym N Ω Y is clearly not the zero map. Consequently,we obtain a non-trivial map A →
Sym N Ω Y , finishing the proof of Theorem 3.1under the Simplifying Assumptions 3.9. (cid:3) The proof outlined above uses the dual of the Kodaira-Spencer map as avehicle to transport the positivity which exists in f ∗ (cid:0) ω ⊗ X/Y (cid:1) into the sheaf Ω Y ofdifferential forms on Y . If f was a family of surfaces rather than a family of curves,then Serre duality cannot easily be used to identify the dual of R f ∗ ( T X/Y ) with apush-forward sheaf of type f ∗ (cid:0) ω ⊗• X/Y (cid:1) , or any with other sheaf whose positivity iswell-known. To overcome this problem, Viehweg suggested to replace the Kodaira-Spencer map κ by sequences of more complicated sheaf morphisms τ p,q and τ k ,constructed in Sections 3.2.3 and 3.2.4 below. To motivate the slightly involvedconstruction of these maps, we recall without proof a description of the classicalKodaira-Spencer map. Construction . Under the Simplifying Assumptions 3.9, consider the standardsequence of relative differential forms on X ,(3.15) 0 → f ∗ Ω Y → Ω X → Ω X/Y → , and its twist with the invertible sheaf ω ∗ X/Y ,0 → f ∗ Ω Y ⊗ ω ∗ X/Y → Ω X ⊗ ω ∗ X/Y → Ω X/Y ⊗ ω ∗ X/Y (cid:124) (cid:123)(cid:122) (cid:125) ∼ = O X → . f ∗ ( O X ) = O Y , the first connecting morphism associated with thissequence then reads(3.16) O Y → Ω Y ⊗ R f ∗ ( ω ∗ X/Y ) =: F . The sheaf F is naturally isomorphic to the sheaf Hom (cid:0) T Y , R f ∗ ( T X/Y ) (cid:1) . To give amorphism O Y → F is thus the same as to give a map T Y → R f ∗ ( T X/Y ), and themorphism obtained in (3.16) is the same as the Kodaira-Spencer map discussed inTheorem 3.11.Observe also that Serre duality yields a natural identification of F with thesheaf Hom (cid:0) f ∗ ( ω ⊗ X/Y ) , Ω Y (cid:1) . To give a morphism O Y → F it is thus the same asto give a map f ∗ ( ω ⊗ X/Y ) → Ω Y . The morphism obtained in this way from (3.16)is of course the morphism κ ∗ of Equation (3.13). τ p,q In the general set-ting of Theorem 3.1 where the simplifying Assumptions 3.9 do not generally hold,the starting point of the Viehweg-Zuo construction is the standard sequence of rel-ative logarithmic differentials associated to the flat morphism f (cid:48) which generalisesSequence (3.15) from above,(3.17) 0 → ( f (cid:48) ) ∗ Ω Y (cid:48) (log D (cid:48) ) → Ω X (cid:48) (log ∆ (cid:48) ) → Ω X (cid:48) /Y (cid:48) (log ∆ (cid:48) ) → . We refer to [EV90, Sect. 4] for a discussion of Sequence (3.17), and for a proof ofthe fact that the cokernel Ω X (cid:48) /Y (cid:48) (log ∆ (cid:48) ) is locally free. By [Har77, II, Ex. 5.16],Sequence (3.17) defines a filtration of the p th exterior power,Ω pX (cid:48) (log ∆ (cid:48) ) = F ⊇ F ⊇ · · · ⊇ F p ⊇ F p +1 = { } , with F r /F r +1 ∼ = ( f (cid:48) ) ∗ (cid:0) Ω rY (cid:48) (log D (cid:48) ) (cid:1) ⊗ Ω p − rX (cid:48) /Y (cid:48) (log ∆ (cid:48) ). Take the first sequenceinduced by the filtration,0 −→ F −→ F −→ F /F −→ , modulo F , and obtain(3.18) 0 −→ ( f (cid:48) ) ∗ (cid:0) Ω Y (cid:48) (log D (cid:48) ) (cid:1) ⊗ Ω p − X (cid:48) /Y (cid:48) (log ∆ (cid:48) ) −→ F /F −→−→ Ω pX (cid:48) /Y (cid:48) (log ∆ (cid:48) ) −→ . Setting L := Ω nX (cid:48) /Y (cid:48) (log ∆ (cid:48) ), twisting Sequence (3.18) with L − and pushing down,the connecting morphisms of the associated long exact sequence give maps τ p,q : F p,q −→ F p − ,q +1 ⊗ Ω Y (cid:48) (log D (cid:48) ) , where F p,q := R q f (cid:48)∗ (Ω pX (cid:48) /Y (cid:48) (log ∆ (cid:48) ) ⊗ L − ) / torsion. Set N p,q := ker( τ p,q ).tefan Kebekus 17 τ p,q The morphisms τ p,q and τ p − ,q +1 can be com-posed if we tensor the latter with the identity morphism on Ω Y (cid:48) (log D (cid:48) ). Morespecifically, we consider the following morphisms, τ p,q ⊗ Id Ω Y (cid:48) (log D (cid:48) ) ⊗ q (cid:124) (cid:123)(cid:122) (cid:125) =: τ p,q : F p,q ⊗ (cid:0) Ω Y (cid:48) (log D (cid:48) ) (cid:1) ⊗ q → F p − ,q +1 ⊗ (cid:0) Ω Y (cid:48) (log D (cid:48) ) (cid:1) ⊗ q +1 , and their compositions(3.19) τ n − k +1 ,k − ◦ · · · ◦ τ n − , ◦ τ n, (cid:124) (cid:123)(cid:122) (cid:125) =: τ k : F n, → F n − k,k ⊗ (cid:0) Ω Y (cid:48) (log D (cid:48) ) (cid:1) ⊗ k . τ k and N p,q Theorem 3.1 is shown byrelating the morphism τ p,q with the structure morphism of a Higgs-bundle comingfrom the variation of Hodge structures associated with the family f ◦ . Viehweg’spositivity results of push-forward sheaves of relative differentials, as well as Zuo’sresults on the curvature of kernels of generalised Kodaira-Spencer maps are themain input here. Rather than recalling the complicated line of argumentation, wesimply state two central results from the argumentation of [VZ02]. Fact . For any k , the morphism τ k factors via the symmetric differentials Sym k Ω Y (cid:48) (log D (cid:48) ) ⊆ (cid:0) Ω Y (cid:48) (log D (cid:48) ) (cid:1) ⊗ k . More precisely, the morphism τ k takes its image in F n − k,k ⊗ Sym k Ω Y (cid:48) (log D (cid:48) ). (cid:3) Consequence . Using Fact 3.20 and the observation that F n, ∼ = O Y (cid:48) , we cantherefore view τ k as a morphism τ k : O Y (cid:48) −→ F n − k,k ⊗ Sym k Ω Y (cid:48) (log D (cid:48) ) . While the proof of Fact 3.20 is rather elementary, the following deep resultis at the core of Viehweg-Zuo’s argument. Its role in the proof of Theorem 3.1 iscomparable to that of the Positivity Theorem 3.10 discussed in Section 3.2.2.
Fact N p,q , [VZ02, Claim 4.8]) . Given any numbers p and q , there exists a number k and an invertible sheaf A (cid:48) ∈ Pic( Y (cid:48) ) of Kodaira-Iitaka dimension κ ( A (cid:48) ) ≥ Var( f ) such that ( A (cid:48) ) ∗ ⊗ Sym k (cid:0) ( N p,q ) ∗ (cid:1) is genericallygenerated. (cid:3) To end the sketch of proof, we follow [VZ02, p. 315] almostverbatim. By Fact 3.22, the trivial sheaf F n, ∼ = O Y (cid:48) cannot lie in the kernel N n, of τ = τ n, . We can therefore set 1 ≤ m to be the largest number with τ m ( F n, ) (cid:54) = { } . Since m is maximal, τ m +1 = τ n − m,m ◦ τ m ≡ τ m ) ⊆ ker( τ n − m,m ) = N n − m,m ⊗ Sym m Ω Y (cid:48) (log D (cid:48) ) . In other words, τ m gives a non-trivial map τ m : O Y (cid:48) ∼ = F n, −→ N n − m,m ⊗ Sym m Ω Y (cid:48) (log D (cid:48) ) . τ m as a non-trivial map(3.23) τ m : ( N n − m,m ) ∗ −→ Sym m Ω Y (cid:48) (log D (cid:48) ) . By Fact 3.22, there are many morphisms A (cid:48) → Sym k (cid:0) ( N n − m,m ) ∗ (cid:1) , for k large enough. Together with (3.23), this gives a non-zero morphism A (cid:48) → Sym k · m Ω Y (cid:48) (log D (cid:48) ).We have seen in Remark 3.8 that the sheaf A (cid:48) ⊆ Sym k · m Ω Y (cid:48) (log D (cid:48) ) extendsto a sheaf A ⊆
Sym k · m Ω Y (log D ) with κ ( A ) = κ ( A (cid:48) ) ≥ Var( f ◦ ). This ends theproof of Theorem 3.1. (cid:3) In spite of its importance, little is known about further properties that theViehweg-Zuo sheaves A might have. Question . For families of higher-dimensional manifolds, how does theViehweg-Zuo construction behave under base change? Does it satisfy any uni-versal properties at all? If not, is there a “natural” positivity result for basespaces of families that does satisfy good functorial properties?In the setup of Theorem 3.1, if Z ◦ ⊂ Y ◦ is any closed submanifold, then theassociated Viehweg-Zuo sheaves A , constructed for the family f ◦ : X ◦ → Y ◦ , and A Z , constructed for the restricted family f ◦ Z : X ◦ × Y ◦ Z ◦ → Z ◦ , may differ. Inparticular, it is not clear that A Z is the restriction of A , and the sheaves A and A Z may live in different symmetric products of their respective Ω ’s.One likely source of non-compatibility with base change is the choice of thenumber m in Section 3.2.6 (“largest number with τ m ( F n, ) (cid:54) = { } ”). It seemsunlikely that this definition behaves well under base change. Question . For families of higher-dimensional manifolds, are there distin-guished subvarieties in moduli space that have special Viehweg-Zuo sheaves, per-haps contained in particularly high/low symmetric powers of Ω ? Does the lackof a restriction morphism induce a geometric structure on the moduli space?The refinement of the Viehweg-Zuo Theorem, presented in Theorem 3.6above, turns out to be important for the applications that we have in mind. Itis, however, not clear to us if the sheaf B which appears in Theorem 3.6 is reallyoptimal. Question . Prove that the sheaf
B ⊆ Ω Y (log D ) is the smallest sheaf possiblefor which Theorem 3.6 holds, or else find the smallest possible sheaf. For instance,does Theorem 3.6 admit a natural improvement if we replace Ω Y (log D ) by asuitable sheaf of orbifold differentials, using Campana’s language of geometricorbifolds?tefan Kebekus 19
4. Techniques II: Reflexive differentials on singular spaces
To motivate the resultspresented in this section, let f ◦ : X ◦ → Y ◦ be a smooth, projective family ofcanonically polarised varieties over a smooth, quasi-projective base manifold, andassume that the family f ◦ is of maximal variation, i.e., that Var( f ◦ ) = dim Y ◦ .As before, choose a smooth compactification Y ⊇ Y ◦ such that D := Y \ Y ◦ is adivisor with only simple normal crossings.To prove Viehweg’s conjecture, we need to show that the logarithmic Kodairadimension of Y ◦ is maximal, i.e., that κ ( Y ◦ ) = dim Y ◦ . In particular, we need torule out the possibility that κ ( Y ◦ ) = 0. As we will see in the proof of Proposi-tion 4.1 below, a relatively elementary argument exists in cases where the Picardnumber of Y is one, ρ ( Y ) = 1. We refer the reader to [HL97, Sect. I.1] for thenotion of semistability and for a discussion of the Harder-Narasimhan filtrationused in the proof. Proposition 4.1 (Partial answer to Viehweg’s conjecture in case ρ ( Y ) = 1) . Inthe setup described above, if we additionally assume that ρ ( Y ) = 1 , then κ ( Y ◦ ) (cid:54) = 0 .Proof. We argue by contradiction and assume that both κ ( Y ◦ ) = 0 and that ρ ( Y ) = 1. Let A ⊆
Sym m Ω Y (log D ) be the big invertible sheaf whose existence isguaranteed by the Viehweg-Zuo construction, Theorem 3.1. Since ρ ( Y ) = 1, thesheaf A is actually ample.As a first step, observe that the log canonical bundle K Y + D must be torsion,i.e., that there exists a number m (cid:48) ∈ N + such that O Y (cid:0) m (cid:48) · ( K Y + D ) (cid:1) ∼ = O Y . Thisfollows from the assumption that κ ( K Y + D ) = 0 and from the observation thaton a projective manifold with ρ = 1, any invertible sheaf which admits a non-zerosection is either trivial or ample. In particular, we obtain that the divisor K Y + D is numerically trivial.Next, let C ⊆ Y be a general complete intersection curve in the sense ofMehta-Ramanathan, cf. [HL97, Sect. II.7]. The numerical triviality of K Y + D will then imply that( K Y + D ) .C = c (cid:0) Ω Y (log D ) (cid:1) .C = c (cid:0) Sym m Ω Y (log D ) (cid:1) .C = 0 . On the other hand, since A is ample, we have that c ( A ) .C >
0. In summary,we obtain that the symmetric product sheaf Sym m Ω Y (log D ) is not semistable.Since we are working in characteristic zero, this implies that the sheaf of K¨ahlerdifferentials Ω Y (log D ) will likewise not be semistable, and contains a destabilisingsubsheaf B ⊆ Ω Y (log D ) with c ( B ) .C >
0, cf. [HL97, Cor. 3.2.10]. Since theintersection number c ( B ) .C is positive, the rank r of the sheaf B must be strictlyless than dim Y , and its determinant is an ample invertible subsheaf of the sheaf0 Differential forms, MMP, and Hyperbolicityof logarithmic r -forms, det B ⊆ Ω rY (log D ) . This, however, contradicts the Bogomolov-Sommese Vanishing Theorem 2.17 andtherefore ends the proof. (cid:3)
The assumption that ρ ( Y ) =1 is not realistic. In the general situation, where ρ ( Y ) can be arbitrarily large, wewill apply the minimal model program to the pair ( Y, D ). As we will sketch inSection 5, assuming that the standard conjectures of minimal model theory holdtrue, a run of the minimal model program for a suitable choice of a boundarydivisor will yield a diagram, Y λ minimal model program (cid:47) (cid:47) Y λ fibre space π (cid:15) (cid:15) Z λ , where λ : Y (cid:57)(cid:57)(cid:75) Y λ is a birational map whose inverse does not contract anydivisors, and where either ρ ( Y λ ) = 1 and Z λ is a point, or where Y λ has thestructure of a proper Mori– or Kodaira fibre space. In the first case, we can try tocopy the proof of Proposition 4.1 above. In the second case, we can use the fibrestructure and try to argue inductively.The main problem that arises when adopting the proof of Proposition 4.1is the presence of singularities. Both the space Y λ and the cycle-theoretic imagedivisor D λ ⊂ Y λ will generally be singular, and the pair ( Y λ , D λ ) will generally bedlt. This leads to two difficulties.(1) The sheaf Ω Y λ (log D λ ) of logarithmic K¨ahler differentials is generally notpure in the sense of [HL97, Sect. 1.1]. Accordingly, there is no good no-tion of stability that would be suitable to construct a Harder-Narasimhanfiltration.(2) The Viehweg-Zuo construction does not work for singular varieties. Theauthor is not aware of any method suitable to prove positivity results forK¨ahler differentials, or prove the existence of sections in any symmetricproduct of Ω Y λ (log D λ ).The aim of the present Section 4 is to show that that both problems canbe overcome if we replace the sheaf Ω Y λ (log D λ ) of K¨ahler differentials by itsdouble dual Ω [1] Y λ (log D λ ) := (cid:0) Ω Y λ (log D λ ) (cid:1) ∗∗ . We refer to [Rei87, Sect. 1.6] for adiscussion of the double dual in this context, and to [OSS80, II. Sect. 1.1] for athorough discussion of reflexive sheaves. The following notation will be useful inthe discussion. Notation 4.2 (Reflexive tensor operations) . Let X be a normal variety and A acoherent sheaf of O X -modules. Given any number n ∈ N , set A [ n ] := ( A ⊗ n ) ∗∗ , tefan Kebekus 21Sym [ n ] A := (Sym n A ) ∗∗ . If π : X (cid:48) → X is a morphism of normal varieties, set π [ ∗ ] ( A ) := (cid:0) π ∗ A (cid:1) ∗∗ . In a similar vein, set Ω [ p ] X := (cid:0) Ω pX (cid:1) ∗∗ and Ω [ p ] X (log D ) := (cid:0) Ω pX (log D ) (cid:1) ∗∗ . Notation 4.3 (Reflexive differential forms) . A section in Ω [ p ] X or Ω [ p ] X (log D ) willbe called a reflexive form or a reflexive logarithmic form , respectively.Fact . Reflexive sheavesare torsion free and therefore pure. In particular, a Harder-Narasimhan filtrationexists for Ω [ p ] X (log D ) and for the symmetric products Sym [ n ] Ω X (log D ). Fact . If X is a normal space, if A is any reflexivesheaf on X and if Z ⊂ X any set of codim X Z ≥
2, then the restriction map H (cid:0) X, A (cid:1) → H (cid:0) X \ Z, A (cid:1) is in fact isomorphic. We say that “sections in A extend over the small set Z ”.If U := X \ Z is the complement of Z , with inclusion map ι : U → X , itfollows immediately that A = ι ∗ ( A| U ). In a similar vein, if B U is any locally freesheaf on U , its push-forward sheaf ι ∗ ( B U ) will always be reflexive. It follows almost by definition that sheaves ofreflexive differentials have very good push-forward properties. In Section 4.2 wewill use these properties to overcome one of the problems mentioned above and toproduce Viehweg-Zuo sheaves of reflexive differentials on singular spaces. Perhapsmore importantly, we will in Section 4.3 recall extension results for log canonicalvarieties. These results show that reflexive differentials often admit a pull-backmap, similar to the standard pull-back of K¨ahler differentials. A generalisation ofthe Bogomolov-Sommese vanishing theorem to log canonical varieties follows as acorollary.In Section 4.4, we recall that some of the most important constructions knownfor logarithmic differentials on snc pairs also work for reflexive differentials ondlt pairs. This includes the existence of a residue sequence. For our purposes,this makes reflexive differentials almost as useful as regular differentials in thetheory of smooth spaces. As we will roughly sketch in Section 5, these results willallow to adapt the proof of Proposition 4.1 to the singular setup, and will givea proof of Viehweg’s Conjecture 2.8, at least for families over base manifolds ofdimension ≤
3. Section 4.5 gives a brief sketch of the proof of the pull-back resultof Section 4.3. We end by mentioning a few open problems and conjectures.Some of the material presented in the current Section 4, including Sec-tion 4.4 and all the illustrations, is taken without much modification from thepaper [GKKP11]. Section 4.2 follows the paper [KK08c].2 Differential forms, MMP, and Hyperbolicity
Fact 4.5 implies that any Viehweg-Zuo sheaf which exists on a pair ( Z, ∆)of a smooth variety and a reduced divisor with simple normal crossing supportimmediately implies the existence of a Viehweg-Zuo sheaf of reflexive differentialson any minimal model of ( Z, ∆), and that the Kodaira-Iitaka dimension onlyincreases in the process. To formulate the result precisely, we briefly recall thedefinition of the Kodaira-Iitaka dimension for reflexive sheaves. Definition 4.6 (Kodaira-Iitaka dimension of a sheaf, [KK08c, Not. 2.2]) . Let Z be a normal projective variety and A a reflexive sheaf of rank one on Z . If h (cid:0) Z, A [ n ] (cid:1) = 0 for all n ∈ N , then we say that A has Kodaira-Iitaka dimension κ ( A ) := −∞ . Otherwise, set M := (cid:8) n ∈ N | h (cid:0) Z, A [ n ] (cid:1) > (cid:9) , recall that the restriction of A to the smooth locus of Z is locally free and considerthe natural rational mapping φ n : Z (cid:57)(cid:57)(cid:75) P (cid:0) H (cid:0) Z, A [ n ] (cid:1) ∗ (cid:1) for each n ∈ M. The Kodaira-Iitaka dimension of A is then defined as κ ( A ) := max n ∈ M (cid:0) dim φ n ( Z ) (cid:1) . With this notation, the main result concerning the push-forward is then for-mulated as follows.
Proposition 4.7 (Push forward of Viehweg-Zuo sheaves, [KK08c, Lem. 5.2]) . Let ( Z, ∆) be a pair of a smooth variety and a reduced divisor with simple normalcrossing support. Assume that there exists a reflexive sheaf A ⊆
Sym [ n ] Ω Z (log ∆) of rank one. If λ : Z (cid:57)(cid:57)(cid:75) Z (cid:48) is a birational map whose inverse does not contractany divisor, if Z (cid:48) is normal and ∆ (cid:48) is the (necessarily reduced) cycle-theoreticimage of ∆ , then there exists a reflexive sheaf A (cid:48) ⊆ Sym [ n ] Ω Z (cid:48) (log ∆ (cid:48) ) of rankone, and of Kodaira-Iitaka dimension κ ( A (cid:48) ) ≥ κ ( A ) .Proof. The assumption that λ − does not contract any divisors and the normalityof Z (cid:48) guarantee that λ − : Z (cid:48) (cid:57)(cid:57)(cid:75) Z is a well-defined embedding over an open sub-set U ⊆ Z (cid:48) whose complement has codimension codim Z (cid:48) ( Z (cid:48) \ U ) ≥
2, cf. Zariski’smain theorem [Har77, V 5.2]. In particular, ∆ (cid:48) | U = (cid:0) λ − | U (cid:1) − (∆). Let ι : U → Z (cid:48) denote the inclusion and set A (cid:48) := ι ∗ (cid:0) ( λ − | U ) ∗ A (cid:1) —this sheaf is reflexive byFact 4.5. We obtain an inclusion of reflexive sheaves, A (cid:48) ⊆ Sym [ n ] Ω Z (cid:48) (log ∆ (cid:48) ).By construction, we have that h (cid:0) Z (cid:48) , A (cid:48) [ m ] (cid:1) ≥ h ( Z, A [ m ] ) for all m >
0, hence κ ( A (cid:48) ) ≥ κ ( A ). (cid:3) Given the importance of the Viehweg-Zuo construction, Theorem 3.1, we willcall the sheaves A which appear in Proposition 4.7 “Viehweg-Zuo sheaves”.tefan Kebekus 23 Notation 4.8 (Viehweg-Zuo sheaves) . Let ( Z, ∆) be a pair of a smooth varietyand a reduced divisor with simple normal crossing support, and let n ∈ N be anynumber. A reflexive sheaf A ⊆
Sym [ n ] Ω Z (log ∆) of rank one will be called a“Viehweg-Zuo sheaf ”. K¨ahler differentials are characterised by a number of universal properties,one of the most important being the existence of a pull-back map: if γ : Z → X isany morphism of algebraic varieties and if p ∈ N , then there exists a canonicallydefined sheaf morphism(4.9) dγ : γ ∗ Ω pX → Ω pZ . The following example illustrates that for sheaves of reflexive differentials onnormal spaces, a pull-back map does not exist in general.
Example . Let X be a normal Gorenstein variety of dimension n , and let γ : Z → X beany resolution of singularities. Observing that the sheaf of reflexive n -forms isprecisely the dualizing sheaf, Ω [ n ] X (cid:39) ω X , it follows directly from the definition ofcanonical singularities that X has canonical singularities if and only if a pull-backmorphism dγ : γ ∗ Ω [ n ] X → Ω nZ exists.Together with Daniel Greb, S´andor Kov´acs and Thomas Peternell, the authorhas shown that a pull-back map for reflexive differentials always exists if the targetis log canonical. Theorem 4.11 (Pull-back map for differentials on lc pairs, [GKKP11, Thm. 4.3]) . Let ( X, D ) be an log canonical pair, and let γ : Z → X be a morphism from anormal variety Z such that the image of Z is not contained in the reduced boundaryor in the singular locus, i.e., γ ( Z ) (cid:54)⊆ ( X, D ) sing ∪ supp (cid:98) D (cid:99) . If ≤ p ≤ dim X is any index and ∆ := largest reduced Weil divisor contained in γ − (cid:0) non-klt locus (cid:1) , then there exists a sheaf morphism, dγ : γ ∗ Ω [ p ] X (log (cid:98) D (cid:99) ) → Ω [ p ] Z (log ∆) , that agrees with the usual pull-back morphism (4.9) of K¨ahler differentials at allpoints p ∈ Z where γ ( p ) (cid:54)∈ ( X, D ) sing ∪ supp (cid:98) D (cid:99) .Remark . If follows from the definition of klt, [KM98, Def. 2.34], that thecomponents of D which appear with coefficient one are always contained in thenon-klt locus of ( X, D ). In particular, the divisor ∆ defined in Theorem 4.11always contains the codimension-one part of γ − (cid:0) supp (cid:98) D (cid:99) (cid:1) .4 Differential forms, MMP, and HyperbolicityThe assertion of Theorem 4.11 is rather general and perhaps a bit involved.For klt spaces, the statement reduces to the following simpler result. Theorem 4.13 (Pull-back map for differentials on klt spaces) . Let X be a normalklt variety , and let γ : Z → X be a morphism from a normal variety Z such thatthe image γ ( Z ) is not entirely contained in the singular locus of X . If ≤ p ≤ dim X is any index then there exists a sheaf morphism, dγ : γ ∗ Ω [ p ] X → Ω [ p ] Z , that agrees on an open set with the usual pull-back morphism of K¨ahler differen-tials. (cid:3) Extension properties of differential forms that are closely related to the ex-istence of pull-back maps have been studied in the literature, mostly consider-ing only special values of p . Using Steenbrink’s generalization of the Grauert-Riemenschneider vanishing theorem as their main input, similar results were shownby Steenbrink and van Straten for varieties X with only isolated singularities andfor p ≤ dim X −
2, without any further assumption on the nature of the singu-larities, [SvS85, Thm. 1.3]. Flenner extended these results to normal varieties,subject to the condition that p ≤ codim X sing −
2, [Fle88]. Namikawa proved theextension properties for p ∈ { , } , in case X has canonical Gorenstein singular-ities, [Nam01, Thm. 4]. In the case of finite quotient singularities similar resultswere obtained in [dJS04]. For a log canonical pair with reduced boundary divisor,the cases p ∈ { , dim X − , dim X } were settled in [GKK10, Thm. 1.1].A related setup where the pair ( X, D ) is snc, and where π : (cid:101) X → X is thecomposition of a finite Galois covering and a subsequent resolution of singularitieshas been studied by Esnault and Viehweg. In [EV82] they obtain in their specialsetting similar results and additionally prove vanishing of higher direct imagesheaves.A brief sketch of the proof of Theorem 4.11 is given in Section 4.5 below. Theproof uses a strengthening of the Steenbrink vanishing theorem, which follows fromlocal Hodge-theoretic properties of log canonical singularities, in particular fromthe fact that log canonical spaces are Du Bois. These methods are combined withresults available only for special classes of singularities, such as the recent progressin minimal model theory and partial classification of singularities that appear inminimal models. Theorem 4.11 has many applications useful for modulitheory. We mention two applications which will be important in our context.The first application generalises the Bogomolov-Sommese vanishing theorem tosingular spaces. More precisely, we should say “Let X be a normal variety such that the pair ( X, ∅ ) is klt. . . ” tefan Kebekus 25 Corollary 4.14 (Bogomolov-Sommese vanishing for lc pairs, [GKKP11,Thm. 7.2]) . Let ( X, D ) be a log canonical pair, where X is projective. If A ⊆ Ω [ p ] X (log (cid:98) D (cid:99) ) is a Q -Cartier reflexive subsheaf of rank one, then κ ( A ) ≤ p .Remark . The number κ ( A ) appearing in thestatement of Corollary 4.14 is the generalised Kodaira-Iitaka dimension introducedin Definition 4.6. A reflexive sheaf A is rank one is called Q -Cartier if there existsa number n ∈ N + such that the n th reflexive tensor product A [ n ] is invertible. Proof of Corollary 4.14 in a special case.
We prove Corollary 4.14 only in the spe-cial case where the sheaf
A ⊆ Ω [ p ] X (log (cid:98) D (cid:99) ) is invertible. The reader interested ina full proof is referred to the original reference [GKKP11].Let γ : Z → X be any resolution of singularities, and let ∆ ⊂ Z be thereduced divisor defined in Theorem 4.11 above. Theorem 4.11 will then assert theexistence of an inclusion γ ∗ ( A ) → Ω pZ (log ∆) , and the standard Bogomolov-Sommese vanishing result, Theorem 2.17, applies togive that κ (cid:0) γ ∗ ( A ) (cid:1) ≤ p . Since A is invertible, and since γ is birational, it is clearthat κ (cid:0) γ ∗ ( A ) (cid:1) = κ ( A ), finishing the proof. (cid:3) Warning . Taking the double dual of a sheaf does generally not commute withpulling back. Since reflexive tensor products were used in Definition 4.6 to definethe Kodaira-Iitaka dimension of a sheaf, it is generally false that the Kodaira-Iitaka dimension stays invariant when pulling a sheaf A back to a resolution ofsingularities. The proof of Corollary 4.14 which is given in the simple case where A is invertible does therefore not work without substantial modification in thegeneral setup where A is only Q -Cartier.The second application of Theorem 4.11 concerns rationally chain connectedsingular spaces. Rationally chain connected manifolds are rationally connected,and do not carry differential forms. Building on work of Hacon and McKernan,[HM07], we show that the same holds for reflexive forms on klt pairs. Corollary 4.17 (Reflexive differentials on rationally chain connected spaces,[GKKP11, Thm. 5.1]) . Let ( X, D ) be a klt pair. If X is rationally chain connected,then X is rationally connected, and H (cid:0) X, Ω [ p ] X (cid:1) = 0 for all p ∈ N , ≤ p ≤ dim X .Proof. Choose a log resolution of singularities, π : (cid:101) X → X of the pair ( X, D ).Since klt pairs are also dlt, a theorem of Hacon-McKernan, [HM07, Cor. 1.5(2)],applies to show that X and (cid:101) X are both rationally connected. In particular, itfollows that H (cid:0) (cid:101) X, Ω p (cid:101) X ) = 0 for all p > X, D ) is klt, Theorem 4.11 asserts that there exists a pull-back mor-phism dπ : π ∗ Ω [ p ] X → Ω p (cid:101) X . As π is birational, dπ is generically injective and since6 Differential forms, MMP, and HyperbolicityΩ [ p ] X is torsion-free, this means that the induced morphism on the level of sectionsis injective: π ∗ : H (cid:0) X, Ω [ p ] X (cid:1) → H (cid:0) (cid:101) X, Ω p (cid:101) X (cid:1) = 0 . The claim then follows. (cid:3)
Logarithmic K¨ahler differentials on snc pairs are canonically defined. Theyare characterised by strong universal properties and appear accordingly in a num-ber of important sequences, filtered complexes and other constructions. Firstexamples include the following:(1) the pull-back property of differentials under arbitrary morphisms,(2) relative differential sequences for smooth morphisms,(3) residue sequences associated with snc pairs, and(4) the description of Chern classes as the extension classes of the first residuesequence.Reflexive differentials do in general not enjoy the same universal properties asK¨ahler differentials. However, we have seen in Theorem 4.11 that reflexive differen-tials do have good pull-back properties if we are working with log canonical pairs.In the present Section 4.4, we would like to make the point that each of the otherproperties listed above also has a very good analogue for reflexive differentials, aslong as we are working with dlt pairs. This makes reflexive differential extremelyuseful in practise. In a sense, it seems fair to say that “reflexive differentials anddlt pairs are made for one another”.
Here we recallthe generalisation of the standard sequence for relative differentials, [Har77,Prop. II.8.11], to the logarithmic setup. For this, we introduce the notion ofan snc morphism as a logarithmic analogue of a smooth morphism. Although relatively snc divisors have long been used in the literature, cf. [Del70, Sect. 3],we are not aware of a good reference that discusses them in detail. Recall that apair (
X, D ) is called an “snc pair” if X is smooth, and if the divisor D is reducedand has only simple normal crossing support. Notation 4.18 (Intersection of boundary components) . Let ( X, D ) be a pair ofa normal space X and a divisor D , where D is written as a sum of its irreduciblecomponents D = α D + . . . + α n D n . If I ⊆ { , . . . , n } is any non-empty subset,we consider the scheme-theoretic intersection D I := ∩ i ∈ I D i . If I is empty, set D I := X .Remark . In the setup of Notation 4.18, it is clearthat the pair (
X, D ) is snc if and only if all D I are smooth and of codimensionequal to the number of defining equations: codim X D I = | I | for all I where D I (cid:54) = ∅ .tefan Kebekus 27 Definition 4.20 (Snc morphism, relatively snc divisor, [VZ02, Def. 2.1]) . If ( X, D ) is an snc pair and φ : X → T a surjective morphism to a smooth va-riety, we say that D is relatively snc , or that φ is an snc morphism of the pair( X, D ) if for any set I with D I (cid:54) = ∅ all restricted morphisms φ | D I : D I → T aresmooth of relative dimension dim X − dim T − | I | .Remark . If (
X, D ) is an snc pair and φ : X → T is any surjective snc morphism of ( X, D ), it is clear from Remark 4.19 that if t ∈ T is any point, with preimages X t := φ − ( t ) and D t := D ∩ X t then the pair ( X t , D t )is again snc. Remark . If (
X, D ) is an snc pair and φ : X → T is any surjective morphism, it is clear from generic smoothness thatthere exists a dense open set T ◦ ⊆ T , such that D ∩ φ − ( T ◦ ) is relatively snc over T ◦ . Let ( X, D ) be a reduced snc pair, and φ : X → T an snc morphism of ( X, D ),as introduced in Definition 4.20. In this setting, the standard pull-back morphismof 1-forms extends to yield the following exact sequence of locally free sheaves on X ,(4.23) 0 → φ ∗ Ω T → Ω X (log D ) → Ω X/T (log D ) → , called the “relative differential sequence for logarithmic differentials”. We referto [EV90, Sect. 4.1] [Del70, Sect. 3.3] or [BDIP02, p. 137ff] for a more detailedexplanation. For forms of higher degrees, the sequence (4.23) induces filtration(4.24) Ω pX (log D ) = F (log) ⊇ F (log) ⊇ · · · ⊇ F p (log) ⊇ { } with quotients(4.25) 0 → F r +1 (log) → F r (log) → φ ∗ Ω rT ⊗ Ω p − rX/T (log D ) → r . We refer to [Har77, Ex. II.5.16] for the construction of (4.24).The main result of this section, Theorem 4.26, gives analogues of (4.23)–(4.25) in case where ( X, D ) is dlt. In essence, Theorem 4.26 says that all propertiesof the relative differential sequence still hold on dlt pairs if one removes from X aset Z of codimension codim X Z ≥ Theorem 4.26 (Relative differential sequence on dlt pairs, [GKKP11,Thm. 10.6]) . Let ( X, D ) be a dlt pair with X connected. Let φ : X → T bea surjective morphism to a normal variety T . Then, there exists a non-emptysmooth open subset T ◦ ⊆ T with preimages X ◦ = φ − ( T ◦ ) , D ◦ = D ∩ X ◦ , and afiltration (4.27) Ω [ p ] X ◦ (log (cid:98) D ◦ (cid:99) ) = F [0] (log) ⊇ · · · ⊇ F [ p ] (log) ⊇ { } on X ◦ with the following properties. The filtrations (4.24) and (4.27) agree wherever the pair ( X ◦ , (cid:98) D ◦ (cid:99) ) is snc,and φ is an snc morphism of ( X ◦ , (cid:98) D ◦ (cid:99) ) . (2) For any r , the sheaf F [ r ] (log) is reflexive, and F [ r +1] (log) is a saturatedsubsheaf of F [ r ] (log) . (3) For any r , there exists a sequence of sheaves of O X ◦ -modules, → F [ r +1] (log) → F [ r ] (log) → φ ∗ Ω rT ◦ ⊗ Ω [ p − r ] X ◦ /T ◦ (log (cid:98) D ◦ (cid:99) ) → , which is exact and analytically locally split in codimension . (4) There exists an isomorphism F [ p ] (log) (cid:39) φ ∗ Ω pT ◦ .Remark . If S is any complex variety, wecall a sequence of sheaf morphisms,(4.29) 0 → A → B → C → , “exact and analytically locally split in codimension 2” if there exists a closedsubvariety C ⊂ S of codimension codim S C ≥ S \ C by subsets( U i ) i ∈ I which are open in the analytic topology, such that the restriction of (4.29)to S \ C is exact, and such that the restriction of (4.29) to any of the open sets U i splits. We refer to Footnote 2 on Page 9 for references concerning the notionof a “dlt pair”. Idea of proof of Theorem 4.26.
We give only a very rough and incomplete idea ofthe proof of Theorem 4.26. To construct the filtration in (4.27), one takes thefiltration (4.24) which exists on the open set X \ X sing wherever the morphism φ is snc, and extends the sheaves to reflexive sheaves that are defined on all of X . Itis then not very difficult to show that the sequences (4.26.3) are exact and locallysplit away from a subset Z ⊂ X of codimension codim X Z ≥
2. The main point ofTheorem 4.26 is, however, that it suffices to remove from X a set of codimensioncodim X Z ≥
3. For this, a careful analysis of the codimension-two structure of dltpairs, cf. [GKKP11, Sect. 9], proves to be key. (cid:3)
A very impor-tant feature of logarithmic differentials is the existence of a residue map. In itssimplest form consider a smooth hypersurface D ⊂ X in a manifold X . Theresidue map is then the cokernel map in the exact sequence0 → Ω X → Ω X (log D ) → O D → . More generally, consider a reduced snc pair (
X, D ). Let D ⊆ D be any irreduciblecomponent and recall from [EV92, 2.3(b)] that there exists a residue sequence,0 → Ω pX (log( D − D )) (cid:47) (cid:47) Ω pX (log D ) ρ p (cid:47) (cid:47) Ω p − D (log D c ) → , where D c := ( D − D ) | D denotes the “restricted complement” of D . Moregenerally, if φ : X → T is an snc morphism of ( X, D ) we have a relative residuetefan Kebekus 29smooth curve D (cid:72)(cid:72)(cid:72)(cid:72)(cid:106) X (singular surface) Figure 1.
A setup for the residue map on singular spaces.sequence(4.30) 0 → Ω pX/T (log( D − D )) (cid:47) (cid:47) Ω pX/T (log D ) ρ p (cid:47) (cid:47) Ω p − D /T (log D c ) → . The sequence (4.30) is not a sequence of locally free sheaves on X , and its re-striction to D will never be exact on the left. However, an elementary argument,cf. [KK08a, Lem. 2.13.2], shows that restriction of (4.30) to D induces the fol-lowing exact sequence(4.31) 0 → Ω pD /T (log D c ) i p −→ Ω pX/T (log D ) | D ρ pD −−→ Ω p − D /T (log D c ) → , which is very useful for inductive purposes. We recall without proof the followingelementary fact about the residue sequence. Fact . If σ ∈ H (cid:0) X, Ω pX/T (log D ) (cid:1) is any reflexive form, then σ ∈ H (cid:0) X, Ω pX/T (log( D − D )) (cid:1) if and only if ρ p ( σ ) = 0.If the pair ( X, D ) is not snc, no residue map exists in general. However,if (
X, D ) is dlt, then [KM98, Cor. 5.52] applies to show that D is normal, andan analogue of the residue map ρ p exists for sheaves of reflexive differentials. Toillustrate the problem we are dealing with, consider a normal space X that containsa smooth Weil divisor D = D , similar to the one sketched in Figure 1. One caneasily construct examples where the singular set Z := X sing is contained in D andhas codimension 2 in X , but codimension one in D . In this setting, a reflexiveform σ ∈ H (cid:0) D , Ω [ p ] X (log D ) | D (cid:1) is simply the restriction of a logarithmic formdefined outside of Z , and the form ρ [ p ] ( σ ) is the extension of the well-definedform ρ p ( σ | D \ Z ) over Z , as a rational form with poles along Z ⊂ D . If the0 Differential forms, MMP, and Hyperbolicitysingularities of X are bad, it will generally happen that the extension ρ [ p ] ( σ ) haspoles of arbitrarily high order. Theorem 4.33 asserts that this does not happenwhen ( X, D ) is dlt.
Theorem 4.33 (Residue sequences for dlt pairs, [GKKP11, Thm. 11.7]) . Let ( X, D ) be a dlt pair with (cid:98) D (cid:99) (cid:54) = ∅ and let D ⊆ (cid:98) D (cid:99) be an irreducible component.Let φ : X → T be a surjective morphism to a normal variety T such that therestricted map φ | D : D → T is still surjective. Then, there exists a non-emptyopen subset T ◦ ⊆ T , such that the following holds if we denote the preimages as X ◦ = φ − ( T ◦ ) , D ◦ = D ∩ X ◦ , and the “complement” of D ◦ as D ◦ ,c := (cid:0) (cid:98) D ◦ (cid:99) − D ◦ (cid:1) | D ◦ . (1) There exists a sequence → Ω [ r ] X ◦ /T ◦ (log( (cid:98) D ◦ (cid:99) − D ◦ )) → Ω [ r ] X ◦ /T ◦ (log (cid:98) D ◦ (cid:99) ) ρ [ r ] −−→ Ω [ r − D ◦ /T ◦ (log D ◦ ,c ) → which is exact in X ◦ outside a set of codimension at least . This sequencecoincides with the usual residue sequence (4.30) wherever the pair ( X ◦ , D ◦ ) is snc and the map φ ◦ : X ◦ → T ◦ is an snc morphism of ( X ◦ , D ◦ ) . (2) The restriction of Sequence (4.33.1) to D induces a sequence → Ω [ r ] D ◦ /T ◦ (log D ◦ ,c ) → Ω [ r ] X ◦ /T ◦ (log (cid:98) D ◦ (cid:99) ) | ∗∗ D ◦ ρ [ r ] D ◦ −−→ Ω [ r − D ◦ /T ◦ (log D ◦ ,c ) → which is exact on D ◦ outside a set of codimension at least and coincideswith the usual restricted residue sequence (4.31) wherever the pair ( X ◦ , D ◦ ) is snc and the map φ ◦ : X ◦ → T ◦ is an snc morphism of ( X ◦ , D ◦ ) . (cid:3) As before, the proof of Theorem 4.33 relies on our knowledge of thecodimension-two structure of dlt pairs. Fact 4.32 and Theorem 4.33 togetherimmediately imply that the residue map for reflexive differentials can be used tocheck if a reflexive form has logarithmic poles along a given boundary divisor.
Remark . In the setting ofTheorem 4.33, if σ ∈ H (cid:0) X, Ω [ p ] X (log (cid:98) D (cid:99) ) (cid:1) is any reflexive form, then σ ∈ H (cid:0) X, Ω [ p ] X (log (cid:98) D (cid:99) − D ) (cid:1) if and only if ρ [ p ] ( σ ) = 0. -forms Let X be a smooth variety and D ⊂ X a smooth, irreducible divisor. The first residue sequence (4.30) of the pair ( X, D )then reads 0 → Ω D → Ω X (log D ) | D ρ −→ O D → , and we obtain a connecting morphism of the long exact cohomology sequence, δ : H (cid:0) D, O D (cid:1) → H (cid:0) D, Ω D (cid:1) . tefan Kebekus 31In this setting, the standard description of the first Chern class in terms of theconnecting morphism, [Har77, III. Ex. 7.4], asserts that(4.35) c (cid:0) O X ( D ) | D (cid:1) = δ ( D ) ∈ H (cid:0) D, Ω D (cid:1) , where D is the constant function on D with value one. Theorem 4.36 generalisesIdentity (4.35) to the case where ( X, D ) is a reduced dlt pair with irreducibleboundary divisor.
Theorem 4.36 (Description of Chern classes, [GKKP11, Thm. 12.2]) . Let ( X, D ) be a dlt pair, D = (cid:98) D (cid:99) irreducible. Then, there exists a closed subset Z ⊂ X with codim X Z ≥ and a number m ∈ N such that mD is Cartier on X ◦ := X \ Z ,such that D ◦ := D ∩ X ◦ is smooth, and such that the restricted residue sequence (4.37) 0 → Ω D → Ω [1] X (log D ) | ∗∗ D ρ D −→ O D → defined in Theorem 4.33 is exact on D ◦ . Moreover, for the connecting homomor-phism δ in the associated long exact cohomology sequence δ : H (cid:0) D ◦ , O D ◦ (cid:1) → H (cid:0) D ◦ , Ω D ◦ (cid:1) we have (4.38) δ ( m · D ◦ ) = c ( O X ◦ ( mD ◦ ) | D ◦ ) . The proof of Theorem 4.11 is rather involved. To illustrate the idea of theproof, we concentrate on a very special case, and give only indications what needsto be done to handle the general setup.
The following sim-plifying assumptions will be maintained throughout the present Section 4.5.
Simplifying Assumptions . The space X has dimension n := dim X ≥
3. Itis klt, has only one single isolated singularity x ∈ X , and the divisor D is empty.The morphism γ : Z → X is a resolution of singularities, whose exceptional set E ⊂ Z is a divisor with simple normal crossing support.To prove Theorem 4.11, we need to show in essence that reflexive differentialforms σ ∈ H (cid:0) X, Ω [ p ] X (cid:1) pull back to give differential forms (cid:101) σ ∈ H (cid:0) Z, Ω pZ (cid:1) . Thefollowing observation, an immediate consequence of Fact 4.5, turns out to be key. Observation . To give a reflexive differential σ ∈ H (cid:0) X, Ω [ p ] X (cid:1) , it is equivalentto give a differential form σ ◦ ∈ H (cid:0) X \ X sing , Ω pX (cid:1) , defined on the smooth locus of X . Since the resolution map identifies the open subvarieties Z \ E and X \ X sing ,we see that to give a reflexive differential σ ∈ H (cid:0) X, Ω [ p ] X (cid:1) , it is in fact equivalentto give a differential form (cid:101) σ ◦ ∈ H (cid:0) Z \ E, Ω pZ (cid:1) .2 Differential forms, MMP, and HyperbolicityIn essence, Observation 4.40 says that to show Theorem 4.11, we need toprove that the natural restriction map(4.41) H (cid:0) Z, Ω pZ (cid:1) → H (cid:0) Z \ E, Ω pZ (cid:1) is in fact surjective. In other words, we need to show that any differential formon Z , which is defined outside of the γ -exceptional set E , automatically extendsacross E , to give a differential form defined on all of Z . This is done in two steps.We first show that the restriction map(4.42) H (cid:0) Z, Ω pZ (log E ) (cid:1) → H (cid:0) Z \ E, Ω pZ (log E ) (cid:1) = H (cid:0) Z \ E, Ω pZ (cid:1) is surjective. In other words, we show that any differential form on Z , definedoutside of E , extends as a form with logarithmic poles along E . Secondly, weshow that the natural inclusion map(4.43) H (cid:0) Z, Ω pZ (cid:1) → H (cid:0) Z, Ω pZ (log E ) (cid:1) is likewise surjective. In other words, we show that globally defined differentialsforms on Z , which are allowed to have logarithmic poles along E , really do nothave any poles. Surjectivity of the morphisms (4.42) and (4.43) together will thenimply surjectivity of (4.41), finishing the proof of Theorem 4.11.The arguments used to prove surjectivity of (4.42) and (4.43), respectively,are of rather different nature. We will sketch the arguments in Sections 4.5.2 and4.5.3 below. (4.42) Under the SimplifyingAssumptions 4.39, surjectivity of the map (4.42) has essentially been shown bySteenbrink and van Straten, [SvS85]. We give a brief synopsis of their line ofargumentation and indicate additional steps of argumentation required to handlethe general setting. To start, recall from [Har77, III ex. 2.3e] that the map (4.42)is part of the standard sequence that defines cohomology with supports,(4.44) · · · → H (cid:0) Z, Ω pZ (log E ) (cid:1) → H (cid:0) Z \ E, Ω pZ (log E ) (cid:1) →→ H E (cid:0) Z, Ω pZ (log E ) (cid:1) → · · · We aim to show that the last term in (4.44) vanishes. There are two main ingre-dients to the proof: formal duality and Steenbrink’s vanishing theorem.
Theorem 4.45 (Formal duality theorem for cohomology with support, [Har70,Chapt. 3, Thm. 3.3]) . Under the Assumptions 4.39, if F is any locally free sheafon Z and ≤ j ≤ dim Z any number, then there exists an isomorphism (cid:0) ( R j γ ∗ F ) x (cid:1) (cid:98) ∼ = H n − jE (cid:0) Z, F ∗ ⊗ ω Z (cid:1) ∗ , where (cid:98) denotes completion with respect to the maximal ideal m x of the point x ∈ X , and where n = dim X = dim Z . (cid:3) tefan Kebekus 33A brief introduction to formal duality, together with a readable, self-containedproof of Theorem 4.45 is found in [GKK10, Appendix A] while Hartshorne’s lecturenotes [Har70] are the standard reference for these matters. Theorem 4.46 (Steenbrink vanishing, [Ste85, Thm. 2.b]) . If p , q are any twonumbers with p + q > dim Z , then R q γ ∗ (cid:0) J E ⊗ Ω pZ (log E ) (cid:1) = 0 . (cid:3) Remark . Steenbrink’s vanishing theorem is proven using local Hodge theoryof isolated singularities. For p = n , the sheaves Ω nZ and J E ⊗ Ω nZ (log E ) areisomorphic. In this case, the Steenbrink vanishing theorem reduces to Grauert-Riemenschneider vanishing, [GR70].Setting F := J E ⊗ Ω n − pZ (log E ) and using that F ∗ ⊗ ω Z ∼ = Ω pZ (log E ), formalduality and Steenbrink vanishing together show that H E (cid:0) Z, Ω pZ (log E ) (cid:1) = 0, for p < dim Z −
1, proving surjectivity of (4.42) for these values of p . The other casesneed to be treated separately. case p = np = np = n : After passing to an index-one cover, surjectivity of (4.42) incase p = n follows almost directly from the definition of klt, cf. [GKK10,Sect. 5]. case p = n − p = n − p = n − : In this case one uses the duality between Ω n − Z and the tan-gent sheaf T Z , and the fact that any section in the tangent sheaf of X al-ways lifts to the canonical resolution of singularities, cf. [GKK10, Sect. 6]. General case
The argument outlined above, using formal duality and Steen-brink vanishing, works only because we were assuming that the singularities of X are isolated. In the general case, where the Simplifying Assumptions 4.39 do notnecessarily hold, this is not necessarily the case. In order to deal with non-isolatedsingularities, one applies a somewhat involved cutting-down procedure, as indi-cated in Figure 2. This way, it is often possible to view non-isolated log canonicalsingularities a family of isolated singularities, where surjectivity of (4.42) can beshown on each member of the family. To conclude that it holds on all of Z , thefollowing strengthening of Steenbrink vanishing is required. Theorem 4.48 (Steenbrink-type vanishing for log canonical pairs, [GKKP11,Thm. 14.1]) . Let ( X, D ) be a log canonical pair of dimension n ≥ . If γ : Z → X is a log resolution of singularities with exceptional set E and ∆ := supp (cid:0) E + γ − (cid:98) D (cid:99) (cid:1) , then R n − γ ∗ (cid:0) Ω pZ (log ∆) ⊗ O Z ( − ∆) (cid:1) = 0 for all ≤ p ≤ n . (cid:3) The proof of Theorem 4.48 essentially relies on the fact that log canonicalpairs are Du Bois, [KK10]. The Du Bois property generalises the notion of rationalsingularities. For an overview, see [KS09].4 Differential forms, MMP, and Hyperbolicity Z , resolution of singularities divisor E (cid:0)(cid:0)(cid:9) divisor E (cid:72)(cid:72)(cid:89) γ resolution map (cid:47) (cid:47) singular space X point γ ( E ) (cid:8)(cid:8)(cid:8)(cid:8)(cid:25) • curve γ ( E ) (cid:64)(cid:64)(cid:73) The figure sketches a situation where X is a threefold whose singular locus is a curve.Near the general point of the singular locus, the variety X looks like a family of isolatedsurfaces singularities. The exceptional set E of the resolution map γ contains twoirreducible divisors E and E . Figure 2.
Non-isolated singularities (4.43) Let σ ∈ H (cid:0) Z, Ω pZ (log E ) (cid:1) be any differential form on Z that is allowed to have logarithmic poles along E .To show surjectivity of the inclusion map (4.43), we need to show that σ reallydoes not have any poles along E . To give an idea of the methods used to provethis, we consider only the case where p >
1. We discuss two particularly simplecases first.
The case where E is irreducible Assume that E is irreducible. To show that σ does not have any logarithmic poles along E , recall from Fact 4.32 that it sufficesto show that σ is in the kernel of the residue map ρ p : H (cid:0) Z, Ω pZ (log E ) (cid:1) → H (cid:0) E, Ω p − E (cid:1) . On the other hand, we know from a result of Hacon-McKernan, [HM07,Cor. 1.5(2)], that E is rationally connected, so that H (cid:0) E, Ω p − E (cid:1) = 0. Thisclearly shows that σ is in the kernel of ρ p and completes the proof when E isirreducible. The case where ( Z, E ) admits a simple minimal model program In gen-eral, the divisor E need not be irreducible. Let us therefore consider the nextdifficult case where E is reducible with two components, say E = E ∪ E . Theresolution map γ will then factor via a γ -relative minimal model program of thepair ( Z, E ), which we assume for simplicity to have the following particularly spe-cial form, sketched also in Figure 3. The computer code used to generate the images in Figure 3 is partially taken from [Bau07]. tefan Kebekus 35snc surface pair (
Z, E + E ) divisor E (cid:88)(cid:88)(cid:122) divisor E (cid:88)(cid:88)(cid:88)(cid:88)(cid:122) λ contracts E (cid:47) (cid:47) resolution map γ (cid:39) (cid:39) dlt surface pair ( Z , E , ) divisor E , (cid:67)(cid:67)(cid:67)(cid:79) λ contracts E , (cid:15) (cid:15) klt surface X This sketch shows a resolution of an isolated klt surface singularity, and thedecomposition of the resolution map given by the minimal model program of the sncpair (
Z, E + E ). Figure 3.
Resolution of an isolated klt surface singularity Z = Z λ contracts E to a point (cid:47) (cid:47) Z λ contracts E , := ( λ ) ∗ ( E ) to a point (cid:47) (cid:47) X. In this setting, the arguments outlined above apply verbatim to show that σ hasno poles along the divisor E . To show that σ does not have any poles along theremaining component E , observe that it suffices to consider the induced reflexiveform on the possibly singular space Z , say σ ∈ H (cid:0) Z , Ω [ p ] Z (log E , ) (cid:1) , where E , := ( λ ) ∗ ( E ), and to show that σ does not have any poles along E , .For that, we follow the same line of argument once more, accounting for thesingularities of the pair ( X , E , ).The pair ( X , E , ) is dlt, and it follows from adjunction that the divisor E , is necessarily normal, [KM98, Cor. 5.52]. Using the residue map for reflexivedifferentials on dlt pairs that was constructed in Theorem 4.33, ρ [ p ] : H (cid:0) X , Ω [ p ] Z (log E , ) (cid:1) → H (cid:0) E , , Ω [ p − E , (cid:1) , ρ [ p ] ( σ ) = 0. Because themorphism λ contracts the divisor E , to a point, the result of Hacon-McKernanwill again apply to show that E , is rationally connected. Even though there arenumerous examples of rationally connected spaces that carry non-trivial reflexiveforms, we claim that in our special setup we do have the vanishing(4.49) H (cid:0) E , , Ω [ p − E , (cid:1) = 0 . For this, recall from the adjunction theory for Weil divisors on normal spaces,[Kol92, Chapt. 16 and Prop. 16.5] and [Cor07, Sect. 3.9 and Glossary], thatthere exists a Weil divisor D E on the normal variety E , which makes the pair( E , , D E ) klt. Now, if we knew that the extension theorem would hold for thepair ( E , , D E ), we can prove the vanishing statement (4.49), arguing exactly asin the proof of Corollary 4.17, where we show the non-existence of reflexive formson rationally connected klt spaces as a corollary of the Pull-Back Theorem 4.11.Since dim E , < dim X , this suggests an inductive proof, beginning with easy-to-prove extension theorems for reflexive forms on surfaces, and working our wayup to higher-dimensional varieties. The proof in [GKKP11] follows this inductivepattern. The general case
To handle the general case, where the Simplifying Assump-tions 4.39 do not necessarily hold true, we need to work with pairs (
X, D ) where D is not necessarily empty, the γ -relative minimal model program might involve flips,and the singularities of X need not be isolated. All this leads to a slightly pro-tracted inductive argument, heavily relying on cutting-down methods and outlinedin detail in [GKKP11, Sect. 19]. In view of the Viehweg-Zuo construction, it would be very interesting toknow if a variant of the Pull-Back Theorem 4.11 holds for symmetric powersof Ω [1] X (log D ), or for other tensor powers. As shown by examples, cf. [GKK10,Ex. 3.1.3], the na¨ıve generalisation of Theorem 4.11 is wrong. Still, it seems con-ceivable that a suitable generalisation, perhaps formulated in terms of Campana’sorbifold differentials, might hold. However, note that several of the key ingredi-ents used in the proof of Theorem 4.11, including Steenbrink’s vanishing theorem,rely on (local) Hodge theory, for which no version is known for tensor powers ofdifferential forms. Question . Is there a formulation of the Pull-Back Theorem 4.11 that holdsfor symmetric and other tensor powers of differential forms?Examples suggest that the Pull-Back Theorem 4.11 is optimal, and that theclass of log canonical pairs is the natural class of spaces where a pull-back theoremcan hold.tefan Kebekus 37
Question . To what extend is the Pull-Back Theorem 4.11 optimal? Is therea version of the pull-back theorem that does not require the log canonical divisor K X + D to be Q -Cartier? If we are interested only in special values of p , is thedivisor ∆ the smallest possible?The last question concerns the generalisation of the Bogomolov-Sommesevanishing theorem. One of the main difficulties with its current formulation isthe requirement that the sheaf A be Q -Cartier. We have seen in Section 4.1how interesting reflexive subsheaves A ⊆ Ω [ p ] X can often be constructed using theHarder-Narasimhan filtration. Unless the space X is Q -factorial, there is, however,no way to guarantee that a sheaf constructed this way will actually be Q -Cartier.The property to be Q -factorial, however, is not stable under taking hyperplanesections and difficult to guarantee in practise. Question . Is there a version of the generalised Bogomolov-Sommese vanishingtheorem, Corollary 4.14, that does not require the sheaf A to be Q -Cartier?
5. Viehweg’s conjecture for families over threefolds, sketch ofproof
We conclude this paper by sketching a proof of the Viehweg Conjecture 2.8in one special case, illustrating the use of the methods introduced in Sections 3and 4. As in Section 4.1 we consider a family f ◦ : X ◦ → Y ◦ of canonically po-larised varieties over a quasi-projective threefold. Assuming that f ◦ is of maximalvariation, we would like to show that the logarithmic Kodaira dimension κ ( Y ◦ )cannot be zero. Proposition 5.1 (Partial answer to Viehweg’s conjecture) . Let f ◦ : X ◦ → Y ◦ bea smooth, projective family of canonically polarised varieties over a smooth, quasi-projective base manifold of dimension dim Y ◦ = 3 . Assume that the family f ◦ isof maximal variation, i.e., that Var( f ◦ ) = dim Y ◦ . Then κ ( Y ◦ ) (cid:54) = 0 . The proof of Proposition 5.1 follows the line of argumentation outlined inSection 4.1. We prove that the Picard number of a suitable minimal model can-not be one, thereby exhibiting a fibre space structure to which induction can beapplied. The presentation follows [KK08c, Sect. 9].
In essence, we follow the line of argument sketched in Section 4.1. We argueby contradiction, i.e., we maintain the assumptions of Proposition 5.1 and assumein addition that κ ( Y ◦ ) = 0.8 Differential forms, MMP, and Hyperbolicity As before, choose a smooth compactification Y ⊇ Y ◦ such that D := Y \ Y ◦ is a divisor with only simple normal crossings. Let λ : Y (cid:57)(cid:57)(cid:75) Y λ be the rational map obtain by a run of the minimal model programfor the pair ( Y, D ) and set D λ := λ ∗ ( D ). The following is then known to hold.(1) The variety Y λ is normal and Q -factorial.(2) The variety Y λ is log terminal. The pair ( Y λ , D λ ) is dlt.(3) There exists a number m (cid:48) such that m (cid:48) (cid:0) K Y λ + D λ (cid:1) ≡
0. In particular, thedivisor K Y λ + D λ is numerically trivial.By Viehweg-Zuo’s Theorem 3.1, there exists a number m > A ⊆
Sym m Ω Y (log D ). As we have seen in Proposition 4.7, this induces areflexive sheaf A λ ⊆ Sym [ m ] Ω Y λ (log D λ ) of rank one and Kodaira-Iitaka dimension κ ( A λ ) = dim Y λ . Ω [1] Y λ (log D λ ) As in Section 4.1above, we employ the Harder-Narasimhan filtration to obtain additional informa-tion about the space Y λ . Claim 5.2.
The divisor D λ is not empty.Proof. For simplicity, we prove Claim 5.2 only in case where the canonical divisor K Y λ is Cartier, and where the space Y λ therefore has only canonical singularities.For a proof in the general setup, the same line of argumentation applies afterpassing to a global index-one cover. We argue by contradiction and assume that D λ = 0.As before, let C ⊆ Y λ be a general complete intersection curve in the sense ofMehta-Ramanathan, cf. [HL97, Sect. II.7]. Since the general complete intersectioncurve C avoids the singular locus of Y λ , we obtain that the restricted sheaf ofK¨ahler differentials Ω Y λ | C as well as its dual T Y λ | C , the restriction of the tangentsheaf, are locally free. Further, the numerical triviality of K Y λ ≡ K Y λ + D λ impliesthat K Y λ .C = c (cid:0) Ω [1] Y λ (log D λ ) (cid:1) .C = c (cid:0) Sym [ m ] Ω Y λ (log D λ ) (cid:1) .C = 0 . On the other hand, since A λ is big, we have that c ( A λ ) .C >
0. As in the proofof Proposition 4.1, this implies that the restricted sheaves Ω Y λ | C as well as itsdual T Y λ | C , are not semistable. The maximal destabilising subsheaf of T Y λ | C issemistable and of positive degree, hence ample. In this setup, a variant [KST07,Cor. 5] of Miyaoka’s uniruledness criterion [Miy87, Cor. 8.6] applies to give theuniruledness of Y λ . For more details on this criterion, see the survey [KS06].To finish the argument, let r : W → Y λ be a resolution of singularities. Sinceuniruledness is a birational property, the space W is uniruled and therefore hasKodaira-dimension κ ( W ) = −∞ . On the other hand, since Y λ has only canonicalsingularities, the Q -linear equivalence class of the canonical bundle K W is giventefan Kebekus 39as K W ≡ r ∗ ( K Y λ ) + (effective, r -exceptional divisor) . But because K Y λ is Q -linearly equivalent to the trivial divisor, we obtain that κ ( W ) ≥
0, a contradiction. (cid:3)
Claim 5.2 implies that K Y λ ≡ − D λ and it followsthat for any rational number 0 < ε < κ (cid:0) K Y λ + (1 − ε ) D λ (cid:1) = κ (cid:0) εK Y λ (cid:1) = κ (cid:0) Y λ (cid:1) = −∞ . Now choose one ε and run the log minimal model program for the dlt pair (cid:0) Y λ , (1 − ε ) D λ (cid:1) . This way one obtains morphisms and birational maps as follows Y λ µ minimal model program (cid:47) (cid:47) Y µ π Mori fibre space (cid:47) (cid:47) Z. Again, let D µ := µ ∗ ( D λ ) be the cycle-theoretic image of D λ . The main propertiesof Y µ and D µ are summarised as follows.(1) The variety Y µ is normal and Q -factorial.(2) The variety Y µ is log terminal. The pair (cid:0) Y µ , (1 − ε ) D µ (cid:1) is dlt.(3) The divisor K Y µ + D µ is numerically trivial.(4) There exists a reflexive sheaf A µ ⊆ Sym [ m ] Ω Y µ (log D µ ) of rank one andKodaira-Iitaka dimension κ ( A µ ) = dim Y µ .In fact, more is true. Claim 5.4.
The pair ( Y µ , D µ ) is log canonical.Proof. Since K Y λ + D λ ≡
0, some positive multiples of K Y λ and − D λ are nu-merically equivalent. For any two rational numbers 0 < ε (cid:48) , ε (cid:48)(cid:48) <
1, the divisors K Y λ + (1 − ε (cid:48) ) D λ and K Y λ + (1 − ε (cid:48)(cid:48) ) D λ are thus again numerically equivalent upto a positive rational multiple.The birational map µ is therefore a minimal model program for the pair (cid:0) Y λ , (1 − ε ) D λ (cid:1) , independently of the number ε chosen in its construction. Itfollows that (cid:0) Y µ , D µ (cid:1) is a limit of dlt pairs and therefore log canonical. (cid:3) Y µ Another application of the “Harder-Narasimhan-trick” exhibits a fibre structure of Y µ . Claim 5.5.
The Picard-number ρ ( Y µ ) is larger than one. In particular, the map Y µ → Z is a proper fibre space whose fibres are proper subvarieties of Y µ .Proof. As before, let C ⊆ Y µ be a general complete intersection curve. Again,the existence of the Viehweg-Zuo sheaf A µ implies that the sheaf of reflexivedifferentials Ω [1] Y µ (log D µ ) is not semistable, and contains a destabilising subsheaf B µ ⊆ Ω [1] Y µ (log D µ ) with c ( B µ ) .C >
0. Since the intersection number c ( B µ ) .C r of the sheaf B µ must be strictly less than dim Y µ , and itsdeterminant is a subsheaf of the sheaf of logarithmic r -forms,det B µ ⊆ Ω [ r ] Y µ (log D µ ) with c (det B µ ) .C > r < dim Y µ . If ρ ( Y µ ) = 1, then the sheaf det B µ would necessarily be Q -ample, violatingthe Bogomolov-Sommese vanishing theorem for log canonical pairs, Corollary 4.14.This finishes the proof of Claim 5.5. (cid:3) Now, if F ⊂ Y µ is a general fibre of π and D F := D µ ∩ F , then F is a normalcurve or surface, and the pair ( F, D F ) is log canonical and has Kodaira dimension κ ( K F + D F ) = 0. By [KM98, Prop. 4.11], the variety F is even Q -factorial. Itis then possible to argue by induction: assuming that Viehweg’s conjecture holdsfor families over surfaces, one obtains that the restriction of the family f ◦ to thestrict transform ( µ ◦ λ ) − ∗ ( F ) cannot be of maximal variation. Since the fibresdominate the variety, this contradicts the assumption that the family f ◦ is ofmaximal variation, and therefore finishes the sketch of proof of Proposition 5.1.The reader interested in more details is referred to [KK08c, Sect. 9], wherea stronger statement is shown, proving that any family over a base manifold with κ ( Y ◦ ) = 0 is actually isotrivial. References [Ara71]
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