Algebraic approximations of compact Kähler threefolds of Kodaira dimension 0 or 1
aa r X i v : . [ m a t h . AG ] O c t ALGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRADIMENSION 0 OR 1 by Hsueh-Yung Lin
Résumé . —
We prove that every compact Kähler threefold X of Kodaira dimension κ = Q -factorialbimeromorphic model X ′ with at worst terminal singularities such that for each curve C ⊂ X ′ , the pair ( X ′ , C ) admitsa locally trivial algebraic approximation such that the restriction of the deformation of X ′ to some neighborhoodof C is a trivial deformation. As an application, we prove that every compact Kähler threefold with κ = κ =
2, it su ffi ces to prove that of an elliptic fibration over a surface. From the point of view of the Hodge theory, compact Kähler manifolds can be considered as a naturalgeneralization of smooth complex projective varieties. While an arbitrarily small deformation as a complexvariety of a smooth complex projective variety might no longer be projective, a su ffi ciently small defor-mation of a Kähler manifold remains Kähler. The so-called Kodaira problem asks wether it is possible toobtain all compact Kähler manifolds through (arbitrarily small) deformations of projective varieties. Problem 1.1 (Kodaira problem) . —
Given a compact Kähler manifold X, does X always admit an (arbitrarilysmall) deformation to some projective variety ?
In dimension 1, compact complex curves are already projective. For surfaces, Problem 1.1 is knownto have a positive answer, first due to Kodaira using the classification of compact complex surfaces [ ],then to N. Buchdahl [ ] proving that any compact Kähler surface has an algebraic approximation using M.Green’s density criterion ( cf. Theorem 4.3). We refer to [
6, 9, 16, 7, 19 ] for other positive results.As for negative answers, C. Voisin constructed in each dimension ≥ ], thus answered inparticular negatively the Kodaira problem. Later on, she constructed in each even dimension ≥ ].For threefolds, the Kodaira problem remains open at present. There are nevertheless positive resultsconcerning a bimeromorphic variant of the Kodaira problem. Theorem 1.2 ( κ = : [ ] , κ = : [ ] ) . — Let X be a compact Kähler threefold of Kodaira dimension κ = or .There exists a Q -factorial bimeromorphic model X ′ of X with at worst terminal singularities such that X ′ has a locallytrivial algebraic approximation. LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 In order to prove Theorem 1.2, thanks to the minimal model program (MMP) for Kähler threefolds [ ],we can choose X ′ to be a minimal model of X , and this is what we did in most of the cases. Geometricdescriptions of these varieties X ′ can be obtained as an output of the abundance conjecture [ ] applied to X ′ , which is enough to prove the existence of a locally trivial algebraic approximation for X ′ .The aim of this article is to prove the following stronger version of Theorem 1.2 by further exploitingthe geometry of X ′ . We refer to Section 2.1 for the terminologies used in the statement of Theorem 1.3. Theorem 1.3 . —
Let X be a compact Kähler threefold of Kodaira dimension κ = or . There exists a Q -factorialbimeromorphic model X ′ with at worst terminal singularities such that whenever C ⊂ X ′ is a curve or empty, thepair ( X ′ , C ) has a locally trivial and C-locally trivial algebraic approximation. We will also prove a result relating the type of algebraic approximation that X ′ has in Theorem 1.3 andthe algebraic approximation of X . Proposition 1.4 . —
Let X be a compact Kähler threefold and X ′ a normal bimeromorphic model of X. If ( X ′ , C ) hasa locally trivial and C-locally trivial algebraic approximation whenever C ⊂ X ′ is a curve or empty, then X has analgebraic approximation. We refer to Corollary 2.4 for a more general statement. Since Q -factorial varieties are normal by definition,putting Proposition 1.4 together with Theorem 1.3 yields immediately the following result. Theorem 1.5 . —
Every compact Kähler threefold of Kodaira dimension 0 or 1 has an algebraic approximation.
As for threefolds of Kodaira dimension 2, since minimal models of such varieties are elliptic fibrations,the existence of algebraic approximations of these varieties is related to the following question.
Question 1.6 . —
Let f : Y → B be an elliptic fibration where Y is a compact Kähler and the base B is smoothand projective. Assume that the locus D ⊂ B parameterizing singular fibers of f is normal crossing, does Y have analgebraic approximation ?
We will see that a positive solution of Question 1.6 will eventually solve the Kodaira problem forthreefolds of Kodaira dimension 2.
Proposition 1.7 . —
If Question 1.6 has a positive answer in the case where B is a surface, then every compactKähler threefold of Kodaira dimension 2 has an algebraic approximation.
In view of [ , Theorem 1.1] and [ , Theorem 1.6], it is plausible that Question 1.6 would have a positiveanswer. It is a work in progress of Claudon and Höring toward an answer to Question 1.6.The article is organized as follows. We will first introduce in Section 2 some deformation-theoreticterminologies including those appearing in Theorem 1.3 then prove some general results. In particular, wewill prove Corollary 2.4 and deduce Proposition 1.4 from it. Next, we will turn to describing minimal modelsof a compact Kähler threefold of Kodaira dimension 0 or 1 in Section 3. According to these descriptions, wewill choose some threefolds X and prove in Section 4 that whenever C ⊂ X is a curve or empty, the pair ( X , C )always has a C -locally trivial algebraic approximation. Based on these results, the proof of Theorem 1.3will be concluded in Section 5, where we also prove Proposition 1.7. LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Let X be a complex variety. A deformation of X is a surjective flat holomorphic map π : X → ∆ containing X as a fiber. We say that a deformation π : X → ∆ is locally trivial if for every x ∈ X , there existsa neighborhood x ∈ U ⊂ X of x such that if U : = π − ( π ( x )) ∩ U , then U is isomorphic to U × π ( U ) over π ( U ).In this article, a fibration is a surjective holomorphic map f : X → B with connected fibers. A deformation π : X → ∆ of X is called strongly locally trivial with respect to the fibration structure f : X → B if π has afactorization of the form X ∆ × B ∆ q π pr such that the restriction of q to X onto its image coincides with f , and that for every ( t , b ) ∈ ∆ × B , thereexist neighborhoods b ∈ U ⊂ B and t ∈ V ⊂ ∆ such that q − ( V × U ) is isomorphic to q − ( { t } × U ) × V over V .Let X be a complex variety and C ⊂ X a subvariety of X . A C-locally trivial deformation of ( X , C ) is adeformation ( X , C ) → ∆ of the pair ( X , C ) such that the deformation ( U , C ) → ∆ restricted to someneighborhood U ⊂ X of C is isomorphic to the trivial deformation ( U × ∆ , C × ∆ ) → ∆ with U : = U ∩ X .An algebraic approximation of the pair ( X , C ) is a deformation ( X , C ) → ∆ of ( X , C ) such that there existsa sequence of points ( t i ) i ∈ N in ∆ parameterizing algebraic members and converging to o , the point whichparameterizes ( X , C ).If X is endowed with a G -action where G is a group and C is a G -invariant subvariety, then a G-equivariantdeformation of the pair ( X , C ) is a deformation ( X , C ) → ∆ of ( X , C ) such that the G -action on X extends toan action on X preserving each fiber of X → ∆ and C . The following lemma concerns the behaviour of C -locally trivial deformations of a pair ( X , C ) underbimeromorphic transformations. Lemma 2.1 . —
Let f : X → Y be a map between complex varieties and assume that there exists a subvariety C ⊂ Ysuch that f maps X \ D isomorphically onto Y \ C where D : = f − ( C ) . Then for every C-locally trivial deformation π : ( Y , C ) → ∆ of Y, there exists a D-locally trivial deformation ( X , D ) → ∆ of the pair ( X , D ) together with a mapF : X → Y over ∆ such that F − ( C ) = D and that F | X \ D is an isomorphism onto Y \ C .Proof . — Let U ⊂ Y be a neighborhood of C such that there exists an isomorphism over ∆ of the pairs( U , C ) ≃ ( U × ∆ , C × ∆ ) (2.1)where U : = Y ∩ U . So we can write Y ≃ (( Y \ C ) ⊔ ( U × ∆ )) / ∼ where ∼ glues the two pieces of the union using isomorphism (2.1). Isomorphism (2.1) also implies thatsince f maps X \ D isomorphically onto Y \ C , we have over ∆ U \ C ≃ f − ( U \ C ) × ∆ . (2.2) LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 We define X : = (cid:16) ( Y \ C ) ⊔ ( f − ( U ) × ∆ ) (cid:17) . ∼ and D : = D × ∆ ⊂ X where ∼ glues the two pieces of the union using isomorphism (2.2). One easily checks that X is Hausdor ff so that X is a complex variety. The map π ′ : Y \ C → ∆ and the projection π ′′ : f − ( U ) × ∆ → ∆ give rise toa map π X : ( X , D ) → ∆ which, by construction, is a D -locally trivial deformation of the pair ( X , D ). Finally the restriction of f to f − ( U ) defines an obvious map F : X → Y satisfying the property that F − ( C ) = D and that F | X \ D : X \ D ≃ Y \ C . (cid:3) Remark 2.2 . — We can also show that given a D -locally trivial deformation ( X , D ) → ∆ of the pair ( X , D ),there exists a C -locally trivial deformation ( X , C ) → ∆ of the pair ( X , C ) together with a map F : X → Y over ∆ such that F − ( C ) = D and that F | X \ D is an isomorphism onto Y \ C . This can be proven by exchangingthe role of C and D in the proof of Lemma 2.1.Let X be a compact Kähler manifold. Assume that X is bimeromorphic to a compact Kähler variety Y .After a sequence of blow-ups of X along smooth centers, we obtain a resolution X Z Y νη (2.3)of the bimeromorphic map X d Y . Let C ⊂ Y be the image of the exceptional set of ν . The following lemmashows in particular that a C -locally trivial deformation of the pair ( Y , C ) always induces a deformation of X . Lemma 2.3 . —
Suppose that π : ( Y , C ) → ∆ is a C-locally trivial deformation of the pair ( Y , C ) . Then up toshrinking ∆ , the deformation π induces a deformation X Z Y of (5.1) .Proof . — Since ν maps ν − ( Y \ C ) isomorphically onto Y \ C and since ( Y , C ) → ∆ is a C -locally trivialdeformation of the pair ( Y , C ), by Lemma 2.1 there exists a deformation Z → ∆ of Z and a map F : Z → Y over ∆ whose restriction to the central fiber is ν : Z → Y .As η ∗ O Z ≃ O X and R η ∗ O Z = η is a composition of blow-ups along smooth centers, by [ ,Theorem 2.1] the deformation Z → ∆ of Z induces a deformation Z → X of the morphism Z → X over ∆ up to shrinking ∆ . (cid:3) The following is an immediate consequence of Lemma 2.3.
Corollary 2.4 . —
With the same notation as above, if Y has a C-locally trivial algebraic approximation, then Xalso has an algebraic approximation. In particular, if Y is normal and satisfies the property that for every subvarietyC ⊂ Y whose irreducible components are all of codimension ≥ , the pair ( Y , C ) has a C-locally trivial algebraicapproximation, then X also has an algebraic approximation. LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Proof . — Let Y → ∆ be a C -locally trivial algebraic approximation of Y and let X Z Y be the induced deformation of (5.1) as in Lemma 2.3. Up to shrinking ∆ we can suppose that for each t ∈ ∆ , the fibers Z t → X t and Z t → Y t of the maps Z → X and Z → Y over t are both bimeromorphic.Therefore if over a point t ∈ ∆ the variety Y t is algebraic, then X t is also algebraic.For the last statement of Corollary 2.4, the normality of Y implies that each irreducible componentof the image in Y of the exceptional set E of ν is of codimension ≥
2. Thus ( Y , ν ( E )) has a ν ( E )-locallytrivial algebraic approximation by assumption. We conclude by the first part of Corollary 2.4 that X has analgebraic approximation. (cid:3) Proof of Proposition 1.4 . — Assume that X ′ satisfies the hypothesis made in the proposition. Let C : = ( C ⊔ C ) ⊂ X ′ be a subvariety of dimension ≤ C i denotes the union of the irreducible components of C of dimension i . Since dim C =
0, a locally trivial deformation of X ′ induces in particular a C -locally trivialdeformation of ( X ′ , C ). Hence by assumption, the pair ( X ′ , C ) has a C -locally trivial algebraic approximation.It follows from the second part of Corollary 2.4 that X has an algebraic approximation. (cid:3) G -equivariant locally trivial deformations The following lemma shows that given a G -equivariant C -locally trivial deformation ( X , C ) → ∆ of( X , C ), there always exists a G-equivariant trivialization of some neighborhood of C . This will imply that thequotient ( X / G , C / G ) → ∆ is a C / G -locally trivial deformation of ( X / G , C / G ). Lemma 2.5 . —
Let X be a smooth complex variety and G a finite group acting on X. Let C be a G-invariantsubvariety of X and assume that there exists a G-equivariant deformation of π : X → ∆ of X over a one-dimensionalbase ∆ . Assume also that there exists an open subset V ⊂ X and an isomorphism V ≃ V × ∆ over ∆ whereV : = V ∩ X such that V contains C (this hypothesis holds for instance, when π induces a G-equivariant C-locallytrivial deformation of ( X , C ) ), then up to shrinking ∆ , there exist C ⊂ X , a G-invariant neighborhood U of C , anda G-equivariant isomorphism ( U , C ) ≃ ( U × ∆ , C × ∆ ) over ∆ where U : = U ∩ X.In particular, π : ( X , C ) → ∆ is a G-equivariant C-locally trivial deformation of ( X , C ) and the quotient ( X / G , C / G ) → ∆ is a locally trivial and C / G-locally trivial deformation of ( X / G , C / G ) . Before proving Lemma 2.5, let us first prove a technical lemma.
Lemma 2.6 . —
Let G be a finite group acting on a variety X and let π : X → ∆ be a G-equivariant deformation ofX over a one-dimensional base. Let V ⊂ X be an open subset such that there exists an isomorphism V ≃ V × ∆ over ∆ where V : = V ∩ X. Let V G : = \ ∈ G ( V ) . Then for every G-invariant relatively compact subset U ⊂ V G : = V G ∩ X, up to shrinking ∆ there exists a G-invariantsubset U of V G and a G -equivariant isomorphism U ≃ U × ∆ over ∆ .Proof . — We may assume that V G , ∅ . Since V G is open by finiteness of G , after shrinking ∆ we canalso assume that the restriction of π to V G is surjective and that ∆ is isomorphic to the open unit disc LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 B (0 , ⊂ C such that 0 parameterizes the central fiber X . Fix a generator ∂∂ t of the space of constant vectorfields Γ ( ∆ , T ∆ ) const ≃ C on ∆ . For z ∈ C , let z ∂∂ t ∈ Γ ( ∆ , T ∆ ) const denote the corresponding vector field.By identifying V G with a subset of V × ∆ through the isomorphism V ≃ V × ∆ , we can define thehomomorphism of Lie algebras ξ : C → Γ ( V G , T V G ) z X ∈ G ∗ (cid:16) χ ( z ) | V G (cid:17) , (2.4)where χ ( z ) is the vector field on V × ∆ which projects to z ∂∂ t in ∆ and to 0 in V . By [ , Satz 3] (see also [ ,Theorem 5.3]), there exists a local group action Φ : Θ → V G of C on V G inducing ξ , where Θ ⊂ C × V G is a neighborhood of { } × V G . We recall that the meaning of alocal group action is the following.i) For all x ∈ V G , the subset Θ ∩ ( C × { x } ) is connected.ii) Φ (0 , • ) is the identity map on V G .iii) Φ ( h , x ) = Φ ( , Φ ( h , x )) whenever it is well-defined.iv) The morphism of Lie algebras C → Γ ( V G , T V G ) induced by Φ coincides with ξ .Since the vector field ξ ( z ) is G -invariant for all z ∈ C by construction, the map Φ is also G -equivariant(where G acts trivially on C ). Also since G acts on V G → ∆ in a fiber-preserving way, the projection of ξ ( z )in Γ ( V G , π ∗ T ∆ ) equals | G | · p ∗ (cid:16) z ∂∂ t (cid:17) . Hence if Φ ∆ denotes the local group action on ∆ defined by Φ ∆ : (Id C × π )( Θ ) → ∆ ( x , b ) b + | G | · x then we have the following commutative diagram. Θ V G (Id C × π )( Θ ) ∆ Φ π Φ ∆ (2.5)By the relative compactness of U inside V G , there exists ε > U : = B (0 , ε ) × U ⊂ Θ . The restriction of Φ to U is isomorphic onto its image. We verify easily with the help of (2.5) and theproperties ii) and iii) that the inverse of Φ : U → Φ ( U ) is Ψ : Φ ( U ) → U v π ( v ) | G | , Φ − π ( v ) | G | , v !! . LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Let U : = Φ (cid:16) B (cid:16) , ε | G | (cid:17) × U (cid:17) ⊂ V G . We have U : = U ∩ X by ii) and up to replacing ∆ by B (0 , ε ), we havethus by construction an isomorphism U × ∆ ∼ −→ U ( x , t ) Φ (cid:18) t | G | , x (cid:19) , over ∆ , which is moreover G -equivariant since Φ is G -equivariant. (cid:3) Proof of Lemma 2.5 . — Since C is G -invariant and since the subset V G : = T ∈ G ( V ) is a finite intersectionso is an open subset, V G : = V G ∩ X is a G -invariant neighborhood of C . Let U ⊂ V G be a G -invariantneighborhood of Y which is relatively compact in V G . By applying Lemma 2.6 to V and to U , we deducethat up to shrinking ∆ , there exists a G -invariant subset U ⊂ V G together with a G -equivariant isomorphism U × ∆ ≃ U over ∆ . As C is a G -invariant subset of U , the image C ⊂ U of C × ∆ under the above isomorphism isalso G -invariant. This proves that the G -equivariant isomorphism U ≃ U × ∆ induces a G -equivariantisomorphism of the pairs ( U , C ) ≃ ( U × ∆ , C × ∆ ), which is the main statement of the lemma.It follows by definition that π : ( X , C ) → ∆ is a G -equivariant C -locally trivial deformation of ( X , C ).Since X is smooth, up to further shrinking ∆ we can assume that X → ∆ is a smooth deformation, so thatthe quotient X / G → ∆ is a locally trivial deformation [ , Proposition 8.2]. As( U / G , C / G ) ≃ (( U / G ) × ∆ , ( C / G ) × ∆ )over ∆ , the deformation ( X / G , C / G ) → ∆ of the pair ( X / G , C / G ) is C / G -trivial. (cid:3) The following lemma is a special case of Lemma 2.5.
Lemma 2.7 . —
Let f : X → B be a G-equivariant fibration where G is a finite group. Let X ∆ × B ∆ q π pr be a G-equivariant strongly locally trivial deformation of f over a one-dimensional base ∆ . Suppose that C is aG-invariant subvariety of X and that f ( C ) is a finite set of points, then the deformation π : X → ∆ induces aG-equivariant C-locally trivial deformation ( X , C ) → ∆ of the pair ( X , C ) .Proof . — Let { p , . . . , p n } : = f ( C ) ⊂ B . By definition, up to shrinking ∆ , for each i there exists a neighborhood p i ∈ V i ⊂ B of p i such that the restriction of π : X → ∆ to V i : = q − ( ∆ × V i ) is isomorphic to ( V i ∩ X ) × ∆ over ∆ . Up to shrinking the V i ’s, we can assume that they are pairwise disjoint, so that V : = ⊔ ni = V i is isomorphicto V × ∆ over ∆ where V : = V ∩ X . Applying Lemma 2.5 to the G -equivariant deformation π : X → ∆ , the G -invariant subvariety C , and V yields Lemma 2.7. (cid:3) Remark 2.8 . — For simplicity, Lemma 2.6 is stated and proven under the assumption that dim ∆ = ∆ = LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 The reader is referred to [ ] for a survey of the minimal model program (MMP) for Kähler threefolds.Let X be a compact Kähler threefold with non-negative Kodaira dimension κ ( X ). By running the MMPon X , we obtain a Q -factorial bimeromorphic model X min of X with at worst terminal singularities (whichare isolated, since dim X =
3) whose canonical line bundle K X min is nef. Such a variety X min is called a minimal model of X . By the abundance conjecture, which is known to be true for Kähler threefolds, thereexists m ∈ Z > such that mK X min is base-point free and that the surjective map f : X min → B defined by thelinear system | mK X min | is a fibration satisfying dim B = κ ( B ) = κ ( X ). The fibration f : X min → B is called the canonical fibration of X min and a general fiber F of f satisfies O ( mK F ) ≃ O F by the adjunction formula.The aim of this section is to describe minimal models of non-algebraic compact Kähler threefolds ofKodaira dimension κ = κ = Proposition 3.1 . —
Let X be a non-algebraic compact Kähler threefold with κ ( X ) = and let X min be a minimalmodel of X. Then X min is isomorphic to a quotient e X / G by a finite group G where e X is either a -torus or a productof a K3 surface and an elliptic curve.Proof . — Since κ ( X ) =
0, there exists m ∈ Z > such that O ( mK X min ) ≃ O X min . Let π : e X min → X min be the index1 cover of X min : this is a finite cyclic cover étale over X \ Sing ( X ) such that K e X min ≃ O e X min [ , p. 159]. As X min has at worst terminal singularities, by [ , Corollary 5.21 (2)], the variety e X min has also at worst terminalsingularities. Since X is assumed to be non-algebraic, by [ , Theorem 6.1] e X min is smooth. Thus by theBeauville-Bogomolov decomposition theorem [ , Théorème 1], there exists a finite étale cover X ′ → e X min such that X ′ is either a 3-torus or a product of a K3 surface and an elliptic curve (as X ′ is non-algebraic, X ′ cannot be a Calabi-Yau threefold) ; let τ : X ′ → X min denote the composition of X ′ → e X min with π .The finite map τ is étale over X \ Sing ( X ). Let e X ◦ → X ′ \ Z → X \ Sing ( X ) be the Galois closure of τ | X ′ \ Z where Z : = τ − (Sing ( X )) and let G : = Gal (cid:16)e X ◦ / ( X \ Sing ( X )) (cid:17) . Since Sing ( X ) and hence Z are finite sets of points, we have π ( X ′ \ Z ) ≃ π ( X ′ ). It follows that e X ◦ → X ′ \ Z extends to e X → X ′ which is the finite étale cover associated to the subgroup Gal (cid:16)e X ◦ / ( X ′ \ Z ) (cid:17) < π ( X ′ \ Z ) ≃ π ( X ′ ). The variety e X is still a 3-torus or a product of a K3 surface and an elliptic curve. As e X \ e X ◦ is a set ofisolated points, the G -action on e X ◦ extends to a G -action on e X whose quotient is X min . (cid:3) Remark 3.2 . — The group G constructed in the proof of Proposition 3.1 acts freely outside of a finite set ofpoints of e X .For quotients ( S × E ) / G of the product of a non-algebraic K3 surface S and an elliptic curve E , we canshow that the G -action is necessarily diagonal. Lemma 3.3 . —
Let G be a group acting on S × E where S is a non-algebraic K3 surface and E is an elliptic curve.Then this G-action is the product of a G-action on S and a G-action on E.Proof . — For each ∈ G and each fiber F of the second projection p : S × E → E , since h , ( F ) < h , ( E ), itfollows that ( F ) is still a fiber of p . So the G -action on S × E induces a G -action on E . Suppose that thereexist ∈ G and a fiber E t of the first projection p : S × E → S such that ( E t ) is not contracted by p , then ifwe vary t ∈ S , we have a two-dimensional covering family of curves { E ′ t : = p ( ( E t )) } t ∈ S LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 on S generically of geometric genus 1. Since algebraic equivalence coincides with linear equivalence forcurves on a K3 surface and since there is only one-dimensional families of curves of geometric genus 1 ineach linear system, { E ′ t } t ∈ S is in fact a one-dimensional family of curves, say parameterized by some propercurve T . As S is non-algebraic, the family { E ′ t } t ∈ T is an elliptic fibration and there exists t ∈ T such that thenormalization e E ′ t of E ′ t is P .Let C ⊂ S be a curve such that for each p ∈ C , we have E ′ p = E ′ t . Since the curves ( E p ) ⊂ S × E are mutuallydisjoint for p ∈ C , their strict transformations ] ( E p ) in the normalization e E ′ t × E of E ′ t × E are also disjointfrom each other. It follows that [ ] ( E p )] = H ( e E ′ t × E , Z ) and since e E ′ t ≃ P , the curve ] ( E p ) has to be afiber of e E ′ t × E → e E ′ t . The latter is in contradiction with the assumption that ( E p ) is not contracted by p . (cid:3) Next we turn to varieties with κ = Theorem 3.4 . —
Let X be a non-algebraic compact Kähler threefold with κ ( X ) = . Let X min be a minimal model ofX and X min → B the canonical fibration of X min . Then X min → B satisfies one of the following descriptions :i) If a general fiber F of X min → B is algebraic, then F is either an abelian surface or a bielliptic surface ;ii) If F is not algebraic, then F is either a K3 surface or a 2-torus, and there exists a finite Galois cover e B → B of Band a smooth fibration e X → e B whose fibers are all isomorphic to F, such that e X is bimeromorphic to X min × B e Bover e B. Moreover, the monodromy action of π ( e B ) on F preserves the holomorphic symplectic form. Finally ifeither F is a K3 surface or X min contains a curve which dominates B, then there exists a finite Galois base changeas above such that e X → e B is isomorphic to the standard projection F × e B → e B.Proof . — Since X min has only isolated singularities, a general fiber F of X min → B is a connected smoothsurface. As K F is torsion, the classification of surfaces shows that F is either a K3 surface, an Enriquessurface, a 2-torus, or a bielliptic surface. Since X , and thus X min is non-algebraic, if F is algebraic then byFujiki’s result [ , Proposition 7] F is irregular, so F can only be an abelian surface or a bielliptic surface,which proves i ).Assume that F is not algebraic, then F is either a K3 surface or a 2-torus and by [ ], the fibration X min → B is isotrivial. By [ , Lemma 4.2], there exists some finite map e B → B of B and a smooth fibration e X → e B allof whose fibers are isomorphic to F , such that e X is bimeromorphic to X min × B e B over e B . Up to taking theGalois closure of e B → B , we can assume that e B → B is Galois.Since ˜ f is smooth and isotrivial, the fundamental group π ( e B ) acts on F by monodromy transformations.Since e X is assumed to be non-algebraic, we have H ( X , Ω X ) ,
0. Hence by the global cycle invariant theorem,the π ( e B )-action on F is symplectic.As e X is Kähler, again by the global cycle invariant theorem there exists a Kähler class on F fixed by theinduced monodromy action on H ( F , R ). It follows that the map π ( e B ) → Aut( F ) / Aut ( F ) has finite imagewhere Aut ( F ) denotes the identity component of Aut( F ) [ , Proposition 2.2].In the case where F is a K3 surface, Aut ( F ) is trivial, so π ( e B ) acts as a finite group on F . Accordinglyafter some finite base change of ˜ f : e X → e B , the fibration ˜ f becomes a trivial. Now assume that F is a 2-torusand that X min contains a curve dominating B . After another finite base change of ˜ f : e X → e B we can assumethat ˜ f has a section σ : e B → e X , namely ˜ f is a Jacobian fibration. Recall that π ( e B ) → Aut( F ) / Aut ( F ) hasfinite image, so after a further finite base change of ˜ f : e X → e B , we can assume that the monodromy actionof π ( e B ) on H ( F , Z ) is trivial. As ˜ f : e X → e B is a Jacobian fibration, we conclude that e X ≃ F × e B and that ˜ f isisomorphic to the projection F × e B → e B . LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 As before, both in the case where F is a K3 surface or a 2-torus, up to taking the Galois closure of e B → B we can assume that e B → B is Galois. (cid:3) In this section, we will prove for some compact Kähler threefolds X endowed with a G -action thatfor every G -invariant curve C ⊂ X , there exists a G -equivariant C -locally trivial algebraic approximationof the pair ( X , C ). Results in Section 3 show that the quotients X / G of these varieties cover all compactKähler threefolds of Kodaira dimension 0 or 1 up to bimeromorphic transformations, hence will allow usto conclude the proof of Theorem 1.3 in Section 5.Before dealing with threefolds, we start by proving analogue statements concerning the existence of a G -equivariant C -locally trivial algebraic approximation for fibrations admitting a strongly locally trivialalgebraic approximation and for surfaces in the next two subsections. Lemma 4.1 . —
Let X be a non-algebraic compact Kähler variety and f : X → B a surjective map onto a curvewith algebraic fibers. Suppose that X has a strongly locally trivial algebraic approximation π : X → ∆ with respectto f , then for any subvariety C ⊂ X, up to shrinking ∆ the deformation π induces a C-locally trivial algebraicapproximation of ( X , C ) .If moreover there exists a finite group G acting f -equivariantly on X and on B and the algebraic approximationof X in the assumption above is G-equivariant, then the induced C-locally trivial algebraic approximation is alsoG-equivariant for every G-invariant subvariety C.Proof . — Since X is non-algebraic and since the base and the fibers of f are algebraic, by Campana’scriterion [ , Corollaire in p.212] every subvariety of X (in particular C ) is contained in a finite number offibers of f . We can thus apply Lemma 2.7 to conclude. (cid:3) Corollary 4.2 . —
Let X be a non-algebraic compact Kähler variety and f : X → B a surjective map onto a curve. LetG be a finite group acting f -equivariantly on X and on B. Assume that a general fiber of f is an abelian variety, thenfor every G-invariant subvariety C ⊂ X, the pair ( X , C ) has a G-equivariant C-locally trivial algebraic approximation.Proof . — By [ , Theorem 1.6], the fibration f has a G -equivariant strongly locally trivial algebraic approxi-mation. Hence Corollary 4.2 follows from Lemma 4.1. (cid:3) First we recall some Hodge-theoretical criteria for the existence of an algebraic approximation.
Theorem 4.3 (Green’s criterion [ , Proposition 1] ) . — Let π : X → B be a family of compact Kähler manifoldsover a smooth base. If a fiber X = π − ( b ) satisfies the property that the composition of the Kodaira-Spencer map andthe contraction with some Kähler class [ ω ] ∈ H ( X , Ω X ) µ [ ω ] : T B , b H ( X , T X ) H ( X , T X ⊗ Ω X ) H ( X , O X ) KS ⌣ [ ω ] is surjective, then there exists a sequence of points in B parameterizing algebraic members which converges to b. The following is a variant of Theorem 4.3 when the variety X is endowed with a finite group action. LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Theorem 4.4 ( [ , Theorem 9.1] ) . — Let X be a compact Kähler manifold with an action of a finite group G. Supposethat the universal deformation space of X is smooth. If there exists a G-invariant Kähler class [ ω ] ∈ H ( X , Ω X ) suchthat the following composition of maps µ [ ω ] : H ( X , T X ) H ( X , T X ⊗ Ω X ) H ( X , O X ) ⌣ [ ω ] is surjective, then X has a G-equivariant algebraic approximation. The following is an easy application of Theorem 4.4.
Lemma 4.5 . —
Let S be a non-algebraic compact Kähler surface and G a finite group acting on S. If K S ≃ O S ,namely if S is either a K3 surface or a 2-torus, then S has a G-equivariant algebraic approximation.Proof . — Since S is a surface with trivial K S , the universal deformation space of S is smooth. Also, we havethe isomorphism T S ≃ Ω S defined by the contraction with a fixed holomorphic symplectic form. So for a G -invariant Kähler class [ ω ], the map µ [ ω ] defined in Theorem 4.4 with X → B replaced by the family ofK3 surfaces S U → U has the factorization µ [ ω ] : H ( S , T S ) ≃ H ( S , Ω S ) H ( S , Ω S ) ≃ H ( X , O X ) . ⌣ [ ω ] (4.1)Since [ ω ] ,
0, the map µ [ ω ] is non-zero. Moreover since h ( S , O S ) =
1, the map µ [ ω ] has to be surjective.Hence Lemma 4.5 is a consequence of Theorem 4.4. (cid:3) Lemma 4.6 and 4.7 concern C -locally trivial algebraic approximations of a pair ( S , C ) for K -trivial surfaces. Lemma 4.6 . —
Let S be a non-algebraic 2-torus and let G be a finite group acting on S. Let C ⊂ S be a G-invariantcurve. Then the pair ( S , C ) has a G-equivariant C-locally trivial algebraic approximation.Proof . — Since S is a non-algebraic 2-torus containing a curve, it is a smooth isotrivial elliptic fibration f : S → B and the only curves of S are fibers of f . As the G -action sends curves to curves, the fibration f is G -equivariant. We thus conclude by Corollary 4.2 that ( S , C ) admits a G -equivariant C -trivial algebraicapproximation. (cid:3) Lemma 4.7 . —
Let S be a non-algebraic K3 surface and let G be a finite group acting on S. Let C ⊂ S be a G-invariantcurve. Then ( S , C ) has a G-equivariant C-locally trivial algebraic approximation. When the algebraic dimension a ( S ) of S is zero, more precisely the deformation S → ∆ of S over the Noether-Lefschetz locus preserving the classes ofeach irreducible component of C in the universal deformation of S preserving the G-action is a G-equivariant C-locallytrivial algebraic approximation.Proof . — First we note that since H ( S , Z ) G is a sub- Z -Hodge structure of H ( S , Z ) of weight 2, if the G -actiondoes not preserve the holomorphic symplectic form, then H ( S , Z ) G is concentrated in bi-degree (1 , H , ( S ) G with the Kähler cone K S ⊂ H ( S , C ) is not 0, we deduce that H ( S , Z ) G containsa Kähler class, which is in contradiction with the hypothesis that S is non-algebraic. We deduce that the G -action preserves the holomorphic symplectic form of S .Since S is assumed to be non-algebraic, according to whether a ( S ) = a ( S ) = S is a disjoint union of trees of smooth ( − a ( S ) = S is an elliptic fibration f : S → B and the G -action sends fibers to fibers. LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 In the second situation, we can apply Corollary 4.2 to get a G -equivariant C -locally trivial algebraicapproximation of ( S , C ) as we did in the proof of Lemma 4.6.In the first situation, let us write C = ∪ i ∈ I C i where the C i ’s are irreducible components of C . Since theuniversal deformation space of S is smooth, its locus preserving the G -action can be identified with an opensubset of H ( S , T S ) G . As the group action G on S is symplectic, the isomorphism T S ≃ Ω S defined by thecontraction with a fixed holomorphic symplectic form induces an isomorphism H ( S , T S ) G ≃ H ( S , Ω S ) G . Under this identification, the universal deformation space ∆ of S preserving the G -action and the curveclasses [ C i ] can be identified with an open subset U of V : = H ( S , Ω S ) G ∩ < [ C i ] > ⊥ i ∈ I where < [ C i ] > i ∈ I denotes the linear subspace of H ( S , Ω S ) spanned by the classes [ C i ] and the orthogonalityis defined with respect to the cup product. Since < [ C i ] > i ∈ I is G -invariant and since the G -action preservesthe cup product, the orthogonal < [ C i ] > ⊥ i ∈ I is also G -invariant. Therefore V = < [ C i ] > ⊥ i ∈ I .Since S is not algebraic, the curve classes [ C i ] cannot generate the whole H ( S , Ω S ), hence V , v be a non-zero element in V . As C i < i , by the Hodge index theorem v >
0. If [ ω ] is aKähler class, then again by the Hodge index theorem we have v · [ ω ] ,
0. Using the factorization (4.1),we see again that since h ( S , O S ) =
1, the map µ [ ω ] defined in Theorem 4.3 with X → B replaced by the G -equivariant deformation S → ∆ of S over the Noether-Lefschetz locus ∆ , is surjective. Therefore byTheorem 4.3, S → ∆ is an algebraic approximation of S . Since the curve classes [ C i ] ∈ H ( S , C ) remainsof type (1 , S → ∆ induces for each i , a deformation ( S , C i ) of the pair ( S , C i ). It remains to show that( S , C : = ∪ i ∈ I C i ) → ∆ is a C -locally trivial deformation.Let us decompose C = ⊔ mi = C ′ i into its connected components. As we mentioned before, each C ′ i is a treeof smooth ( − ∆ , if C = ⊔ mi = C ′ i denotes thedecomposition of C into its connected components, then up to reordering the indices i , each fiber of C ′ i → ∆ is still a tree of ( − C ′ i .Since a tree of smooth ( − ν : S → S ′ over ∆ such that for each fiber S t of S → ∆ , the restriction of ν to S t is the contraction of C ′ i ∩ S t to a rational double point [ , Theorem 2]. Since fibers of C ′ i → ∆ are allisomorphic, the singularity type of ν ( C ′ i ∩ S t ) ⊂ S t does not depend on t ∈ ∆ . As the germs of a rationaldouble point of a fixed type on a surface are all isomorphic, up to shrinking ∆ there exists a neighborhood U i ⊂ S ′ of ν ( C ′ i ) such that the pair (cid:16) U i , ν ( C ′ i ) (cid:17) is isomorphic over ∆ to the trivial product (cid:16) U i × ∆ , ν ( C ′ i ) (cid:17) with U i : = U i ∩ ν ( S ). It follows that ( S ′ , ν ( C )) → ∆ is a ν ( C )-locally trivial deformation of ( ν ( S ) , ν ( C )), hence( S , C ) → ∆ is C -locally trivial by Lemma 2.1. (cid:3) For the sake of completeness, we conclude the present subsection by the following proposition whichwill not be used latter in the article. It is the generalization of [ , Lemma 5.1] in the G -equivariant setting. Proposition 4.8 . —
Let S be a compact Kähler surface and G a finite group acting on S. Whenever C ⊂ S is a curveor empty, the pair ( S , C ) has a G-equivariant C-locally trivial algebraic approximation.Proof . — We may assume that S is non-algebraic. If the algebraic dimension a ( S ) of S is 1, then S is anelliptic fibration and we can use Corollary 4.2 to conclude. If a ( S ) =
0, then the minimal model S ′ of S iseither a 2-torus or a K3 surface and the map ν : S → S ′ is G -equivariant. By Lemma 4.5, 4.6, and 4.7, the LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 pair ( S ′ , ν ( C )) has a G -equivariant ν ( C )-locally trivial algebraic approximation. Hence by Lemma 2.1, ( S , C )has a G -equivariant C -locally trivial algebraic approximation. (cid:3) Lemma 4.9 . —
Let X : = S × B where S is a non-algebraic K3 surface and B is a smooth projective curve. Let G be afinite group acting on B and on S and let G act on X by the product action. Whenever C ⊂ X is a G-invariant curveor empty, the pair ( X , C ) has a G-equivariant C-locally trivial algebraic approximation.Proof . — Let p : S × B → S denote the first projection. As the G -action on S × B is a product action, theimage C ′ : = p ( C ) is a G -invariant curve. By Lemma 4.5 and 4.7, there exists a G -equivariant C ′ -locallytrivial algebraic approximation π : ( S , C ′ ) → ∆ of the pair ( S , C ′ ). Let U ⊂ S be a neighborhood of C ′ such that there exists an isomorphism U ≃ U × ∆ over ∆ , so U × B × ∆ ≃ U × B over ∆ . Since U × B is a neighborhood of C and since C is G -invariant, Lemma 2.5 implies that the algebraic approximation Π : X : = S × B → ∆ of X defined by the composition of π with the first projection S × B → S induces a C -locally trivial algebraic approximation of ( X , C ). (cid:3) Before we study the existence of ( C -)locally trivial algebraic approximations of a pair ( X , C ) in the case of2-torus fibrations, let us first prove a statement concerning the existence of multisections of a torus fibration via strongly locally trivial perturbation. Lemma 4.10 . —
Let f : X → B be a smooth torus fibration whose total space X is compact Kähler. There exists anarbitrarily small strongly locally trivial deformation f ′ : X ′ → B of f such that f ′ has a multisection. Moreover if fis endowed with an f -equivariant G-action for some finite group G, then one can choose the above deformation to beG-equivariant.Proof . — The construction of an arbitrarily small deformation of f possessing a multi-section alreadyappeared in [ ]. We will recall how this deformation is constructed and prove that it is strongly locallytrivial along the way.Let J → B be the Jacobian fibration associated to f and J its sheaf of sections. The sheaf can be definedby the exact sequence 0 H Z E J H Z : = R − f ∗ Z and E : = H / H , − : = R − f ∗ C / R − f ∗ Ω X / B . To each isomorphism class of J -torsor : Y → B , one can associate in a biunivocal way, an element η ( ) ∈ H ( B , J ) satisfying the property that has a multisection if and only if η ( ) is torsion ( cf. [ , Section 2.2]). Moreover, if exp : V : = H ( B , E ) → H ( B , J ) denotes the morphism induced by the quotient E → J , then there exists a family X V × BV q π pr (4.2)of J -torsor such that for each v ∈ V , the element in V associated to the J -torsor π − ( v ) → B is η ( f ) + exp( v ). LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Concretely, the above family is constructed as follows. The map V → H ( B , J ) v η ( f ) + exp( v ) , defines an element η V ∈ Map (cid:16) V , H ( B , J ) (cid:17) ≃ H ( V , O V ) ⊗ H ( B , J ) ≃ H ( V × B , pr ∗ J )where Map (cid:16) V , H ( B , J ) (cid:17) denotes the space of holomorphic maps between V and H ( B , J ). So one canfind a covering ∪ ni = U i = B of B by open subsets such that η V represents a ˇCech 1-cocycle η Vij ∈ Γ ( V × U ij , pr ∗ J ) ≃ Map (cid:16) V , Γ ( U ij , J ) (cid:17) where U ij : = U i ∩ U j . Let us write X i : = f − ( U i ) and X ij : = f − ( U ij ) for all i and j . The 1-cocycle ( η Vij ) i , j definesthe transition maps V × X ij → V × X ij which are translations by η Vij and the family X → V × B is obtainedby glueing ( V × X i → V × U i ) i together using these transition maps. Since q − ( V × U i ) ≃ V × X i over V forall i , the family π : X → V is strongly locally trivial.If f : X → B is endowed with an f -equivariant G -action for some finite group G , then this G -actioninduces an action on E and on J . The restriction to the G -invariant subspace V G ⊂ V of (4.2) is adeformation of the J -torsor f : X → B preserving the equivariant G -action [ , Proposition 2.10]. The proofthat V G contains a dense subset of points parameterizing J -torsors having a multi-section is contained inthe proof of [ , Theorem 1.1], which we sketch now and provide necessary references for the detail.By Deligne’s theorem, W : = H ( B , H Z ) is a pure Hodge structure of degree 2 and concentrated inbi-degrees ( − , + , ), and ( + , −
1) [ , Section 2]. Let W K : = W ⊗ K for any field K . If F • W C denotes the Hodge filtration, then V is isomorphic to W C / F W C [ , Section 2]. Let µ : W R → V denote thecomposition µ : W R ֒ → W C → V . Using the Hodge theory we see easily that µ is surjective, so µ ( W Q ) is dense in V . Since G is finite, we have µ ( W G Q ) ⊗ R = µ ( W Q ) G ⊗ R = V G . Therefore µ ( W G Q ) is dense in V G .Using the assumption that X is Kähler, one can prove that the image of the G -equivariant class η G ( f ) ∈ H G ( B , J ) associated to X (which is a refinement of η ( f ), cf. [ , Section 2.4]) under the connection morphism H G ( B , J ) → H G ( B , H Z )is torsion [ , Proposition 2.11]. It follows that there exists m ∈ Z > and v ∈ V G such that m η ( f ) = exp( v ).Therefore η ( f ) + exp (cid:16) v − v m (cid:17) is torsion for each v ∈ µ ( W G Q ), so each of the fibrations X v → B parameterizedby the subset µ ( W G Q ) − v m ⊂ V G in the family (4.2) has a multisection. As we saw that µ ( W G Q ) ⊂ V G is dense, we conclude that the restrictionof (4.2) to V G is a deformation of f : X → B containing a dense subset of members having a multisection. (cid:3) Lemma 4.11 . —
Let f : X → B be a smooth isotrivial 2-torus fibration over a smooth projective curve B. Let G be afinite group acting f -equivariantly on X and on B such that X → B coincides with the base change of X / G → B / G by
LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 B → B / G. Whenever C ⊂ X is a G-invariant curve or empty, the pair ( X / G , C / G ) has a C / G-locally trivial algebraicapproximation.Proof . — First we assume that f does not have any multisection. In particular, the curve C is containedin a finite union of fibers of f . Using Lemma 4.10 there exists an arbitrarily small strongly locally trivial,so in particular C -locally trivial, deformation of f to some fibration which has a multisection. Thus up toreplacing f by this arbitrarily small deformation, we can assume that f has a multisection.Since f has a multisection, there exists a finite base change ˜ f : e X → e B of X → B such that e X ≃ S × e B where S is a fiber of f and that ˜ f is the second projection. After base changing with the Galois closure of e B → B / G , we can assume that e B → B / G is Galois whose Galois group is denoted by e G acting on e B and on S by monodromy transformations.Let e C be the pre-image of C under the map e X → X , which is e G -invariant by assumption. By Lemma 2.5,it su ffi ces to show that the pair ( e X , e C ) has a e G -equivariant e C -locally trivial algebraic approximation. Since S × e B → e B is isomorphic to the base change of e X / G → e B / G by e B → B / G , the e G -action on S × e B inducesa e G -action on S such that the first projection p : S × e B → S is e G -equivariant. As C is e G -invariant, thecurve C ′ : = p ( C ) is also e G -invariant. By Lemma 4.5 and 4.6, there exists a e G -equivariant C ′ -locally trivialalgebraic approximation ( S , C ′ ) → ∆ of the pair ( S , C ′ ). By repeating the same argument as in the proofof Lemma 4.9, we conclude that the deformation Π : S × e B → ∆ induces a e G -equivariant e C -locally trivialalgebraic approximation of the pair ( e X , e C ). (cid:3) Lemma 4.12 . —
Let X be a non-algebraic 3-torus and G a finite group acting on X. Then for every G-invariantcurve C ⊂ X, the pair ( X , C ) has a G-equivariant algebraic approximation.Proof . — First we assume that there exists a generically injective morphism ν : C ′ → X from a smoothcurve of geometric genus ≥ X . Since ν factorizes through C ′ → J ( C ′ ) j −→ X where J ( C ′ ) denotes theJacobian associated to C ′ , the 3-torus X contains an abelian variety of dimension ≥ j ( J ( C ′ )) ⊂ X .As X is non-algebraic, we have dim j ( J ( C ′ )) = X is a smooth isotrivial fibration f : X → B inabelian surfaces. As X is assumed to be non-algebraic, the G -action on X preserves the fibers of f . Hencewe can apply Corollary 4.2 to conclude that ( X , C ) has a G -equivariant algebraic approximation.Now assume that X does not contain any curve of geometric genus ≥
2, then C is a union of smoothelliptic curves. It follows that X is a smooth isotrivial elliptic fibration f : X → S . Moreover, the fibration f does not have any proper curve other than the fibers of f . Indeed, if such a curve C ′ exists, then for anyfiber F of f the image of α : C ′ × F → X defined by α ( x , y ) : = x + y is an algebraic surface, so necessarilycontains a curve of geometric genus ≥ X are fibers of f , the curve C is a union of fibers of f . It also follows that the G -action preserves the fibers of f , so induces a G -action on S . By Lemma 4.10, there exists an arbitrarilysmall G -equivariant strongly locally trivial, hence C -locally trivial, deformation f ′ : ( X ′ , C ) → B of f havinga multisection. For such an X ′ , we already saw that X ′ contains at least one curve of geometric genus 2, sothat ( X ′ , C ), and hence ( X , C ), have a G -equivariant C -locally trivial algebraic approximation. (cid:3) We can now conclude the proof of Theorem 1.3.
LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 Proof of Theorem 1.3 . — Let X be a non-algebraic compact Kähler threefold and let X ′ be a bimeromorphicmodel of X for which we wish to prove that whenever C ⊂ X ′ is a curve or empty, the pair ( X ′ , C ) has alocally trivial and C -locally trivial algebraic approximation. If the choice of X ′ is isomorphic to the quotient e X / G of some smooth variety e X by a finite group G , then first of all e X / G is Q -factorial. To prove that ( e X / G , C )has a locally trivial and C -locally trivial algebraic approximation, it su ffi ces by Lemma 2.5 to prove thatthe pair ( e X , e C ) has a G -equivariant e C -locally trivial algebraic approximation ( f X , e C ) → ∆ where e C is thepre-image of C under the quotient map e X → e X / G .If κ ( X ) =
0, we choose X ′ to be a minimal model of X . In particular, X ′ is Q -factorial and has at worstterminal singularities. By Proposition 3.1, the variety X ′ is a quotient e X / G by a finite group where e X iseither a non-algebraic 3-torus or the product of a non-algebraic K3 surface and an elliptic curve. If C = ∅ ,since X ′ is minimal, the existence of a locally trivial algebraic approximation of X ′ results from [ , Theorem1.4]. If C is a curve, the existence of a G -equivariant e C -locally trivial algebraic approximation of ( e X , e C ) is aconsequence of Lemma 4.12 or of Lemma 4.9 together with Lemma 3.3, according to wether e X is a 3-torusor the product of a non-algebraic K3 surface and an elliptic curve.If κ ( X ) =
1, then by Theorem 3.4 there are two cases to be distinguished. If we are in the first case ofTheorem 3.4, with the same notation therein we take X ′ = X min , so in particular X ′ is Q -factorial with atworst terminal singularities. Since the canonical fibration X min → B has a strongly locally trivial algebraicapproximation [ , Theorem 1.6, Corollary 6.2], we can apply Lemma 4.1 and deduce that ( X ′ , C ) has alocally trivial and C -locally trivial algebraic approximation for every curve C ⊂ X ′ . If we are in the secondcase of Theorem 3.4, with the same notation therein we take X ′ = e X / G where G : = Gal( e B / B ). By [ ,Proposition 4.7], the variety X ′ has at worst terminal singularities. The existence of a G -equivariant e C -locally trivial algebraic approximation of ( e X , e C ) is a consequence of Lemma 4.9 or Lemma 4.11, accordingto wether e X → e B is a fibration in K3 surfaces or 2-tori. (cid:3) As was mentioned in the introduction, the combination of Proposition 1.4 and Theorem 1.3 provesTheorem 1.5, the existence of an algebraic approximation of any compact Kähler threefold of Kodairadimension 0 or 1.Finally we prove Proposition 1.7, which concerns threefold of Kodaira dimension 2.
Proof of Proposition 1.7 . — As an output of the Kähler MMP for threefolds and the abundance theorem ( cf .the beginning of Section 3), a compact Kähler threefold X with κ ( X ) = X ′ → B ′ with X ′ being normal and B ′ a projective surface. Let X Y ′ X ′ νµ (5.1)be a resolution of the bimeromorphic map X d X ′ where µ is bimeromorphic. Since X ′ is normal, thereexists a subvariety C ⊂ X ′ of dimension at most 1 such that the restriction of ν to Y ′ \ ν − ( C ) is an isomorphismonto X ′ \ C . Accordingly f ′ : Y ′ → B ′ is still an elliptic fibration. Let D ′ ⊂ B ′ denote the locus parameterizingsingular fibers of f ′ and let ( B , D ) → ( B ′ , D ′ ) be a log-resolution of the pair ( B ′ , D ′ ). Let ν ′ : Y → Y ′ × B ′ B be a desingularization of Y ′ × B ′ B . As U : = Y ′ × B ′ ( B \ D ) → B \ D and hence U are smooth, we can assumethat the restriction of ν ′ to the Zariski open ν ′− ( U ) is an isomorphism onto U . It follows that Y → B is anelliptic fibration whose locus of singular fibers is contained in the normal crossing divisor D .Let η : Y → Y ′ × B ′ B → Y ′ → X denote the composition, which is bimeromorphic. Since both Y and X are smooth, we have η ∗ O Y = O X and R η ∗ O Y =
0. We can therefore apply [ , Theorem 2.1] as in the proof LGEBRAIC APPROXIMATIONS OF COMPACT KÄHLER THREEFOLDS OF KODAIRA DIMENSION 0 OR 1 of Lemma 2.3 to conclude that if Question 1.6 has a positive answer for the elliptic fibration Y → B , then X has an algebraic approximation by [ , Theorem 2.1]. (cid:3) Acknowledgement
The author is supported by the SFB / TR 45 "Periods, Moduli Spaces and Arithmetic of Algebraic Varieties"of the DFG (German Research Foundation). He would like to thank F. Gounelas, C.-J. Lai, S. Schreieder, A.Soldatenkov, and C. Voisin for questions, remarks, and general discussions on various subjects related tothis work.
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