aa r X i v : . [ m a t h . AG ] F e b MOISHEZON MORPHISMS
J ´ANOS KOLL ´AR
Abstract.
We try to understand which morphisms of complex analytic spacescome from algebraic geometry. We start with a series of conjectures, and thengive some partial solutions.
Contents
1. Open questions 12. Moishezon spaces 33. Moishezon morphisms 44. 1-parameter families 95. Approximating Moishezon morphisms 126. Inversion of adjunction 14References 17A proper, irreducible, reduced analytic space X is Moishezon if it is bimeromor-phic to a projective variety X p , and a proper morphism of analytic spaces g : X → S is Moishezon if it is bimeromorphic to a projective morphism g p : X p → S ; see (7)and (10–11) for details.The aim of this note is to discuss a series of questions about Moishezon mor-phisms, and give partial solutions to some of them.We start with a list of conjectures in Section 1. Sections 2–3 are mostly review;new results are in Sections 4–6.1. Open questions
The theory of Moishezon spaces can be viewed as a special chapter of the theoryof algebraic spaces (and later stacks). However, a deformation of a Moishezon spaceneed not be Moishezon, thus we get a theory that is not algebraic. The questionwe consider is the following. • Which morphisms of complex amalytic spaces come from algebraic geome-try, up to bimeromorphism?The main open problem may be Conjecture 1 and its special case Conjecture 2.I have very little evidence supporting them and several experts thought that theyare likely wrong.Conjectures 3–4 about the deformation theory of Moishezon spaces go back atleast to Hironaka’s unpublished thesis.Conjecture 5 can be viewed as a geometric form of deformation invariance ofplurigenera [RT20, Thm.1.2], see also (34).
Conjecture 1.
A proper morphism of analytic spaces g : X → S is locally Moishe-zon (11) iff every irreducible component of every fiber is Moishezon.Comments 1.1. We check in (16) that the fibers of a Moishezon morphism areMoishezon. If g is smooth, a positive answer is in [RT21, Thm.1.4] and [RT20,Thm.1.2]; see (22).If π ( S ) is finite and S is either Stein or quasi-projective, then maybe g is alsoglobally Mosihezon. Easy examples (13) show that finiteness of π ( S ) is necessaryfor the global variant.Note that we do not assume that g is flat or that X, S are smooth. It is, however,quite likely that the above generality does not matter. The semi-stable reductiontheorem [AK00] suggests that there is a projective, bimeromorphic morphism S ′ → S such that the main component of X × S S ′ is bimeromorphic to a morphism g ′ : X ′ → S ′ such that g ′ is flat, toroidal and S ′ is smooth.A positive answer for g ′ : X ′ → S ′ does not automatically imply a positiveanswer for g : X → S , but the method may generalize.It is reasonable to start with the case when g is flat with mildly singular fibers.The following may well be the key special case, where D denotes the complex disc. Conjecture 2.
Let X be a smooth analytic space and g : X → D a proper mor-phism. Assume that the fiber X s is Moishezon for s = 0 , and X is a simple normalcrossing divisor whose irreducible components are Moishezon. Then g is Moishezon. For an arbitrary proper morphism, the set of Mosihezon fibers need not be closed(25), but the following could be true.
Conjecture 3.
Let g : X → D be a flat, proper morphism. Assume that X isirreducible with rational singularities and X s is Moishezon for s = 0 . Then X isMoishezon.Comments 3.1. There are 3 preprints [Pop09, Bar17, Pop19] claiming a positiveanswer if X is smooth. These lie outside my expertise, but my understanding isthat not everyone is able to follow the arguments in them.The analogous question for surfaces with cusp singularities has a negative answer;see (14). Cusps are the simplest non-rational surface singularities. This suggestedthat either log terminal or rational singularities may be the right class here. Conjecture 4.
Let X be a smooth analytic space and g : X → D a proper mor-phism. Assume that one of the irreducible components of X is of general type.Then g is Moishezon and all other fibers are of general type (over a possibly smallerdisc).Comments 4.1. Smooth, projective K3 and elliptic surfaces have deformationsthat are not even Moishezon, so general type may be the best one can hope for.We can harmlessly assume that X is a reduced, simple normal crossing divisor.If X is irreducible and smooth, this is posed in [Sun80, p.201]; which in turnbuilds on problems and conjectures in [Iit71, Moi71, Nak75, Uen75].If X is irreducible, projective and has canonical singularities, a positive an-swer is given in [Kol21]. Note, however, that smooth Moishezon spaces can haveunexpected deformations; see [Cam91, LP92].Let g : X → S be a Moishezon morphism. By definition, it is bimeromorphic toa projective morphism g p : X p → S . Thus the fibers X s and X p s are bimeromorphic OISHEZON MORPHISMS 3 to each other for general s ∈ S , but may be quite different for special points s ∈ S .The following conjecture suggests that, over 1-dimensional bases, one can arrange X s and X p s to be bimeromorphic to each other for every s ∈ S . Conjecture 5.
Let g : X → D be a flat, proper, Moishezon morphism. Assume that X has canonical (resp. log terminal) singularities. Then g is fiberwise birational(26) to a flat, projective morphism g p : X p → D such that (1) X p0 has canonical (resp. log terminal) singularities, (2) X p s has terminal singularities for s = 0 , and (3) K X p is Q -Cartier.Comments 5.4. This is where singularities inevitably enter the picture. Even if g is a smooth family of projective surfaces, X p may need to be singular; see forexample [Kol21, Exmp.4].If g is smooth and the fibers are of general type, then [RT20, Thm.1.2] impliesthat the canonical models of the fibers give the optimal choice for g p : X p → D .(Here all fibers can have canonical singularities.)We give a positive answer to the log terminal case, provided X is not uniruled,see (28). The canonical case is discussed in (32). Remark 6.
My aim is to understand how much of the theory of Moishezon spacesfits into algebraic geometry, and especially minimal model theory. The paper [RT20]gave the impetus to try to organize this into a systematic series of questions.Campana pointed out that several of these questions have analogs for compactspaces of Fujiki’s class C , and have a positive answer if we assume that the totalspace is of class C ; see [Cam81].A very different direction studies the place of the Moishezon property in thetheory of compact complex manifolds. Solutions of Conjectures 1–2 are more likelyto come from this approach. See [Pop11] for a survey. Acknowledgments.
I thank D. Abramovich, F. Campana, J.-P. Demailly, T. Mu-rayama, V. Tosatti and C. Xu for helpful comments. I am grateful to S. Rao andI-H. Tsai for detailed remarks, corrections and references. Partial financial supportwas provided by the NSF under grant number DMS-1901855.2.
Moishezon spaces
We give a quick review of the theory of Moishezon spaces.
Definition 7.
A proper, irreducible, reduced analytic space X is Moishezon if itis bimeromorphic to a projective variety X p . That is, there is a closed, analyticsubspace Γ ⊂ X × X p such that the coordinate projections Γ → X and Γ → X p are isomorphisms on Zariski open dense sets.By Chow’s theorem, any 2 such X p are birational to each other, so X acquiresa unique algebraic structure.A proper analytic space X is Moishezon iff the irreducible components of red X are Moishezon . Thus X is Moishezon iff every irreducible component of its nor-malization is Moishezon. (Basic theorems) . Let X be a proper Moishezon space. This is not standard terminology.
J´ANOS KOLL´AR (1) There is a projective variety X ′ and a bimeromorphic morphism X ′ → X (Chow lemma).(2) For every x ∈ X there is a pointed quasi projective scheme ( x ′ , X ′ ) and an´etale morphism ( x ′ , X ′ ) → ( x, X ).(3) If X is normal then there is a proper variety Y and a finite group G acting on Y such that X ∼ = Y /G . (Note that usually Y can not be chosen projective.)(4) If Z → X is finite, then Z is Moishezon.(5) If X → Y is surjective, then Y is Moishezon.(6) Assume that X is smooth. Then the usual Hodge decomposition H i ( X, C ) = ⊕ p + q = i H p ( X, Ω qX ) holds.(7) Hilb( X ) and Chow( X ) are algebraic spaces whose irreducible componentsare proper (but the connected components may have infinitely many irre-ducible components). The connected components of the space of divisorsChow n − ( X ) are proper.(8) If X has rational singularities then it is projective iff it is K¨ahler. Hints of proofs.
Note that (1) is not obvious [Moi66]. It also follows from themore general results of [Hir75], and one can easily modify the arguments in [Sta15,Tag 088U]; the key step is probably [Sta15, Tag 0815].(2) is quite hard; see [Art70].For (3), cover X with finitely many X ′ i → X as in (2). Then normalize X in theGalois closure of the field extensions C ( X ′ i ) / C ( X ). One can then use this to get Z → X as a quotient of a finite morphism Z ′ → X ′ to obtain (4).For (5), using (4) and (1) we may assume that X → Y is generically finite, sayof degree d , and X is projective. Then y [ g − ( y )] ∈ S d X gives a bimeromorphicembedding of Y into the d th symmetric power of X .By direct computation, the existence of a Hodge decomposition is invariant undersmooth blow ups, thus we get (6). A better argument is in [Uen83, Prop.1.3].For (7) see [Art69, Bar75, Cam80, Fuj82] and [Kol96, Sec.I.5].The smooth case of (8) is proved in [Moi66], the singular one in [Nam02]. Remark.
The complements of closed analytic subsets form the open subsets ofthe Zariski topology. Note, however, that 2 open subsets can be biholomorphic toeach other even if they are not birational. This is the main reason why one usuallydoes not define ‘Moishezon’ for non-proper anaytic spaces.3.
Moishezon morphisms
Definition 9 (Projective morphisms) . A proper morphism of analytic spaces g : X → S is projective if X can be embedded into P S := P N × S → S for some N .Note that some authors allow P S → S to be any (locally trivial) P N -bundle. The2 versions are equivalent if S is Stein or quasi-projective (the cases we are mostlyinterested in) but not in general. Definition 10 (Moishezon morphisms) . [Moi74, Fuj82] Assume now that S isreduced. A proper morphism of analytic spaces g : X → S is Moishezon iff thefollowing equivalent conditions hold.(1) g : X → S is bimeromorphic to a projective morphism g p : X p → S .That is, there is a closed subspace Y ⊂ X × S X p such that the coordinateprojections Y → X and Y → X p are bimeromorphic. OISHEZON MORPHISMS 5 (2) There is a projective morphism of algebraic varieties G : X → S and ameromorphic φ S : S S such that X is bimeromorphic to X × S S .Here (2) ⇒ (1) is clear. To see the converse, note that g p : X p → S is flat over adense, open subset S ◦ ⊂ S , thus we get a meromorphic map φ : S Hilb( P N ).The pull-back of the universal family over Hilb( P N ) is then bimeromorphic to X . Comment.
This is the right notion if S is Stein or quasi projective, but, as withprojectivity, there are different versions in general.Assume that X is normal and the maps X φ X p ι ֒ → P S (10 . X → S is Moishezon. Then ι ◦ φ : X P S is defined outside acodimension 2 closed subset, and ( ι ◦ φ ) ∗ O P S (1) extends to a rank 1 reflexive sheaf L on X . This L ‘certifies’ that X is Moishezon. This gives another equivalentcharacterization (in case X is normal, and S is Stein or quasi projective.)(4) There is a rank 1, reflexive sheaf L on X such that the natural map X Proj S ( g ∗ L ) is bimeromorphic onto the closure of its image.We call such a sheaf L very big (over S ) . L is big (over S ) if L [ m ] is very big forfome m >
0, where L [ m ] denotes the reflexive hull of the m th tensor power.Note that L is big (resp. very big) on X → S iff it is big (resp. very big) on X ◦ → S ◦ on some dense, Zariski open S ◦ ⊂ S . Warning.
By contrast it can happen that X ◦ → S ◦ is Moishezon but X → S isnot, since the L ◦ that certifies Moishezonness need not extend to X ; see (14). Definition 11 (Locally Moishezon morphisms) . [Moi74] A proper morphism ofanalytic spaces g : X → S locally Moishezon if S is covered by (Euclidean) opensets S i ⊂ S such that each g − ( S i ) → S i is Moishezon. Comment.
This follows standard usage of ‘locally’ in algebraic geometry and itworks best for the purposes of Conjecture 1. However, it is not equivalent to thedefinition in [Fuj82].
Example 12.
Let g : X → S be a proper morphism of analytic spaces, S Moishe-zon. Then g is Moishezon iff X is Moishezon. Example 13.
Let Z be a normal, projective variety with discrete automorphismgroup. Let g : X → S be a fiber bundle with fiber Z over a connected base S .Then g is Moishezon ⇔ g is projective ⇔ the monodromy is finite.There are rational and K3 surfaces with infinite, discrete automorphism group.These lead to fiber bundles over the punctured disc D ◦ that are locally Moishezonbut not globally Moishezon. Example 14. [Loo81] studies examples where X is an Inoue surface (which isnot Moishezon) with a cusp (which is log canonical), yet X s is a smooth rationalsurface for s = 0.Next we look at fibers of Moishezon morphisms. Lemma 15.
Let g : X → S be a proper, generically finite, dominant morphism ofnormal, complex, analytic spaces. Then Ex( g ) → S is Moishezon. Very big is not standard terminology, but it matches very ample.
J´ANOS KOLL´AR
Proof. We prove the special case when the smooth locus of S is dense in g (cid:0) Ex( g ) (cid:1) .This is a harmless assumption if S is Stein (or quasi-projective), since we cancompose g with a finite S → C dim S (or with a quasi-finite S → P dim S ). A moreheavy handed approach, which works in general, is to use a resolution S ′ → S andreplace X by the normalization of the main component of X × S S ′ .Let E be a g -exceptional divisor. Set ( g : X → S ) := ( g : X → S ) and Z := g ( E ).If g i : X i → S i and E i ⊂ X i are already defined, we set Z i := g i ( E i ). Let S i +1 be the normalization of the blow-up B Z i S i , g i +1 : X i =1 → S i +1 the normalizationof the graph of X i → S i S i +1 and E i +1 ⊂ X i +1 the bimeromorphic transformof E i . (Note that X i +1 → X i is a isomorphism over an open subset of E i .)Let a ( E i , S i ) denote the vanishing order of the Jacobian of g i along E i . By anelementary computation we get that a ( E i +1 , S i +1 ) ≤ a ( E i , S i ) + 1 − codim( Z i ⊂ S i ) . Thus eventually we reach the situation when codim( Z i ⊂ S i ) = 1, hence E i → Z i is generically finite.Note that each Z i +1 → Z i is projective, thus E i → Z is Moishiezon by (8.4),and so is E → S . (cid:3) The following is the easy direction of Conjecture 1.
Corollary 16.
The fibers of a proper, Moishezon morphism are Moishezon.
Proof. Let g : X → S be a proper, Moishezon morphism. It is bimeromorphicto a projective morphism X p → S . We may assume X p to be normal. Let Y bethe normalization of the closure of the graph of X X p .Fix now s ∈ S . Let Z s ⊂ X s be an irreducible component and W s ⊂ Y s anirreducible component that dominates Z s . By (8.5) it is enough to show that W s is Moishezon.If π : Y → X p is generically an isomorphism along W s , then W s is bimeromorphicto an irreducible component of X p s , hence Moishezon. Otherwise W s ⊂ Ex( π ).Now Ex( π ) → X p is Moishezon by (15) and dim Ex( π ) < dim Y = dim X . Now W s is contained in a fiber of Ex( π ) → S , hence Moishezon by induction on thedimension. (cid:3) Remark 17.
More generally, if g : X → S is proper and Moishezon and T → S isa morphism of analytic spaces then X × S T → T is also proper and Moishezon.The rest of this section is a study of the set of Moishezon fibers for arbitraryproper morphisms of analytic spaces. It is mostly a summary of some of the resultsof [RT20], with occasional changes. Definition 18.
Let g : X → S be a proper morphism of normal analytic spacesand L a line bundle on X . Set(1) VB S ( L ) := { s ∈ S : L s is very big on X s } ⊂ S ,(2) GT S ( X ) := { s ∈ S : X s is of general type } ⊂ S ,(3) MO S ( X ) := { s ∈ S : X s is Moishezon } ⊂ S ,(4) PR S ( X ) := { s ∈ S : X s is projective } ⊂ S . Lemma 19.
Let g : X → S be a proper morphism of normal, irreducible analyticspaces and L a line bundle on X . Then VB S ( L ) ⊂ S is OISHEZON MORPHISMS 7 (1) either nowhere dense (in the analytic Zariski topology), (2) or it contains a dense open subset of S , and g : X → S is Moishezon. Proof. By passing to an open subset of S , we may assume that g is flat, g ∗ L islocally free and commutes with restriction to fibers. We get a meromorphic map φ : X P S ( g ∗ L ). There is thus a normal, bimeromorphic model π : X ′ → X such that φ ◦ π : X ′ → P S ( g ∗ L ) is a morphism.After replacing X by X ′ and again passing to an open subset of S , we mayassume that g is flat, g ∗ L is locally free, commutes with restriction to fibers, and φ : X → P S ( g ∗ L ) is a morphism. Let Y ⊂ P S ( g ∗ L ) denote its image and W ⊂ X theZariski closed set of points where π : X → Y is not smooth. Set Y ◦ := Y \ φ ( W )and X ◦ := X \ φ − ( φ ( W )). The restriction φ ◦ : X ◦ → Y ◦ is then smooth andproper.We assume that φ − ( y ) is a single point for a dense set in Y , hence for a denseset in Y ◦ . Since φ ◦ is smooth and proper, it is then an isomorphism. Thus φ isbimeromorphic on every irreducible fiber that has a nonempty intersection with X ◦ . (cid:3) Corollary 20.
Let g : X → S be a proper morphism of normal, irreducible analyticspaces. Then GT S ( X ) ⊂ S is (1) either nowhere dense (in the analytic Zariski topology), (2) or it contains a dense open subset of S , and g : X → S is Moishezon. Proof. Using resolution, we may assume that X is smooth. By passing to anopen subset of S , we may also assume that S and g are smooth. By [HM06] thereis an m (depending only on dim X s ) such that | mK X s | is very big whenever X s isof general type. Thus (19) applies to L = mK X . (cid:3) The following is essentially proved in [RT21, Thm.1.4] and [RT20, Thm.1.2].
Theorem 21.
Let g : X → S be a smooth, proper morphism of normal, irreducibleanalytic spaces. Then MO S ( X ) ⊂ S is (1) either contained in a countable union ∪ i Z i , where Z i ( S are Zariski closed, (2) or MO S ( X ) contains a dense, open subset of S .Furthermore, if R g ∗ O X is torsion free then (2) can be replaced by (3) MO S ( X ) = S and g is locally Moishezon.Remark 21.4. A positive answer to Conjecture 3 for smooth morphisms wouldimply that in fact MO S ( X ) = S always holds in case (21.2); see (22).Proof. Assume first that R g ∗ O X is torsion free.As in [RT20, 3.15], the push-forward of the exponential sequence0 → Z X → O X exp −→ O × X → R g ∗ O × X → R g ∗ Z X e −→ R g ∗ O X . We may pass to the universal cover of S and assume that R g ∗ Z X is a trivial H ( X s , Z )-bundle.Let { ℓ i } be those global sections of R g ∗ Z X such that e ( ℓ i ) ∈ H ( S, R g ∗ O X )is identically 0, and { ℓ ′ j } the other global sections. The ℓ i then lift back to globalsections of R g ∗ O × X , hence to line bundles L i on X . J´ANOS KOLL´AR
If there is an L i such that VB S ( L i ) contains a dense open subset of S , then X → S is Moishezon by (19) and we are done. Otherwise we claim thatMO S ( X ) ⊂ ∪ i VB S ( L i ) [ ∪ j (cid:0) e ( ℓ ′ j ) = 0 (cid:1) . (21 . s / ∈ ∪ j (cid:0) e ( ℓ ′ j ) = 0 (cid:1) . Then every line bundle on X s isnumerically equivalent to some L i | X s . Since being big is preserved by numericalequivalence, we see that X s has a big line bundle ⇔ L i | X s is big for some i ⇔ L i | X s is very big for some i . This completes the case when R g ∗ O X is torsion free.In general, the torsion subsheaf of R g ∗ O X is supported on a Zariski closed,proper subset, hence (21.2) gives that if (21.1) does not hold then MO S ( X ) containsa Zariski dense open subset of S . (cid:3) Corollary 22.
Let g : X → S be a smooth, proper morphism of normal, irreducibleanalytic spaces whose fibers are Moishezon. Then g is locally Moishezon. Proof. If X s is Moishezon, then Hodge theory (8.6) tells us that H i ( X s , C ) → H i ( X s , O X s ) is surjective for every i . Thus R g ∗ O X is locally free by (24), hence(21.3) applies. (cid:3) There are many complex manifolds for which Hodge decomposition holds; theseare called cohomologically K¨ahler manifolds or ∂ ¯ ∂ -manifolds. We also get the fol-lowing variant. Corollary 23.
Let g : X → S be a smooth, proper morphism of normal, irreducibleanalytic spaces. Assume that MO S ( X ) contains a dense, open subset of S and allfibers are cohomologically K¨ahler. Then g is locally Moishezon. (cid:3) We have used the following result of [DJ74]; see also [Nak87, 3.13] and [Kol20,2.64].
Theorem 24.
Let g : X → S be a smooth, proper morphism of analytic spaces.Assume that H i ( X s , C ) → H i ( X s , O X s ) is surjective for every i for some s ∈ S .Then R i g ∗ O X is locally free in a neighborhood of s for every i . (cid:3) (Note that the proof in [DJ74] works by descending induction on i , so althoughwe are interested in the i = 2 case, we need the surjectivity of H i ( X s , C ) → H i ( X s , O X s ) for every i ≥ Example 25. (25.1) Let X → D be a universal family of K3 surfaces. A smooth,compact surface is Moishezon iff it is projective. The projective fibers of X → D correspond to a countable union of hypersurfaces H g ⊂ D .(25.2) Let E ⊂ P be a smooth cubic. Fix m ≥
10 and let X → D be theuniversal family of surfaces obtained by blowing up m distinct points p i ∈ E , andthen contracting the birational transform of E . (So D is open in E m .) Such asurface is projective iff there are positive n i such that P i n i [ p i ] ∼ nH where H isthe line class on P and n = P i n i .Here X → D is Moishezon and the projective fibers correspond to a countableunion of hypersurfaces H i ⊂ D . All fibers have log canonical singularities. OISHEZON MORPHISMS 9 Definition 26.
Let g i : X i → S be a proper morphisms. A bimeromorphic map φ : X X is fiberwise bimeromorphic if φ induces a bimeromorphic map φ s : X s X s for every s ∈ S .If X , X are fiberwise bimeromorphic then X s , X s are bimeromorphic to eachother for every s ∈ S , but this is only a sufficient condition in general.We study whether a flat, proper, Moishezon morphism g : X → D is fiberwisebimeromorphic to a flat, projective morphism g p : X p → D . The next examplessuggest that the answer is • negative if g is very singular, • positive if g is mildly singular, and • even if g is smooth, g p usually can not be chosen smooth. Example 27.
Let g : X → D be a smooth, projective morphism. Assume thatPic( X ) ∼ = Z but rank Pic( X ) ≥ Z ⊂ X be a smooth, ample divisor whose class is not in the image ofPic( X ) → Pic( X ). Blow up Z to get g ′ : X ′ → D . Here X ′ ∼ = X has normalbundle O X ( − Z ), hence it is contractible. We get a non-projective, Moishezonmorphism h : Y → D . Conjecture 27.1.
In most cases, h : Y → D is not fiberwise birational to a flat,projective morphism.The next result is a positive answer to the log terminal case of (5), provided X is not uniruled. See (32) for a discussion of the canonical case. Theorem 28.
Let g : X → D be a flat, proper, Moishezon morphism. Assume that (1) X has log terminal singularities and (2) X is not uniruled.Then g is fiberwise birational to a flat, projective morphism g p : X p → D (possiblyover a smaller disc) such that (3) X p0 has log terminal singularities, (4) X p s is not uniruled and has terminal singularities for s = 0 , and (5) K X p is Q -Cartier.Remark 28.6. Conjecturally we can also achieve that K X p is relatively nef. Themain obstacle is that (algebraic) minimal models are currently known to exist onlyin the general type case. (Proof of (28)) . The basic plan is similar to the proof of properness of the KSBmoduli space; see [KSB88, Sec.5] or [Kol20, Sec.2.5].We take a resolution of singularities Y → X such that Y → D is projective, andthen take a relative minimal model of Y → D . We hope that it gives what we want.There are, however, several obstacles. Next we discuss these, and their solutions,but for all technical details we refer to later sections.(29.1) We need to control the singularities of X . First (39) reduces us to thecase when K X is Q -Cartier. We assume this from now on. Then (30) implies thatthe pair ( X, X ) is plt.(29.2) After a base change z z r we get g r : X r → D . For suitable r , thereis a semi-stable, projective resolution h : Y → D ; we may also choose it to beequivariant for the action of the cyclic group G ∼ = Z r . All subsequent steps will be G -equivariant. We denote by X Y the birational transform of X and by E i theother irreducible components of Y .(29.3) We claim that Y s is not uniruled for s = 0. Indeed, for smooth familiesbeing uniruled is a deformation invariant property, and by Matsusaka’s theorem[Kol96, IV.1.7], we would get that X Y is uniruled. Thus K Y s is pseudo-effective by[BDPP13].(29.4) The required relative minimal model theorem is known only when thegeneral fibers are of general type. To achieve this, let H be an ample, G -equivariantdivisor such that Y + H is snc. For ǫ > Y, ǫH ) whose general fibers( Y s , ǫH s ) are of log general type since K Y s is pseudo-effective. For such algebraicfamilies, relative minimal models are known to exist [BCHM10]. We also know that( Y, Y + ǫH ) is dlt for 0 < ǫ ≪ φ : ( Y, ǫH ) ( Y m , ǫH m ) , and ( Y m , Y m0 + ǫH m ) is dlt. Here H m is Q -Cartier for general choice of ǫ by [Ale15,Lem.1.5.1], thus ( Y m , Y m0 ) is also dlt. Remark.
We have a choice here whether to take the minimal or the canonicalmodel. The minimal model has milder singularities, but it is not unique. Conjec-turally, the canonical model Y c is independent of 0 < ǫ ≪
1, but this is known onlyin dimensions ≤ φ contracts all the E i . Since ( X r , X ) is plt, all the E i have discrepancy > −
1. Thus the E i are contained in the restricted, relative baselocus of K Y + Y by (31.2). For ǫ small enough, the E i are also contained in therestricted, relative base locus of K Y + Y + ǫH by (31.1). Thus any MMP contractsthe E i . On the other hand, X Y can not be contracted, so X Y m is fiberwisebirational.(29.7) Note that h is smooth away from Y , thus ( Y s , ǫH s ) is terminal for s = 0and 0 ≤ ǫ ≪
1. Since H s is ample, we do not contract it, so ( Y m s , ǫH m s ) is stillterminal. Hence so is Y m s , giving (4).(29.8) As we noted, ( Y m , Y m0 ) is dlt, hence plt since Y m0 is irreducible. Thus Y m0 is log terminal by the easy direction of (30). (cid:3) The following results were also used in the proof of (28).
Proposition 30 (Inversion of adjunction I) . Let X be a normal, complex analyticspace, X ⊂ X a Cartier divisor and ∆ an effective R -divisor such that K X + ∆ is R -Cartier. Then ( X, X + ∆) is plt in a neighborhood of X iff ( X , ∆ | X ) is klt. Proof. The proof given in [Kol92, Sec.17] or [KM98, Sec.5.4] applies with minorchanges, using the complex analytic vanishing theorems proved in [Tak85] and[Nak87]. (cid:3) (Divisorial restricted base locus) . The basic theory is in [ELM +
09, 1.12–21] andan extension to the non-projective case is outlined in [FKL16, Sec.5].Let g : X → S be a proper, Moishezon morphism, X normal. The base locus ofa Weil divisor F is B ( F ) := Supp coker (cid:2) g ∗ g ∗ O X ( F ) → O X ( F ) (cid:3) . OISHEZON MORPHISMS 11
Its divisorial part is denoted by B div ( F ); we think of it as a Weil divisor.Let D be an R -divisor on X . Its stable divisorial base locus is the R -divisor B div ( D ) := lim m →∞ m B div ( ⌊ mD ⌋ ) , and its restricted divisorial base locus is B div − ( D ) := sup A B div ( D + A ) , where A runs through all big R -divisors on X that satisfy B div ( A ) = ∅ . This couldbe an infinite linear combination of prime divisors.An important observation of [FKL16] is that all the projective theorems onthe divisorial restricted base locus carry over to proper schemes and Moishezonvarieties. We need 2 properties:(31.1) Let X → S be a proper, Moishezon morphism, D an R -divisor on X ,and A a big R -divisor on X such that B div ( A ) = ∅ . Then, for every prime divisor F ⊂ X , coeff F B div − ( D ) = lim ǫ → coeff F B div − ( D + ǫA ) . (31.2) Let X i → S be proper, Moishezon morphisms, h : X → X a proper,bimeromorhic morphism, D a pseudo-effective, R -Cartier divisor on X , and E aneffective, h -exceptional divisor. Then B div − ( E + h ∗ D ) ≥ E. (Canonical case of Conjecture 5) . An argument similar to (29) should prove thecanonical case, but there are 3 difficulties.The reduction to the case when K X is Q -Cartier again follows from (39). Then weneed to show that the pair ( X, X ) is canonical. This is proved (though not stated)in [Nak04, 5.2]. This is also a special case of the general inversion of adjunction; aquite roundabout proof for Moishezon morphisms is given in (40).In (29) next we run the MMP for K Y + ǫH , which is the same as MMP for K Y + Y + ǫH since Y is numerically relatively trivial.In the canonical case we would need to run the MMP for K Y + X Y + η P E i + ǫH ,where we choose ǫ, η small, positive. The arguments of [KNX18] do not cover thiscase, but I expect that a method similar to [KNX18] would prove this.If X is of general type, then the canonical model of ( Y, X Y ) gives what we want.In general, arguing as in (36) we should get a minimal model g m : ( Y m , Y m0 + ǫH m ) → D . Here η P E m i is omitted since the E i get contracted. General theorytells us that discrep( Y m0 ) ≥ discrep( Y m0 , Diff Y m0 ǫH m ) ≥ − ǫ. We can choose ǫ arbitrarily small, but Y m0 may depend on ǫ , so we can not just takea limit as ǫ →
0. This is a problem that appears even if we start with a projective,algebraic family.At this point we could appeal to one of the ACC conjectures (33) which saysthat, for ǫ small enough, we must have discrep( Y m0 ) ≥
0. That is, Y m0 is canonical.The necessary result is known in dimensions ≤
3, but it is likely to be quitedifficult in general. So an alternate approach to our situation would be better. (A gap conjecture) . The following is a special case of [Sho96, Conj.4.2] (1) For every n ≥ ǫ ( n ) > X is an n -dimensionalvariety and discrep X > − ǫ ( n ), then in fact discrep X ≥ X hascanonical singularities).In dimension 2 this can be read off from the classifiation of log terminal singular-ities (these have discrep X > − ǫ (2) = and equalityholds for C / (1 , ǫ (3) = and the extremal case is the cyclic quotient singularity C / (3 , , Remark 34.
The deformation invariance of plurigenera for smooth, proper mor-phisms with Moishezon fibers is proved in [RT20, Thm.1.2].The canonical case of Conjecture 5 would show that the projective case im-plies the Moishezon case. However, the hard part in [RT20] is to show that g isMoishezon, so using Conjecture 5 would only yield a longer proof.5. Approximating Moishezon morphisms
We discuss 2 ways of approximating a projective (resp. Moishezon) morphism g : X → D by morphisms between projective (resp. Moishezon) varieties. Thisallows us to prove some results for Moishezon morphisms g : X → D . (Algebraic approximation of projective morphisms) . Let g : Y → D be a pro-jective morphism with relatively ample line bundle L . For later purposes we alsospecify a finite set of relative Cartier divisors E i ⊂ Y .Then ( Y , L := L | Y , E i := E i | Y ) is a projective, polarized scheme marked witheffective Cartier divisors. (For now Y can be even nonreduced.)( Y , L , E i ) has a universal deformation space G S : ( Y S , L S , E iS ) → S , where S, Y S are quasi-projective schemes, G S is flat and projective, L S is G S -ample andthe E iS are relative Cartier divisors.The original family gives a holomorphic φ S : D → S . Next we replace S first bythe Zariski closure of φ S ( D ) and then by its resolution. We obtain the followingdata.(1) A smooth C -variety B ,(2) a flat, projective morphism G : ( Y , L , E i ) → B , where L is G -ample, the E i are relative Cartier divisors, and(3) a holomorphic map φ : D → B ( C ),such that,(4) (cid:0) ( Y, L, E i ) → D (cid:1) ∼ = (cid:0) φ ∗ ( Y , L , E i ) → D (cid:1) and(5) φ ( D ) is smooth and Zariski dense in B .We call G : ( Y , L , E i ) → B an algebraic envelope of g : ( Y, L, E i ) → D .Note that we have no control over the dimension of B . However, if Y is smooth,then so is Y .Since B is smooth, the holomorphic curve φ ( D ) ⊂ B can be approximated byalgebraic curves to any order. Thus, for any fixed m > c, C ),(7) a flat, projective morphism ( Y C , L C , E iC ) → C , and(8) an isomorphism ( Y, L, E i ) m ∼ = ( Y C , L C , E iC ) m , OISHEZON MORPHISMS 13 where the subscript m denotes the m th order infinitesimal neighborhood of thecentral fibers.If m = 0 then we only get that the central fibers Y and ( Y C ) are isomorphic.The case m = 1 carries much more information: the smoothness of the total spacealong the central fiber and the normal bundles of the irreducible components of thecentral fiber are also preserved.As in [KNX18], algebraic envelopes can be used to show that MMP works forprojective morphism over Riemann surfaces. Proposition 36.
Let g : ( Y, ∆) → D be projective, locally stable with irreducible,normal general fibers. Assume that K Y + ∆ is big on almost every fiber. Then (1) there is a relative minimal model g m : ( Y m , ∆ m ) → D , and (2) the relative canonical model g c : ( Y c , ∆ c ) → D exists.Assume in addition that ( Y, Y +∆) is dlt and there is an irreducible divisor Y ∗ ⊂ Y such that Y \ Y ∗ is contained in the stable, relative base locus of K Y + ∆ . Then (3) Y ∗ Y m0 → Y c0 are birational, (4) ( Y m , Y m0 + ∆ m ) and ( Y c , Y c0 + ∆ c ) are plt, and (5) ( Y m0 , Diff Y m0 ∆ m ) and ( Y c0 , Diff Y c0 ∆ c ) are klt. (6) If ∆ is R -Cartier then ( Y m , Y m0 ) is plt and Y m0 is log terminal. (7) If the coefficients in ∆ are sufficiently general, then ( Y c , Y c0 ) is plt and Y c0 is log terminal. Proof. Claims (1–2) are basically proved in [KNX18]. Unfortunately, the mainresult [KNX18, Thm.2] is formulated to apply to the Calabi-Yau case. However,[KNX18, Props.8–14] contain a complete proof, though not a clear statement.Any MMP Y Y m contracts the stable base locus of K Y + ∆, thus Y m0 isirreducible and Y ∗ Y m0 is thus bimeromorphic. Also ( Y m , Y m0 + ∆ c ) is dlt,hence plt since Y m0 is irreducible. Since Y m → Y c does not contract Y m0 , we seethat ( Y c , Y c0 + ∆ c ) is also plt. This is (4), and (5) follows by the easy direction ofadjunction [Kol13, 4.8].If ∆ is Q -Cartier then so is ∆ m , hence (6) follows from [KM98, 2.27]. A similarargument works for (7) using [Ale15, Lem.1.5.1]. (cid:3) (Algebraic approximation of Moishezon morphisms) . Let f : X → D be aproper, Moishezon morphism and h : Y → X a proper morphism such that g := f ◦ h : Y → D is projective with relatively ample line bundle L . Also chooserelative Cartier divisors E i on Y . Assume also that X is seminormal (though thisis probably ultimately not necessary).We can apply (35.1–5) to get an algebraic envelope G : ( Y , L , E i ) → B . By[Art70], after an ´etale base change, we may assume that h : Y → X extends to H : Y → X where F : X → B is an algebraic space. Comment 37.1.
General extension theory, as in [Art70, MR71], tells us only thatwe have H : Y = Y h −→ X τ −→ X , where τ is a finite homeomorphism. Then we use that the functor of simultaneousseminormalizations is formally representable. The projective case is discussed in[Kol11]; see [Kol20, 9.61] for algebraic spaces. We assumed that we have a simulta-neous seminormalization over the completion of φ ( D ), which is Zariski dense. Thus X → B has seminormal fibers, hence X ∼ = X , as claimed. Assume next that fibers of f over D ◦ satisfy a property P that is Zariski openin families (for example smooth, normal or reduced). Then general fibers of F alsosatisfy P . As before, φ ( D ) ⊂ B can be approximated by algebraic curves to anyorder. Thus, for any fixed m > c, C ),(3) morphisms h c : ( Y C , L C , E iC ) → X C → C , where(a) g C : ( Y C , L C ) → C is flat, projective,(b) f C : X C → C is a flat algebraic space,(c) general fibers of f C satisfy P , and(4) an isomorphism (cid:0) ( Y, L, E i ) → X (cid:1) m ∼ = (cid:0) ( Y C , L C , E iC ) → X C (cid:1) m ,where the subscript m denotes the m th order infinitesimal neighborhood of thecentral fibers. Corollary 38.
Let f : X → D be a flat, proper, Moishezon morphism, X normal.Assume that it has a resolution h : Y → X where g := f ◦ h : Y → D is projectiveand Y a reduced, snc divisor. Then X has a canonical modification π : X c → X .(That is, X c has canonical singularities and K X c is π -ample.) Proof. Let H : Y → X and F : X → B be an algebraic envelope as in (37).Note that canonical modifications are unique and commute with ´etale morhisms.They exist for quasi-projective varieties over C by [BCHM10], hence every algebraicspace of finite type over C has a canonical modification.Let Π : X c → X denote the canonical modification of X . Since Y → B is locallystable, so is X c → B ; cf. [Kol20, Sec.4.8].By pull-back we get a locally stable morphism π : X c → X → D whose generalfibers are canonical. Since ( X c , X c0 ) is lc and X c0 is a Cartier divisor, we see that X c has canonical singularities. (cid:3) The following extends [KSB88, Sec.3] to Moishezon morphisms, see also [Kol20,Sec.5.5].
Corollary 39.
Let f : X → D be a flat, proper, Moishezon morphism. Assumethat X is log terminal. Then X has a canonical modification π : X c → X , X c0 islog terminal and π is fiberwise birational. Proof. After a ramified base change ˜ D → D with group G := Z /r , we can apply(38) to ˜ X → ˜ D to get ˜ π : ˜ X c → ˜ X .As in [Kol20, 5.32] we get that ˜ X c0 is log terminal and ˜ X c0 → ˜ X is birational.Set X c := ˜ X c /G .The base change group acts trivially on the central fiber ˜ X c0 , hence X c0 ∼ = ˜ X c0 isalso log terminal. Finally the pair (cid:0) ˜ X c , ˜ X c0 (cid:1) is log canonical, hence so is (cid:0) X c , X c0 (cid:1) by [KM98, 5.20]. Thus X c is canonical. (cid:3) Inversion of adjunction
The proof of the general inversion of adjunction theorem given in [Kol13, 4.9]relies on MMP, which is not known for projective morphisms over an analytic base.(See [Nak87] for the first steps and [Kol21] for some special cases.)We go around this for Moishezon morphisms using approximations.
OISHEZON MORPHISMS 15
Theorem 40.
Let g : X → D be a flat, proper, Moishezon morphism and ∆ aneffective Q -divisor on X . Assume that K X + ∆ is Q -Cartier. Then discrep( X, X + ∆ ) = totaldiscrep( X , ∆ ) , where on the left we use only those exceptional divisors E over X whose centers on X have nonempty intersection with X . Note that here the ≤ part is easy [Kol13, 4.8]. The known proofs of the ≥ partuse MMP, and the cases settled in [KNX18] do not seem enough.We start the proof of (40) with a discussion on snc divisors and then with ageneral result which says that discrepancies can be computed from the 1st orderneighborhood of the exceptional set. (Simple normal crossing divisors) . It would be convenient to recognize simplenormal crossing divisors (abbreviated as snc ) from an infinitesimal neighborhoodof the special fiber. At first sight, this seems impossible. Consider for example thefamily g : (cid:0) C , D := ( xy = z m +1 ) (cid:1) → C z , where D is not an snc divisor. The m th order infinitesimal neighborhood of the spe-cial fiber is defined by z m +1 = 0, hence isomorphic to the m th order neighborhoodof the snc family g : (cid:0) C , B := ( xy = 0) (cid:1) → C z . There is also the added problem that snc in the Euclidean topology is not the sameas snc in the Zariski topology. (For example, ( y = x + x is snc in the Euclideantopology but not in the Zariski topology.)We can, however, solve both problems by a simple bookkeeping convention.Let M be a complex manifold and { E i : i ∈ I } (reduced) divisors on M . We saythat ( M, E i : i ∈ I ) is a marked snc pair if for every p ∈ M there are(1) local analytic coordinates z , . . . , z n , and(2) an injection σ : { , . . . , r } ֒ → I for some 0 ≤ r ≤ m ,such that(3) E σ ( i ) = ( z i = 0) near p , and(4) the other E j do not contain p .With this definition we have the following. Claim 41.5.
Let E , . . . , E r , E r +1 , . . . , E m and E ′ r +1 , . . . , E ′ m be reduced divisorson a complex manifold M . Assume that(a) ( M, E + · · · + E m ) is a marked snc pair, and(b) E j and E ′ j have the same restriction on E ∪ · · · ∪ E r for all j > r .Then ( M, E + · · · + E r + E ′ r +1 + · · · + E ′ m ) is also a marked snc pair in a neighborhoodof E ∪ · · · ∪ E r . (cid:3) Lemma 42.
Let ( X, ∆ = P d j D j ) be a normal analytic pair such that K X + ∆ is Q -Cartier. Let B X ⊂ X be a Cartier divisor. Let p : ( Y, P i B i + P j ¯ D j + P ℓ E ℓ ) → ( X, B X + Supp ∆) be a log resolution, where B := P i B i = red p − ( B X ) , ¯ D j is the birational transformof D j and E ℓ are the other p -exceptional divisors.Then the discrepancies a ( ∗ , X, ∆) of all p -exceptional divisors (whose centershave nonempty intersection with B X ) can be computed from (1) B (2) := Spec O Y / O Y ( − B ) , (2) ¯ D j | B and E ℓ | B . Proof. After replacing X by a smaller neighbood of B X , we may assume that B is a deformation retract of Y . In particular, the centers of all p -exceptionaldivisors have nonempty intersection with B X , and numerical equivalence of divisorsis determined by their restriction to B .The discrepancies b i and e ℓ are uniquely determined by the conditions K Y + P ′ i b i B i + P j d j ¯ D j + P ℓ e ℓ E ℓ ≡ p p ∗ ( K X + ∆) , (42 . P ′ i we sum over the p -exceptional divisor in B . Restricting to B andusing adjunction we get P ′ i b i ( B i | B ) + P ℓ e ℓ ( E ℓ | B ) ≡ p − K B + ( B | B ) + ( p | B ) ∗ ( K X + ∆) . (42 . B (2) determines the B i | B and hence B | B . Thus the right hand side isknown and the b i , e ℓ are the unique solution to (42.4). (cid:3) Corollary 43.
Using the notation of (42), assume that there is another pair witha log resolution p ′ : ( Y ′ , P i B ′ i + P j ¯ D ′ j + P ℓ E ′ ℓ + F ′ ) → ( X ′ , B ′ X ′ + Supp ∆ ′ ) such that there is an isomorphism φ : (cid:0) B ′ (2) ← ֓ B ′ p ′ → B ′ X ′ (cid:1) ∼ = (cid:0) B (2) ← ֓ B p → B X (cid:1) , that sends ¯ D ′ j | B ′ to ¯ D j | B and E ′ ℓ | B ′ to E ℓ | B for every j, ℓ . Then (1) corresponding divisors have the same discrepancies, and (2) divisors in F ′ have discrepancy 0. Proof. Note that (42.4) gives us that P ′ i b i ( B ′ i | B ′ ) + P ℓ e ℓ ( E ′ ℓ | B ′ ) + 0 · F ′ ≡ p ′ − K B ′ + ( B ′ | B ′ ) + ( p ′ | B ′ ) ∗ ( K X ′ + ∆ ′ ) . Since this equation has a unique solution, b i , e ℓ give the discrepancies over X ′ . (cid:3) The following example illustrates the role of the divisor F ′ in (43). Example 44.
Let X = ( x − y + z = t ) ⊂ C , B = ( t = 0) and Y the smallresolution obtained by blowing up ( x − y = z − t = 0). (Here ∆ = 0 and E sempty.) Next set X ′ = ( x − y + z = 0) ⊂ C , B ′ = ( t = 0) and Y ′ the resolutionobtained by blowing up ( x = y = z = 0).The 1st order neigborhoods are isomorphic, but on Y ′ we have an exceptionaldivisor F ′ . Note that if we replace t by t m +2 , we have isomorphisms of m th orderinfinitesimal neighborhoods as well.Thus we can not tell whether a singularity is terminal or canonical by lookingat m th order infinitesimal neighborhoods for some fixed m . (Proof of (40)) . Write ∆ = P d j D j .Let h : ( Y, P ¯ D j ) → ( X, P D j ) be a log resolution of singularities such that Y → D is projective. Let ¯ D j denote the birational transform of D j , and let E i ⊂ Y be the exceptional divisors that dominate D . OISHEZON MORPHISMS 17
By (37.2–4) there are a smooth, pointed algebraic curve ( c, C ), a flat, propermorphism of algebraic spaces X a → C and a projective resolution h a : Y a → X a such that (cid:0) h a : ( Y a , P ¯ D a j + P E a j ) → X a (cid:1) ∼ = (cid:0) h : ( Y, P D j + P E i ) → X (cid:1) . (45 . h a ( E a j ) ∩ X a0 = h ( E j ) ∩ X , thus the E a j are h a -exceptional. (As in (44),there may be other h a -exceptional divisors.)By (41), ( Y a , P ¯ D a j + P E a j ) is also an snc pair, we are thus in the situation of(43). Since inversion of adjunction holds for the algebraic pair ( X a , X a0 + ∆ a ), italso holds for ( X, X + ∆). (cid:3) References [AK00] D. Abramovich and K. Karu,
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