Non-algebraic compact Kähler threefolds admitting endomorphisms
aa r X i v : . [ m a t h . AG ] N ov NON-ALGEBRAIC COMPACT KÄHLER THREEFOLDSADMITTING ENDOMORPHISMS
ANDREAS HÖRING AND THOMAS PETERNELL
Abstract.
We classify non-algebraic compact Kähler threefolds admitting anendomorphism f : X → X of degree at least two. Contents
1. Introduction 12. Notation and basic results 33. Compact Kähler threefolds 124. Torus fibrations 175. Non-uniruled manifolds 216. Uniruled manifolds 27References 381.
Introduction
An endomorphism of a compact complex manifold X is a surjective map f : X → X ,usually assumed to be of degree d at least two, i.e. automorphisms are excluded.Endomorphisms of projective manifolds were intensively studied in the last years[Bea01, Fuj02, Nak02, HM03, Ame03, FN05, FN07, NZ07, Nak08, AKP08, NZ09].For example if X is a smooth projective threefold and f is étale, X is completelyclassified up to étale cover. Also the higher-dimensional case is intensively treatedbut far from being completely understood.In this paper we classify all non-algebraic three-dimensional compact Kähler man-ifolds X admitting an endomorphism f regardless whether f is ramified or not.Before we state our classification results let us give an example how the non-algebraicity assumption gives additional restrictions on the existence of endomor-phisms. Let X be a non-algebraic compact Kähler manifold of dimension n which admits a meromorphic endomorphism f : X X of degree d > . Then κ ( X ) = n − . Date : July 21, 2009.2000
Mathematics Subject Classification.
Key words and phrases. endomorphism, compact Kähler manifold, non-algebraic manifold,torus fibrations, MMP. n fact if κ ( X ) = n − , the Iitaka fibration coincides with the algebraic reduction,which is an elliptic fiber space over a projective manifold of dimension n − . Theexistence of the endomorphism gives rise to a meromorphic multi-section of thefibration, and therefore any two points of X can be joined by a chain of compactcurves. Hence X is projective due to a result of Campana.If the manifold is not uniruled our results can be summarised as follows. Let X be a compact non-algebraic Kähler threefold which is notuniruled. Suppose that X admits an endomorphism f : X → X of degree d > . Then (up to étale cover) one of the following holds:1.) κ ( X ) = 0 : then eithera) X is a torus orb) X is a product Y × E where Y is bimeromorphic to a torus or K3 surfaceand E an elliptic curve.2.) κ ( X ) = 1 : then eithera) X is a product C × A where C is a curve of general type and A is atwo-dimensional torus of algebraic dimension at most one orb) X is a product E × S where E is an elliptic curve and S an ellipticsurface of algebraic dimension one. If f is ramified, it is a basic fact that K X is not pseudoeffective. If X is algebraic,then X is uniruled (cf. [BDPP04] which is based on a theorem of Miyaoka-Moriusing characteristic p methods). In the Kähler case the uniruledness is only knownin dimension three due to a remarkable result of Brunella [Bru06]. Let X be a compact non-algebraic Kähler threefold of algebraicdimension a ( X ) which is uniruled. Suppose that X admits an endomorphism f : X → X of degree d > . If f is ramified, then (up to étale cover) one of the followingholds.1.) a ( X ) = 0 : then X is a projectivised bundle P ( E ) over a torus A of algebraicdimension zero, f induces an endomorphism on A of degree at least two,and E is a direct sum of line bundles.2.) a ( X ) = 1 : then eithera) X is a product S × P , where S is a compact Kähler surface of algebraicdimension zero and f induces an automorphism on S orb) X is a projectivised bundle P ( E ) over a torus A of algebraic dimensionat most one, f induces an endomorphism on A of degree at least two,and E is a direct sum of line bundles.3.) a ( X ) = 2 : then eithera) X is a product Y × P where Y is a surface of algebraic dimension oneand f induces an automorphism on Y orb) X is a projectivised bundle P ( E ) over a torus A of algebraic dimensionone and f induces an endomorphism g of degree at least two on A .If f is étale, then X is (up to étale cover) a projectivised bundle P ( E ) over anon-algebraic torus A and c ( E ) = 4 c ( E ) . In the projective case Mori theory is heavily used to pass to minimal models. In theKähler case only rudiments of Mori theory are known (see Section 3.B), but they uffice in the special cases we are interested in. Moreover the algebraic reductionprovides a powerful tool. Guide to the reader.
In Section 2 we recall the basic definitions and gather variousresults which will be used at some point in the paper. In Section 3 we provenew statements in the Mori theory of compact Kähler threefolds which shouldbe interesting in their own right. Together with the results from [Pet98, Pet01]they allow us to establish a MMP for compact Kähler threefolds admitting étaleendomorphisms of degree at least two. A crucial point is that the contractionsnever contract a divisor to a point, so that we always stay in the smooth category.Section 4 provides results that would be trivial in the projective case: based on adiscussion of the fixed point set of torus endomorphisms, we establish the existenceof multisections for torus fibrations commuting with an endomorphism. Using atheorem of Nakayama and Zhang, the proof of Theorem 1.1 is then an easy exercise(cf. page 19). The Theorems 1.2 and 1.3 are proven in the Sections 5 and 6respectively. These two sections are the core of this paper, we advise the reader tostart here and skip the preceding technical sections for the first reading.
Acknowledgements.
We would like to thank T.-C. Dinh, C. Favre, P. Popescu-Pampu and N. Sibony for very helpful discussions on this subject. We also thankthe Research Group “Classification of Algebraic Surfaces and Compact ComplexManifolds” of the Deutsche Forschungsgemeinschaft DFG for financial support.2.
Notation and basic results
For standard definitions in complex geometry we refer to [Har77] or [KK83]. More-over we refer to [BHPVdV04] for basic results on surfaces and to [Fuj83] to theclassification theory of higher-dimensional non-algebraic varieties. Manifolds andvarieties are always supposed to be irreducible. We will always assume implicitlythat a compact Kähler manifold/surface/threefold is smooth. If a certain statementholds for a singular variety, we will mention what types of singularities are allowed.We say that a certain property holds for a general (resp. very general) point x ∈ X if there exists a finite (resp. countable) union of proper subvarieties of X such thatthe property holds for every point in the complement.2.A. Endomorphisms.2.1. Notation.
Let X be a compact complex variety that is normal or Gorenstein.An endomorphism is a holomorphic surjective map f : X → X . It is easy to seethat f is a finite map, so the ramification formula K X = f ∗ K X + R holds and we will call the support of R the ramification locus. The support of thecycle theoretic image B := f ∗ R will be called the branch locus. We will say that f is étale (in codimension one) if R is empty. Remarks.
1. If X is smooth, an endomorphism that is étale in codimension oneis étale in every point.2. Analogously one defines a meromorphic endomorphism of a compact complexvariety as a meromorphic dominant map f : X X . he following well-known results will be used many times in this paper: Let X be a compact Kähler manifold of dimension n , and let f : X → X be an endomorphism of degree d > . Then f is finite. The linear maps f ∗ : H ∗ ( X, Q ) → H ∗ ( X, Q ) and f ∗ : H ∗ ( X, Q ) → H ∗ ( X, Q ) are isomorphisms. More precisely we have f ∗ f ∗ = d Id .If f is étale, we have • χ ( X, O X ) = 0 , • e ( X ) := χ top ( X ) = c n ( X ) = 0 , and • K nX = 0 . Let X be a compact Kähler variety of dimension n , and let f : X → X be an endomorphism of degree d > . Let D be a Cartier divisor on X such that f ∗ D ≡ num mD for some m ∈ N . Then we have D n = 0 or d = m n .In particular if D is an effective divisor that is not contained in the branch locus of f and such that f − ( D ) = D , we have D n = 0 .Proof. Since f ∗ f ∗ = d Id , we see that m n D n = ( f ∗ D ) n = f ∗ ( D n ) = dD n , so the first statement is immediate. For the second statement observe that thehypothesis implies that f ∗ D ≃ D . (cid:3) The next statement shows that from the point of view of endomorphisms, it is quitenatural to treat separately uniruled and non-uniruled manifolds.
Let X be a smooth compact Kähler threefold and f : X → X be a ramified endomorphism of degree d > . Then X is uniruled.Proof. By [AKP08, Thm.4.1] the canonical divisor K X is not pseudo-effective, i.e.the class of K X is not contained in the closure of the Kähler cone. Therefore by[Bru06, Cor.1.2], X is uniruled. (cid:3) Fibrations.
A fibration is a proper surjective morphism ϕ : X → Y withconnected fibres from a complex manifold onto a normal complex variety Y . Afibre is always a fibre in the scheme-theoretic sense and will be denote by ϕ − ( y ) or X y . A set-theoretic fibre is the reduction of the fibre. The ϕ -smooth locusis the largest Zariski open subset Y ∗ ⊂ Y such that for every y ∈ Y ∗ , the fibre ϕ − ( y ) is a smooth variety of dimension dim X − dim Y . The ϕ -singular locus isits complement.Let us recall the rigidity lemma that will be used many times in this paper. Let f : X → Y and g : X → Z be fibrations. Suppose that for every z ∈ Z the fibre g − ( z ) is mapped by f onto a point. Then there exists a holomorphicmap h : Z → Y such that f = h ◦ g .If moreover g is flat the same conclusion holds if at least one g -fibre is contractedby f . .6. Definition. A meromorphic map ϕ : X Y from a compact Kähler man-ifold to a normal Kähler variety is almost holomorphic if there exist non-emptyZariski open subsets X ∗ ⊂ X and Y ∗ ⊂ Y such that ϕ | X ∗ : X ∗ → Y ∗ is a fibration.In particular for y ∈ Y a general point, the fibre ϕ − ( y ) exists in the usual senseand is compact. The importance of almost holomorphic maps is due to the fact that every compactKähler manifold admits such a fibration that separates the rationally connectedpart and the non-uniruled part: the rationally connected quotient . [Kol96, Thm.5.4] , [Cam04, Thm. 1.1] , [GHS03] Let X be acompact Kähler manifold. Then there exists an almost holomorphic fibration ϕ : X Y onto a normal compact Kähler variety Y such that the general fi-bre is rationally connected and the variety Y is not uniruled. This map is uniqueup to meromorphic equivalence of fibrations [Cam04] and will be called the rationallyconnected quotient. Endomorphisms that preserve fibrations.2.8. Definition.
Let X be a compact Kähler manifold, and let f : X → X be anendomorphism of degree d > . Suppose that X admits a fibration ϕ : X → Y ontoa normal Kähler variety Y . If there exists an endomorphism g : Y → Y such that g ◦ ϕ = ϕ ◦ f , we say that f preserves the fibration and g is the endomorphisminduced by f on Y . All the natural fibrations attached to a variety are preserved by an endomorphism.
Let X be a compact Kähler manifold, and let f : X → X be anendomorphism of degree d > .1. Let α : X → Alb ( X ) be the Albanese map. Then there exists an endomorphism g : Alb ( X ) → Alb ( X ) such that g ◦ ϕ = ϕ ◦ f .2. Let ϕ : X Y be the Iitaka fibration. Then there exists a automorphism g : Y → Y such that g ◦ ϕ = ϕ ◦ f .3. Let ϕ : X Y be the rationally connected quotient of X . Then there exists ameromorphic map g : Y Y such that g ◦ ϕ = ϕ ◦ f .Proof. The first statement follows immediately from the universal property of theAlbanese map, the second statement can be shown as in [Fuj02, Prop.2.5] where theprojectiveness assumption is actually not used. For the last statement, note thatup to replacing Y by a Zariski open dense subset, we may suppose that the almostholomorphic map ϕ is holomorphic. By the rigidity lemma it is sufficient to showthat a general ϕ -fibre F is contracted by ϕ ◦ f . Since F is rationally connected and Y is not uniruled, this is trivially true. (cid:3) Before we can prove an analogue of Proposition 2.9 for the algebraic reduction, weneed one more definition. The statement in [Kol96] is in the algebraic setting, but the same proof goes through inthe compact Kähler category: the main technical tool [Cam04, Thm. 1.1] holds in this largergenerality. .10. Definition. Let X be a compact Kähler manifold. An integral effectivedivisor D ⊂ X is polar if there exists a meromorphic function ψ on X such that D ⊂ div ( ψ ) . If X is projective, every divisor is polar. If the algebraic dimensionis zero, every divisor is non-polar. Let X be a compact complex manifold that admits a mero-morphic endomorphism f : X X . Denote by ϕ : X Y the algebraicreduction of X . Then there exists a meromorphic endomorphism g : Y → Y suchthat ϕ ◦ f = g ◦ ϕ . Remark. If X has algebraic dimension one the variety Y is a smooth compactcurve, so the meromorphic endomorphism extends to a holomorphic map. Proof.
Up to replacing X by some bimeromorphic model we may suppose that thealgebraic reduction is holomorphic. The endomorphism f acts by pull-back on themeromorphic function field C ( X ) = C ( Y ) , and we define g to be the meromorphic map corresponding to f ∗ : C ( Y ) → C ( Y ) .Since every polar divisor on X is contained in a pull-back from Y and the pull-backof a polar divisor is polar, one sees easily that ϕ ◦ f = g ◦ ϕ . (cid:3) Let Y be a normal compact Kähler variety and let g : Y → Y be an endomorphism of degree at least two. We say that g is totally ramified in apoint y ∈ Y , if the set-theoretical fibre ( g − ( y )) red is a singleton. Let X be a compact Kähler manifold, and let f : X → X bean endomorphism. Let ϕ : X → Y be a surjective morphism onto a normal Kählervariety such that there exists an endomorphism g : Y → Y such that g ◦ ϕ = ϕ ◦ f .1. For m ∈ N , set T m := { y ∈ Y | dim ϕ − ( y ) > m − } . Then we have (set-theoretically) g − ( T m ) = T m . If T m is finite and g of degree atleast two, then g is totally ramified in every point of T m .2. Set R := { y ∈ Y | ϕ − ( y ) is reducible } . Then we have (set-theoretically) g − ( R ) = R . If R is finite and g of degree at leasttwo, then g is totally ramified in every point of R .3. Let ∆ := { y ∈ Y | ϕ − ( y ) is singular } be the ϕ -singular locus and denote by R the branch locus of g . Then g − (∆) iscontained set-theoretically in the union of ∆ and the branch locus R . If g is étale,then (set-theoretically) g − (∆) = ∆ .In particular if g is an étale map of degree at least two, then T m , R and ∆ areeither of positive dimension or empty. roof.
1. Since g − ( T m ) has at least as many irreducible components as T m , it issufficient to show that g − ( T m ) ⊂ T m . Yet if t ∈ T m and y ∈ g − ( t ) , then ϕ − ( y ) surjects via f on ϕ − ( t ) , so dim ϕ − ( y ) ≥ dim ϕ − ( t ) > m − . If T m is finite, then g − ( T m ) ≥ T m and equality holds if and only if g is totally ramified in everypoint of T m .2. Analogous to 1.3. If t ∈ ∆ and y ∈ g − ( t ) such that y R , then for every x ∈ ϕ − ( y ) we have rk T ϕ,x = rk T g ◦ ϕ,x = rk T ϕ ◦ f,x ≤ rk T ϕ,f ( x ) . Since ϕ − ( t ) is singular, there exists a point f ( x ) ∈ ϕ − ( t ) such that the tangentmap does not have maximal rank. It follows that y ∈ ∆ . (cid:3) Let X be a compact Kähler variety, and let f : X → X be anendomorphism of degree d > . Suppose that there exists a fibration ϕ : X → C onto a smooth curve C and an automorphism g : C → C such that g ◦ ϕ = ϕ ◦ f . If B i is an irreducible component of the branch locus of f , then B i surjects onto C .Proof. By Proposition 2.14 the ϕ -singular locus ∆ satisfies g − (∆) = ∆ . Since ∆ is finite, we can suppose, up to replacing f by f k that g is the identity on ∆ .We argue by contradiction and suppose that ϕ ( B i ) = c , then it is an irreduciblecomponent of the fibre X c ≃ ϕ ∗ c . Suppose first that X c is not a smooth fibre, then c ∈ ∆ which implies g ∗ c = c . Therefore we have X c ≃ ϕ ∗ g ∗ c ≃ f ∗ ϕ ∗ c ≃ f ∗ X c , in particular m i B i ≃ f ∗ B i ≃ B i . Since the ramification index m i is strictly largerthan one, this implies that B i is homologous to zero, a contradiction. If X c is asmooth fibre, then X c = B i and g ∗ c = c ′ with c ′ ∆ . The same computation showsthat X c ′ ≃ f ∗ B i = m i B i . Since X c ′ is smooth, so reduced, we get a contradiction. (cid:3) The following statement generalises [Ame03, Thm.1], its proof is a mere adaptationof Amerik’s proof to our context.
Let X be a compact Kähler manifold, and let f : X → X bean endomorphism of degree d > . Suppose that there exists a fibration ϕ : X → Y onto a normal variety Y such that • there exists an automorphism g such that g ◦ ϕ = ϕ ◦ f , • the general ϕ -fibre is isomorphic to P r , • the ϕ -singular locus ∆ has codimension at least two in every point, and • any finite étale cover Y ′ → Y \ ∆ extends (maybe after a further finite étalecover) to a finite map Y ′ → Y .Then there exists a finite map Y ′ → Y such that X × Y Y ′ is bimeromorphic to Y ′ × P r . In particular we have a ( X ) = a ( Y ) + r . If Y is smooth, the last condition in the proposition is automati-cally satisfied. In fact we have π ( Y \ ∆) ≃ π ( Y ) , so we can even extend by an étale map. roof. Since g is an automorphism, the restriction of f to a general fibre inducesan endomorphism of degree d > and up to replacing f by some f k , we cansuppose without loss of generality that d > r + 1 . Then by [Ame03, Prop.1.1]the space of endomorphism of P r of degree d has an affine geometric quotient R m ( P r , P r ) /P GL ( r + 1) ⊂ C N . Thus the fibration ϕ induces a holomorphic map ( Y \ ∆) → R m ( P r , P r ) /P GL ( r + 1) ⊂ C N . Since Y is normal, the map extends toa holomorphical map Y → C N by Hartog’s theorem. Since Y is compact, this mapis constant. Arguing as in Amerik’s proof of [Ame03, page 22, line7ff], we see thatthere exists an étale cover Y ′ → Y \ ∆ such that the fibre product X × Y \ ∆ Y ′ isisomorphic to Y ′ × P r . By the last condition, we know that (up to replacing Y ′ by some higher étale cover), the étale cover extends to a finite map Y ′ → Y . Byconstruction we then have a holomorphic map Y ′ × P r ֒ → X × Y Y ′ , and since ( Y ′ × P r ) \ ( Y ′ × P r ) has codimension at least two, we can apply [Gun90,Ch.P.,Thm.10] to get a bimeromorphic map Y ′ × P r X × Y Y ′ . (cid:3) Auxiliary results on compact Kähler surfaces.
Recall that by the clas-sification of surfaces a compact complex surface of algebraic dimension zero thatis in the Fujiki class is bimeromorphic to a torus or a K3 surface. The followingtechnical lemma is well-known to experts, but for the convenience of the reader weinclude it and its (easy) proof.
Let S be a normal complex compact surface of algebraic dimensionzero that is in the Fujiki class. Then there exists a bimeromorphic map S → S min onto a normal surface S min that does not contain any curves.If S is bimeromorphic to a torus, then S min is a torus. If S is bimeromorphic to aK3 surface, then S min has at most rational double points.Moreover if D is an effective, non-trivial Cartier divisor on S then D < . Remark. If S is bimeromorphic to a K3 surface, we will call S min a singular K3surface. Although this has a completely different meaning in the theory of latticesof K3 surfaces, we hope that no confusion will arise. Proof.
Suppose first that S is smooth, and let S → S ′ be the minimal model of S .If S ′ is a torus, it has no curves since the algebraic dimension is zero. Thus in thiscase we can just set S min = S ′ .If S ′ is a K3 surface, we proceed as follows: let D be an effective divisor on S ′ .Since the algebraic dimension is zero, we have h ( S ′ , D ) = 1 and h ( S ′ , D ) = 0 bySerre duality, so the Riemann-Roch formula yields − h ( S ′ , D ) = 12 D + 1 . In particular we have D ≤ . On the other hand by the adjunction formula p a ( D ) − D , so the non-negativity of the arithmetic genus p a ( D ) impliesthat D = − . If we apply this to effective reduced divisors D with one, two andthree irreducible components, we see that every irreducible curve is smooth and somorphic to P , two curves are disjoint or meet transversally in exactly one pointand three curves never meet in the same point. Thus by [BHPVdV04, Lemma 2.12]the configuration of curves are of type A-D-E. Since there are only finitely manydivisors on S [FF79, Thm.], we can contract all the divisors by Grauert’s criterionand obtain a normal surface S min with at most rational double points.If S is singular we apply the first step to some desingularisation: thus there existsa meromorphic map µ : S S min . Let Γ ⊂ S × S min be the graph of the map,then the projection Γ → S is bimeromorphic and has connected fibres by Zariski’slemma. Yet any positive-dimensional fibre would be a curve in S min , so Γ ≃ S and µ extends to an isomorphism.Since any effective divisor on S is contracted by µ to a point, the last statement isimmediate from the easy direction of Grauert’s criterion. (cid:3) In the situation of the Lemma 2.18, set S := S min \ { p , . . . , p r } and let µ : S ′ → S be an étale morphism. Then µ extends to a finite map µ : S ′ → S min .Proof. It is sufficient to show that we can extend µ locally. Yet by the lemma, thesurface S min has at most rational double points and these are quotient singularitiesof the form D /G where G is a finite group. Thus if we take an étale cover of D \ (0 , /G , we can lift it to the universal cover D \ (0 , and extend by theinclusion D \ (0 , → D . (cid:3) Since a smooth K3 surface is simply connected, it does not admit a (necessarilyétale) endomorphism of degree at least two. This is no longer true for the singularK3 surface S min . Let A be a two-dimensional torus of algebraic dimension zeroand let S be the corresponding Kummer surface. It is not hard to see that theKummer quotient A/ Z is the surface S min . The multiplication by n ∈ N gives anendomorphism of degree n of the torus A which descends to an endomorphism of A/ Z of degree at least two.The following proposition shows that the example is essentially everything that canhappen. Let S be a singular K3 surface of algebraic dimension zero,and let f : S → S be an endomorphism of degree d > . Then there exists a Galoiscovering ν : A → S by a torus that is étale in codimension one such that f lifts toan endomorphism f A : A → A of degree d . The proof is essentially a reproduction of the arguments used in [Nak08] for thealgebraic case. For the reader’s convenience, we give the basic ideas:
Proof.
By Lemma 2.18, the surface S has only rational double points, so it isGorenstein and has only isolated quotient singularities. Since S contains no curves,the endomorphism f is necessarily étale in codimension one. Thus by [Nak08,Lemma 3.3.2] there exists a finite Galois covering ν : A → S that is étale incodimension one such that A is smooth and e ( A ) = 0 . Moreover κ ( A ) ≥ and A has algebraic dimension zero, so it is a torus by the classification of compact omplex surfaces. Replacing the covering ν by a suitable one, we may suppose that deg ν ≤ deg ν ′ for every Galois covering ν ′ : A ′ → S by a torus A ′ that is étalein codimension one. Arguing as in [NZ07, Lemma 2.6], one sees that such a ν isunique up to an isomorphism over S . Let W be the normalisation of an irreduciblecomponent of the fibre product A × S S such that the natural morphisms f ′ : W → A and ν ′ : W → S are surjective. Then we have a commutative diagram W f ′ (cid:15) (cid:15) ν ′ / / S f (cid:15) (cid:15) A ν / / S and f ′ is étale in codimension one. The variety A being smooth the map f ′ is étale,so W is a torus. By construction, we have deg ν ′ ≤ deg ν so the minimality of ν implies that there exists an isomorphism ψ : A → W such that ν ′ ◦ ψ = ν . Themorphism f A := f ′ ◦ ψ has the stated properties. (cid:3) We would like to thank C. Favre for suggesting to us the proof of the next statement.
Let S be a compact Kähler surface that admits a relativelyminimal elliptic fibration ϕ : S → C ≃ P . Suppose that S admits a meromorphicendomorphism f : S S of degree d > such that there exists an endomorphism g : C → C of degree at least two such that g ◦ ϕ = ϕ ◦ f . Then S is algebraic.Proof. We argue by contradiction and suppose that S has algebraic dimension one.Since g has degree at least two, there are infinitely many ϕ -fibres that are isomorphicelliptic curves. Since the j -invariant yields a holomorphic map j : C → P , it mustbe constant. Thus all the smooth fibres are isomorphic and a look at the localbehaviour of the j -invariant near the singular fibres [BHPVdV04, V.10,Table 6]shows that the singular fibres are multiple elliptic curves. Since S contains nocurve that maps onto C , this implies that S contains no rational curves. Thus themeromorphic endomorphism extends to a holomorphic endomorphism f : S → S of degree d > . Since g is an endomorphism of P it is ramified, so f is ramified.Thus S is uniruled by [AKP08, Thm.4.1], a contradiction. (cid:3) We will need the following generalisation of [AKP08, Thm.4.1] for singular surfaces.
Let X be a compact Kähler threefold, and let f : X → X be anendomorphism of degree d > . Let D ⊂ X be an irreducible divisor such that f − ( D ) = D and such that f D : D → D has degree at least two. If f D is ramified,the surface D is uniruled.Proof. We claim that the canonical divisor K D is not pseudoeffective. Assumingthis for the time being, let us show how to conclude: let ν : ˜ D → D be thenormalisation, then K ˜ D = ν ∗ K D − N where N is an effective divisor. Let π : D ′ → ˜ D be the minimal resolution, then K D ′ = π ∗ K ˜ D − N ′ where N ′ is an effective divisor. Thus K D ′ = π ∗ ν ∗ K D − N ′ − π ∗ N s not pseudoeffective, since K D is not pseudoeffective. Since D ′ is a compactKähler surface, this implies that D ′ is uniruled. Proof of the claim:
We start by establishing a ramification formula on D . Since f − ( D ) = D , we have f ∗ D = mD where m ∈ N is the order of ramification along D . Thus the ramification divisor R of f is of the form R = ( m − D + R ′ where R ′ does not contain D . Since f D is ramified, the restriction R ′ D := R ′ ∩ D isan effective, non-trivial Cartier divisor. By the ramification formula on X we have K X = f ∗ K X + R , so the adjunction formula K D = K X | D + D D implies K D − D D = f ∗ D ( K D − D ) + R D = f ∗ D K D − mD D + ( m − D D + R ′ D , so K D = f ∗ D K D + R ′ D . We proceed now as in the proof of [AKP08, Thm.4.1]: let f m be the m -th iterateof f , then the ramification formula reads K D = f ∗ m K D + f ∗ m − R ′ D + . . . + f ∗ R ′ D + R ′ D . Let ω D be the restriction of a Kähler form ω to D . There exists a c > suchthat for every pseudoeffective line bundle L on D , we have L · ω > c . If K D ispseudoeffective, then f ∗ m K D · ω ≥ , so K D · ω ≥ mc for arbitrary m which is impossible. (cid:3) Let S be a complex Gorenstein surface, i.e. the canonical sheafexists and is locally free. Let f : S → S be an endomorphism such that K S ≃ f ∗ K S .Let ν : ˜ S → S be the normalisation, and denote by N the effective Weil divisor on ˜ S such that K ˜ S = ν ∗ K S − N . There exists an endomorphism ˜ f : ˜ S → ˜ S such that ν ◦ ˜ f = f ◦ ν . Furthermore we have ˜ f − ( N ) = N and ˜ f is étale in codimension onein the complement of N .Proof. The existence of ˜ f is immediate by the universal property of the normalisa-tion.For the second statement, note that we have an equality of Weil divisors ˜ f ∗ ( K ˜ S + N ) = ˜ f ∗ ν ∗ K S = ν ∗ f ∗ S K S = K ˜ S + N. Since by the ramification formula for normal surfaces we also have an equality ofWeil divisors K ˜ s = ˜ f ∗ K ˜ S + R, where R is the ramification divisor, we get an equality of Weil divisors R = f ∗ N − N, so N is a completely invariant divisor of f ˜ D and f ˜ D is étale outside N . (cid:3) . Compact Kähler threefolds
Algebraic connectedness.3.1. Definition.
Let X be a compact Kähler manifold. X is called algebraicallyconnected if there exists a family of curves ( C t ) t ∈ T such that C t is irreducible forgeneral t ∈ T and such that two very general points can be joined by a chain of C t ’s. The following theorem of Campana illustrates the importance of this notion. [Cam81]
An algebraically connected compact Kähler manifold isprojective.
An immediate consequence of this theorem is that if X is a compact Kähler threefoldof algebraic dimension two, then the algebraic reduction of X is almost holomorphic.If the algebraic dimension is one, this no longer holds in general, but only in specialcases: Let X be a compact Kähler non-algebraic threefold. Let ϕ : X C be a meromorphic map onto a curve such that the general fibre is an algebraicsurface. Then ϕ is holomorphic. Indeed the fibres of ϕ are algebraically connected and cover X . Thus they can’tmeet since otherwise X is algebraically connected. This shows that ϕ is almostholomorphic, but an almost holomorphic map onto a curve is holomorphic.The existence of an endomorphism of degree at least two allows us to assure theholomorphicity of the algebraic reduction in another situation. Let X be a compact non-algebraic Kähler threefold of algebraicdimension one, and let f : X → X be an endomorphism of degree d > . Supposethat the general fibre of the algebraic reduction is a compact Kähler surface ofalgebraic dimension zero. Then the algebraic reduction of X is holomorphic.Proof. Let µ : X ′ → X be a bimeromorphic map such that ϕ : X ′ → C is aholomorphic model of the algebraic reduction. The endomorphism f induces ameromorphic endomorphism f ′ : X ′ X ′ . By Proposition 2.12 there exists anendomorphism g : C → C such that g ◦ ϕ = ϕ ◦ f ′ . Since the general ϕ -fibre F isirreducible, we have f ′− ( F ) = F ∪ . . . F δ , where δ is the degree of g . For simplicitydenote by F also the image of a general fibre under the birational map µ . Since f − ( F ) has δ irreducible components and F does not lie in the branch locus of f ,we see that f ∗ [ F ] ≡ num δ [ F ] . Thus Lemma 2.3 shows that F = 0 or d = δ . F = 0 . We want to prove that if F ′ is another generalmember of the family, then F ∩ F ′ = ∅ . Indeed if this is not the case, there exists anontrivial effective Cartier divisor F ′ ∩ F on F such that ( F ′ F ) = 0 , a contradictionto Lemma 2.18. Since F and F ′ are irreducible and homologous, this implies that [ F ] = 0 . Thus a general F does not meet any member of the family. Hence thealgebraic reduction of X is almost holomorphic onto a curve, so it is holomorphic. F = 0 . In this case we have δ = √ d . By [CP00,Cor.7.6.(ii)], the general ϕ -fibres are isomorphic. Thus we can consider the in-duced map f | F : F → F as an endomorphism f F : F → F of degree δ > . Since = 0 , the intersection F ∩ F ′ with another general member is not trivial. Thesurface F has only finitely many divisors [FF79, Thm.], so we may suppose that f − F ( F ∩ F ′ ) = F ∩ F ′ . Yet F is not uniruled, so the endomorphism f F is not ram-ified by Lemma 2.23. Another application of Lemma 2.3 shows that [ F ∩ F ′ ] = 0 .Now [ F ∩ F ′ ] = F · F ′ · F ′ = F gives a contradiction. (cid:3) Mori program for compact Kähler threefolds.
Let X be a compactKähler manifold. A contraction is defined to be a surjective map with connectedfibres ϕ : X → X ′ onto a normal complex variety such that − K X is relatively ampleand b ( X ) = b ( X ′ ) + 1 . In general X ′ might not be a Kähler variety, in particular ϕ is not necessarily a contraction of an extremal ray in the cone N E ( X ) . Theexistence and structure of threefold contractions is assured by following statement. [Pet01, Thm.2] , [Pet98, Main Thm.] Let X be a smooth compactKähler threefold with K X not nef. Then X carries a contraction ϕ : X → X ′ unless(possibly) X is simple with κ ( X ) = −∞ . The contraction is of one of the followingtypes.1.) ϕ is a P - bundle or a conic bundle over a smooth non-algebraic surface,2.) ϕ is bimeromorphic contracting an irreducible divisor E to a point, and E together with its normal bundle N E/X is one of the following ( P , O P ( − , ( P , O P ( − , ( P × P , O P × P ( − , − , ( Q , O Q ( − , where Q is the quadric cone,3.) X ′ is smooth and ϕ is the blow-up of X ′ along a smooth curve.The variety X ′ is (a possibly singular) Kähler space in all cases except possibly3.). Moreover in all cases but possibly 3.), the morphism ϕ is the contraction of anextremal ray in the cone N E ( X ) . In case 3.), the variety X ′ is Kähler if and only if the ray of a fiber l of ϕ is extremalin the dual Kähler cone N A ( X ) . The main defect of the preceding statement (and the main open problem in theMori of compact Kähler threefolds) is that in general it is not clear whether X ′ isKähler. We give an affirmative answer in a number of situations. First we needsome preparation. Let ψ : X → X ′ be the blow-up of a smooth curve B ⊂ X ′ inthe compact manifold X ′ . Assume X is Kähler. Then X ′ is Kähler provided thefollowing condition (*) is satisfied.If T is a positive closed current of bidimension (1 , such that T = dS with somecurrent S, then T = 0 . Proof.
Let b ∈ B and l = ψ − ( b ) . By Theorem 3.5 we need to show that [ l ] isextremal in the dual Kähler cone N A ( X ) . So suppose that [ l ] = a + a with a i ∈ N A ( X ) \ . Now represent a i by a positive closed current T i (see e.g.[OP04, Prop.1.8], which is a consequence of the work of Demailly-Paun [DP04]).So l ∼ T + T , nd therefore ψ ∗ ( T ) + ψ ( T ) ∼ . By our assumption we obtain ψ ∗ ( T ) + ψ ∗ ( T ) = 0 , so that ψ ∗ ( T ) = ψ ∗ ( T ) = 0 . Therefore the a i are proportional to [ l ] and [ l ] is extremal. (cid:3) In the following corollary, χ B denotes the characteristic function of B and T B isthe current given by integration over B. Let ψ : X → X ′ be the blow-up of a smooth curve B ⊂ X ′ inthe compact manifold X ′ , and denote by E the exceptional divisor. Assume X isKähler. Then X ′ is Kähler if one of the following assertions holds.1.) There is no positive closed current λT B + T ′ with λ > and χ E T ′ = 0 which is a boundary dS. B moves in a positive-dimensional family.3.) The normal bundle N B/X ′ is not negative.Proof. (1) By [Siu74], see also [Dem01] we have a decomposition T = λT B + T ′ with T ′ = χ X \ B T and some λ ≥ . In particular T ′ is again closed. Since we assumethat (*) already holds for all T for which λ > , we may assume that λ = 0 andneed to verify (*) for those T. By the proof of 3.6 we need to check that only forthe current ψ ∗ ( T + T ) . But for those currents λ = 0 can actually never happen: if χ B ψ ∗ ( T + T ) = 0 , then χ E ( T + T ) = 0 , where E = ψ − ( B ) . Thus T l ∼ T with χ E T = 0 . Now l · E = − on the other hand χ E T = 0 implies via Demailly’s regularization theorem that E · [ T ] ≥ [Pet98,Sublemma 7.a].(2) By (1) it suffices to show the following. If λT B + T ′ = dS with T ′ a positiveclosed current such that χ B T ′ = 0 and λ ≥ , then λ = 0 . So assume T B + T ′ = dS. Now a multiple of B moves, so we obtain an irreducible curve B ′ = B and a > such that aT B ′ + T ′ = dS ′ . But χ X ′ \ B ( aT B ′ + T ′ ) = 0 , so the arguments of (1)show that a = 0 and T ′ = 0 .(3) Suppose that the normal bundle N B/X ′ is not negative. If X ′ were not Kähler,then by 3.7, we find a positive closed current T with χ B T = 0 such that B + T = dS. We find easily a positive closed current ˜ T on X such that χ E ˜ T = 0 and such that ψ ∗ ( ˜ T ) ∼ T (sse the proof of Prop. 2.1 in [OP04]). Now choose a positive closedcurrent B on E representing the second ray of N E ( E ) (i.e. the one not representedby the ruling line l ). Let R = i ∗ ( B ) where i : E → X is the inclusion. Then up toscaling ψ ∗ ( R ) = B and therefore R + ˜ T ∼ al or some a > . Intersecting with E and observing ˜ T · E = 0 as in the proof of3.7 we obtain B · E < so that the normal bundle N E/X is negative and so does N B/X ′ . (cid:3) We can now prove that the study of compact Kähler threefolds with endomorphismscan be reduced to minimal models, at least if X is not uniruled. Let X be a compact Kähler threefold which is not uniruled, andlet f : X → X be a (necessarily étale) endomorphism of degree d > . Then thereexists a finite sequence of smooth compact Kähler threefolds X = X µ = µ −→ X µ −→ . . . µ n −→ X n such that • the canonical bundle of X n is nef; • for all i ∈ { , . . . , n } , the manifold X i − is the blow-up of X i along a smoothelliptic curve C i ⊂ X i such that the normal bundle is an indecomposablerank two vector bundle of degree zero or a direct sum of numerically trivialline bundles; • up to replacing f =: f by f k for some k ∈ N , there exist endomorphisms f i : X i → X i of degree d such that f i ◦ µ i = µ i ◦ f i − for all i ∈ { , . . . , n } .The curve C i is invariant under f i , i.e. we have f − i ( C i ) = C i .Proof. We will use Theorem 3.5: note first that the case where X is simple with κ ( X ) = −∞ can be easily excluded in our situations: since f is étale by Proposition2.4 we have χ ( X, O X ) = 0 which implies that h ( X, K X ) ≥ or h ( X, Ω X ) ≥ .In the first case we are obviously done and in the second case we can look at theAlbanese map X → Alb ( X ) and conclude that κ ( X ) ≥ by the C n,m -conjecture(which is known for compact Kähler threefolds [Uen87, Thm.2.2, Thm.4.1]).Thus there exists a m ∈ N such that | mK X | is not empty. Let D be the fixedpart of this linear system, then f ∗ K X ≃ K X implies that f ∗ D ≃ D . Since all thecontractions are divisorial by Theorem 3.5, their exceptional loci are contained in D . Since D has only finitely many components, the description of the exceptionalloci implies that there are only finitely many contractions. The endomorphism f permutes the irreducible components of D , so up to replacing f by f k we maysuppose that f − ( E ) = E for the support E of every birational contraction.Choose now a birational contraction µ : X → X with exceptional divisor E . ByLemma 2.3 we have E = 0 , so the description of the normal bundles in Theorem3.5 shows that E is contracted onto a smooth curve C and X is a smooth complexmanifold. The restriction of f to E ≃ P ( N ∗ C /X ) yields an étale endomorphism f : E → E of degree d , so [Nak08, Thm.1.1.] shows that C is elliptic and N ∗ C /X has the prescribed form, at least up to twist by a line bundle. Since c ( N ∗ C /X ) = O P ( N ∗ C /X ) ( − = E = 0 , we see that such a line bundle is numerically trivial. This completes the descriptionof the normal bundle of C . Corollary 3.7 now implies that X is a Kähler manifold.By [AKP08, Rem.3.3,Cor.3.4] there exists a k ∈ N and an an endomorphism f : X → X of degree d such that f ◦ µ = µ ◦ f . ence X satisfies all the assumption made on X and we can argue by induc-tion on the Picard number to get the existence of the sequence satisfying the firsttwo properties. Replacing the k i by a sufficiently divisible k , we obtain the thirdstatement. (cid:3) If X is uniruled the situation gets more complicated. One reason is that the excep-tional divisor of a birational contraction can lie in the branch locus of f . If this isnot the case, we can again use Lemma 2.3, Theorem 3.5 and Corollary 3.7 to showthe next statement. Let X be a compact Kähler threefold, and let f : X → X be anendomorphism of degree d > . Let ψ : X → X ′ be a contraction of birational typesuch that the exceptional divisor E satisfies f − ( E ) = E . Suppose that E is notcontained in the branch locus of f or that the ramification order along E is not √ d .Then E = 0 and ψ contracts the divisor E onto a smooth curve C ′ . Moreover thevariety X ′ is a compact Kähler threefold. Remark.
If the endomorphism f is étale one proves as above that the curve C ′ iselliptic. If f is ramified, C ′ can a-priori be arbitrary. Let X be a compact non-algebraic Kähler threefold, and let f : X → X be an endomorphism of degree d > . Suppose that there exists afibration ϕ : X → C onto a smooth curve C with algebraic general fiber and anautomorphism g : C → C such that g ◦ ϕ = ϕ ◦ f .Suppose that X admits a birational contraction ψ : X → X ′ . Then • the exceptional locus E is contained in a ϕ -fibre, • there exists a fibration ϕ ′ : X ′ → C such that ϕ = ϕ ′ ◦ ψ , • ψ is the blow-up of a smooth curve B in X ′ , • there exists an endomorphism f ′ : X ′ → X ′ of degree d such that f ′ ◦ ψ = ψ ◦ f , and • X ′ is a compact Kähler threefold unless possibly when the curve B hasnegative normal bundle.If ϕ is locally projective, there exists a possibly different contraction ψ ′ : X → X ∗ with locally projective factorization X ∗ → C .Proof. By the classification in Theorem 3.5, the exceptional loci of birational con-tractions are uniruled surfaces and therefore algebraic. Therefore E is contained ina ϕ -fibre, since otherwise X is algebraically connected, hence algebraic by Theo-rem 3.2. The existence of the fibration ϕ ′ is now an immediate consequence of therigidity lemma.Since E is contained in a fibre, it is not contained in the branch locus by Lemma2.15. Therefore Corollary 3.9 shows that E is contracted onto a smooth curve and X ′ is smooth. By what precedes we know that the exceptional loci of birationalcontractions on X are irreducible components of ϕ -fibres. Since there are onlyfinitely many reducible fibres and these have finitely many components, we see thatthere are only finitely many birational contractions. Thus up to replacing f by f k we have f ∗ [Γ] = λ [Γ] , where Γ is a fibre of E → ψ ( E ) . We conclude as in [AKP08,Cor.3.4], the Kähler property follows from Corollary 3.7. hus we are left with the last assertion. Let { x , . . . , x s } ⊂ C be the singular locus of ϕ . Choose small disjoint open neighborhoods U i of x i such that ϕ is projective over U i . Let X i = ϕ − ( U i ) and ϕ i = ϕ | X i . We alreadyknow that there exists some i such that K X is not ϕ i − nef. Hence by [Nak87] thereexists a relative contraction µ : X i → X ′ i such that the induced map X ′ i → U i isprojective. Now µ might be birational or not. We always choose µ birational unlessit is simply not possible, i.e. for all choice of i , the map µ is a fibration. In case µ isbirational we patch things and obtain X ∗ with a locally projective bimeromorphicmap ϕ ′ : X ∗ → C . The map µ extends to ψ : X → X ∗ . In case µ is a fibration, we obtain by deformation of the extremal rational curvesand by our assumption a global relative contraction which is a P − bundle or aconic bundle X → X ∗ with factorization X ∗ → C. (cid:3) Finally let us recall that abundance holds for most minimal Kähler threefolds. [Pet01, Thm.1]
Let X be a normal compact Kähler threefold( Q − factorial with at most terminal singularities) such that K X is nef. Assumethat X is not both simple and non-Kummer. Then K X is semi-ample. Torus fibrations
A torus fibration is a fibration ϕ : X → Y such that the generalfibre is isomorphic to a complex torus. A torus bundle is a smooth torus fibrationthat is locally trivial. If the total space of a torus fibration is not projective, the fibration in general doesnot admit a multisection, i.e. there is no subvariety Z ⊂ X such that ϕ | Z : Z → Y is surjective and generically finite. The main technical statement of this section(Lemma 4.2) shows that if X admits an endomorphism commuting with ϕ , thenthere exists a natural meromorphic factorisation of ϕ which admits a multisection. Let X be a compact normal variety in the Fujiki class that admits atorus fibration ϕ : X → Y . Suppose furthermore that there exists an endomorphism f : X → X of degree d > and an automorphism g : Y → Y of finite order suchthat g ◦ ϕ = ϕ ◦ f . Then (up to replacing f by some power) there exists a compactnormal variety Z that is in the Fujiki class and admits a torus fibration ψ : Z → Y with a multisection and which satisfies the following properties: • there exists an almost holomorphic fibration τ : X Z such that ϕ = ψ ◦ τ . • there exists an endomorphism f : Z → Z of degree d that commutes with ψ and such that f ◦ τ = τ ◦ f .If a very general fibre of ϕ is a simple torus, then ϕ admits a multisection. The proof of this lemma is based on the following easy observation. The statementgeneralises [FN07, Lemma 2.22] whose strategy of proof we follow. .3. Proposition. Let A be a complex torus, and let f : A → A be an endomor-phism of degree d > . Then there exists a (maybe trivial) subtorus T ( A and anendomorphism f : A/T → A/T of degree d such that the set of fixed points of f isnon-empty and finite.In particular if A is simple, the set of fixed points of f is non-empty and finite.Proof of Proposition 4.3. We will argue by induction on the dimension, the case ofdimension one is included in [FN07, Lemma 2.22]. Choose a point ∈ A so thatthe torus A has a group structure. The map h : A → A, x f ( x ) − f (0) − x is a morphism of groups and not zero, since f has degree at least two. Let T bethe connected component of the kernel of h . We make a case distinction. T is trivial In this case h is surjective and has finite kernel. Since { x ∈ A | f ( x ) = x } = { x ∈ A | h ( x ) = f (0) } the statement follows. T has positive dimension. Since h ( x ) = 0 for all x ∈ T , we have f ( x ) = x + f (0) for all x ∈ T . It is thus clear that there exists an endomorphism f : A/T → A/T of degree d such that f ◦ q = q ◦ f , where q : A → A/T is thequotient map. Apply the induction hypothesis to A/T . (cid:3) At first glance the proof of Proposition 4.3 may suggest that therestriction of the endomorphism f to T is a translation. The endomorphism ofdegree nf : E × E × E → E × E × E, ( x , x , x ) ( x + x + x , x + x , nx ) shows that this is not true, since we will have T = E × E × { } . The restriction of f to T is rather a “tower” of translations. Proof of Lemma 4.2.
The automorphism g : Y → Y is assumed to be of finiteorder, so up to replacing f by some multiple we can suppose that g = Id Y . Thestatement claims that there exists a commutative diagram X τ ϕ (cid:23) (cid:23) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵ f ′ / / X τ ~ ~ ϕ (cid:7) (cid:7) ✍✍✍✍✍✍✍✍✍✍✍✍✍✍ Z ψ (cid:15) (cid:15) f / / Z ψ (cid:15) (cid:15) Y Id Y / / Y such that f has degree d and Z → Y has a multisection.We denote by f y : X y → X y the restriction of f to a general fibre X y : this is anendomorphism of degree d . If the fix point set of f y is finite, the fix point set of f isa multisection of ϕ , thus the statement is trivially true. Note that if X y is simple,we are always in this case, which proves the last part of the statement.Suppose now that the fix point set of f y is not finite. Then there exists by Propo-sition 4.3 a torus T y ⊂ X y and an endomorphism f y : X y /T y → X y /T y of degree that commutes with the projection X y → X y /T y and has non-empty finite fixpoint set. The countability of the number of irreducible components of the relativecycle space C ( X/C ) implies that if we choose y very general, the torus T y deformswith X y . Let Z → Y be the normalisation of the component of C ( X/Y ) whosevery general points corresponds to the translates of T y for y ∈ Y very general. Thealmost holomorphic map τ : X Z such that ϕ = ψ ◦ τ is given fibrewise by X y → X y /T y .The (necessarily étale) endomorphism f acts via the push-forward of cycles on C ( X/Y ) . Noting that the general fibre of Z → Y is the quotient torus X y /T y andthe restriction of f ∗ to X y /T y identifies to f y , we see that f ∗ maps the componentcorresponding to Z onto itself. Thus we obtain a holomorphic endomorphism f : Z → Z of degree d that commutes with ψ . The restriction of f to a very generalfibre Z y has non-empty finite fix point set, so the fix point set of f is a multisectionof ψ . By construction of f , it is clear that f ◦ τ = τ ◦ f . (cid:3) Theorem 1.1 is now an immediate application:
Proof of Theorem 1.1.
We argue by contradiction and suppose that X is not pro-jective. Since a ( X ) ≥ κ ( X ) = n − , we see that the algebraic dimension is n − .By Campana’s theorem 3.2 this implies that the algebraic reduction ϕ : X Y is almost holomorphic. Since κ ( X ) = n − it is bimeromorphicly equivalent to theIitaka fibration. By [NZ07, Thm. A] the endomorphism f induces an automor-phism g : Y → Y of finite order. Thus Lemma 4.2 implies that X → Y admits amultisection. Since Y is projective and X y a curve, X is algebraically connected.Hence it is projective by Campana’s theorem, a contradiction. (cid:3) Let X be a compact Kähler manifold that is a torus bundle ϕ : X → Y with fibre A . Suppose furthermore that there exists an endomorphism f : X → X of degree d > and an automorphism g : Y → Y of finite ordersuch that g ◦ ϕ = ϕ ◦ f . Then (up to replacing f by some power) there exists acompact Kähler manifold Z that is a torus bundle ψ : Z → Y with fibre A/T witha multisection and satisfies the following properties: • X is a torus bundle τ : X → Z with fibre T such that ϕ = ψ ◦ τ . • There exists an endomorphism f : Z → Z of degree d that commutes with ψ and such that f ◦ τ = τ ◦ f . • The multisection is given by
F ix ( f ) which is an étale cover of Y .If A is simple, then ϕ admits an étale multisection.Proof. The proof of the first two points is the same as for Lemma 4.2, with onedifference: since X → Y is a bundle, it is clear that for very general y the torus T y ⊂ A does not depend on y ∈ Y . Thus the variety Z can be directly defined asthe quotient bundle Z → Y with fibre A/T .The étaleness of
F ix ( f ) → Y is shown by copying word by word the proof of [FN07,Thm.2.24]. (cid:3) et now X be a complex torus that contains a proper subtorus T ( X . Then X is naturally a torus bundle over X/T with fibre T and if X is projective Poincaré’sirreducibility lemma shows that it is isogenous to X/T × T . In particular the studyof endomorphisms of X is reduced to X/T × T . If X is not projective the situationis more complicated, but the geometric intuition still says that if X → X/T admitsan “interesting” endomorphism, the torus X should be close to being a product.We illustrate this philosophy in two special cases: Let X be a torus that is a torus bundle ϕ : X → Y over atorus Y with fibre A . Suppose that dim Y = 1 or dim Y = 2 , a ( Y ) = 1 . Supposethat X admits an endomorphism f : X → X of degree d > such that there existsan automorphism g : Y → Y such that g ◦ ϕ = ϕ ◦ f . Then there exists a (maybetrivial) subtorus T ( A such that X is isogenous to a torus bundle over E × A/T .In particular if A simple, then X is isogenous to Y × A .Proof. Choose a point ∈ X so that the tori X, Y and A have a group structure.Up to composing f with the translation x → x − f (0) , we can suppose that f (0) = 0 .Thus the automorphism g satisfies g (0) = 0 , i.e. is a group automorphism of Y .Since Y is a curve or a surface of algebraic dimension one, this group is finite (cf.[Fuj88, Prop.3.10] for the surface case), so g is of finite order. By Corollary 4.5,there exists a quotient X → X/T → Y such that f descends to an endomorphism f on X/T . Moreover the covering
F ix ( f ) → Y is étale, so an irreducible componentof F ix ( f ) is a complement to A/T in X/T . The statement follows by [BL99,Prop.6.1]. (cid:3)
Let X be a torus that is a torus bundle ϕ : X → Y over a torus Y with fibre A . Suppose that End ( A ) ≃ Z or dim A = 1 . Suppose that X admits anendomorphism f : X → X of degree d > such that there exists an automorphism g : Y → Y such that g ◦ ϕ = ϕ ◦ f . Then X is decomposable.Proof. We prove the statement in the case where
End ( A ) ≃ Z (the same strategyworks if A is an elliptic curve with complex multiplication, cf. the proof of [Fuj88,Prop.3.10]). We argue by contradiction and suppose that X is indecomposable.Since End ( A ) ≃ Z , the restriction of f to A is the multiplication by an integer n such that n dim A = d . Thus f − n is an endomorphism of X whose restriction to A is constant, in particular f − n is not an isogeny. Thus by [BL99, Prop.7.3], theendomorphism f − n is nilpotent. Hence the induced endomorphism g − n on Y isnilpotent, so its kernel has positive dimension. Thus there exists a positive dimen-sional subtorus T ⊂ Y such that the restriction of g to T equals the multiplicationby n . In particular g | T is not injective, so g is not an automorphism. (cid:3) Combining Proposition 4.3 with Proposition 4.7, we obtain:
Let A be a two-dimensional torus of algebraic dimension one, andlet f : A → A be an endomorphism of degree d > . Then f has a non-empty finiteset of fixed points. . Non-uniruled manifolds
In this section we prove Theorem 1.2. Using the minimal model program for non-uniruled threefolds admitting endomorphisms established in Proposition 3.8 theproof naturally splits into two parts: the first and most difficult part is to classifythe minimal models admitting endomorphisms (Theorem 5.1 below). Based on therather short list obtained in the first step, we then discuss the structure of theblow-ups in Subsection 5.B.5.A.
Minimal models.5.1. Theorem.
Let X be a smooth compact non-algebraic Kähler threefold whichis not uniruled. Suppose that X admits a (necessarily étale) endomorphism f : X → X of degree d > . If K X is nef, then (up to étale cover) one of the followingholds:1.) κ ( X ) = 0 : then eithera) X is a torus orb) X is a product S × E where S is a non-algebraic K3 surface and E anelliptic curve.2.) κ ( X ) = a ( X ) = 1 : then X is a product C × A where C is a curve of generaltype and A a torus of algebraic dimension zero.3.) κ ( X ) = 1 , a ( X ) = 2 : then eithera) X is a product Y × A where Y is of general type and A a torus ofalgebraic dimension one orb) X is a product E × S where E is an elliptic curve and and S a non-algebraic Kähler surface of Kodaira dimension one.Proof. By Theorem 1.1 we have κ ( X ) ≤ , and by Theorem 3.11 the canonicalbundle is semi-ample.If κ ( X ) = 0 this implies mK X = O X for some m > . By the Beauville-Bogomolovdecomposition theorem X admits a finite étale cover by a torus, or a product of anelliptic curve and a K3 surface, or a Calabi-Yau manifold of dimension three. Thelast case is excluded since a Calabi-Yau manifold is simply connected.Thus we are reduced to study the cases κ ( X ) = 1 . These are dealt with in theTheorems 5.3, and 5.4. (cid:3) Let X be a non-algebraic compact Kähler manifold of dimension n ,and let f : X → X be an endomorphism of degree d > . Suppose that X admitsa fibration ϕ : X → Y onto a projective variety Y such that ϕ ◦ f = ϕ and thegeneral fibre X y has Kodaira dimension zero. Then the general fibre X y is (up toétale cover) a two-dimensional torus.1.) If a ( X ) = n − , the fibre X y has algebraic dimension zero and is isomorphicto fixed torus A .2.) If a ( X ) = n − and a ( X y ) = 1 , the fibre X y is isomorphic to fixed torus A .3.) If a ( X ) = n − and a ( X y ) = 2 , there exists a normal projective variety Z that admits an elliptic fibration ψ : Z → Y and satisfies the followingproperties: there exists an almost holomorphic fibration τ : X Z such that ϕ = ψ ◦ τ . • there exists an endomorphism f : Z → Z of degree d that commuteswith ψ and such that f ◦ τ = τ ◦ f .Proof. The general fibre X y has Kodaira dimension zero and the restricted endo-morphism f y : X y → X y has degree d . Thus X y is covered by a torus by [FN05].Moreover if a ( X y ) ≤ , the general fibres are isomorphic to a fixed torus A : apply[CP00, Cor.6.8] to X × Y C → C where C is a general complete intersection curvein Y .Suppose now that a ( X y ) = 1 . Let r y : X y → E y be the algebraic reduction,then there exists an endomorphism f y : E y → E y such that f y ◦ r y = r y ◦ f y .The endomorphism f y can’t be an automorphism since otherwise a ( X y ) = 2 byProposition 4.6. Thus it has degree at least two and if ψ : Z Y denotes therelative algebraic reduction of X → Y , there exists a meromorphic endomorphism f : Z Z that commutes with ψ . By Lemma 4.2 this implies that ψ has amultisection, hence a ( X ) ≥ a ( Z ) = n − .Suppose now that a ( X y ) = 2 . Then the fixed point set of f y is not finite, sinceotherwise f has a multisection and X is projective by Campana’s theorem 3.2.Thus by Lemma 4.2 there exists a compact normal variety Z that is in the Fujikiclass and admits an elliptic fibration ψ : Z → Y with a multisection and satisfiesthe stated properties. Since ψ has a multisection, we have a ( X ) ≥ a ( Z ) = n − and Z is algebraic. (cid:3) Let X be a compact Kähler manifold of dimension n , and let f : X → X be an endomorphism of degree d > . Suppose that κ ( X ) = a ( X ) = n − . Then X is (up to étale cover) a product Y × A where Y is of general type and A atwo-dimensional torus of algebraic dimension zero.Proof. By [NZ09, Sect. 1.4] there exists a “ f -equivariant” resolution of the inde-terminacies µ : X ′ → X of the Iitaka fibration ϕ : X Y such that f lifts to a holomorphic endomorphism f ′ : X ′ → X ′ . If we show that X ′ ≃ Y ′ × A , it is a-posteriori clear that the Iitaka fibration of X is holomorphic and hence X ≃ Y × A .Thus we can suppose without loss of generality that X admits a holomorphic fibra-tion ϕ : X → Y onto a projective variety Y such that the general fibre has Kodairadimension zero. By [NZ09, Thm. A] the endomorphism f induces an automor-phism g : Y → Y of finite order, so up to replacing f by some multiple we cansuppose that g = Id Y . By Lemma 5.2 the general fibre X y is a torus of algebraicdimension one. Since by hypothesis X y is not algebraic, it is isomorphic to a fixedtorus A (ibid).Thus by [CP00, Cor. 6.6] there exists a finite Galois cover Y ′ → Y such that X × Y Y ′ is bimeromorphic over Y ′ to a principal torus bundle ϕ ′ : X ′ → Y ′ . Thefollowing commutative diagram and the universal property of the fibre product how that f lifts to an endomorphism h : X × Y Y ′ → X × Y Y ′ of degree d thatcommutes with the projection on Y ′ . X × Y Y ′ $ $ ■■■■■■■■■ (cid:15) (cid:15) h ∃ univ. prop. fibre prod. / / X × Y Y ′ z z ✉✉✉✉✉✉✉✉✉ (cid:15) (cid:15) X ϕ (cid:15) (cid:15) f / / X ϕ (cid:15) (cid:15) Y Id Y / / YY ′ : : ✉✉✉✉✉✉✉✉✉✉ Id Y ′ / / Y ′ d d ■■■■■■■■■■ Thus there exists a meromorphic endomorphism f ′ : X ′ X ′ of degree d thatcommutes with ϕ ′ . The ϕ ′ -fibre is a two-dimensional torus of algebraic dimensionzero, so it contains no curves. This immediately implies that f ′ extends to aholomorphic endomorphism. Since X y is simple, Corollary 4.5 shows that the fixedpoint set of f gives an étale covering F ix ( f ) → Y ′ . Since X ′ → Y ′ is a principalbundle, we see that after étale base change X ′ ≃ Y ′ × A Moreover since Y ′ isprojective any morphism from Y to A is constant, so f ′ = ( Id Y ′ , f A ) where f A isthe restriction of f to any y × A .Since A contains no curves, the composed map ψ := p A ◦ µ : X × Y Y ′ A extendsto a holomorphic map. Moreover f A ◦ ψ = ψ ◦ h , so Proposition 2.14 implies that ψ is smooth. The natural maps X × Y Y ′ → Y ′ and ψ then define an isomorphismonto Y ′ × A . (cid:3) Let X be a compact Kähler threefold, and let f : X → X be anendomorphism of degree d > . Suppose that K X is semiample and κ ( X ) = 1 , a ( X ) = 2 . Then (up to étale cover) one of the following holds:1.) X is a product C × A where C is a curve of general type and A a two-dimensional torus of algebraic dimension one or2.) X is a product E × S where E is an elliptic curve and S a non-algebraicKähler surface of Kodaira dimension one. The stategy of the proof should also work for compact Kähler manifolds with κ ( X ) = n − , a ( X ) = n − if one is able to show the following statement (due toNakayama [Nak08, Thm.6.2.1] in the surface case). Let X be a normal projective variety of dimension n − , and let f : X → X be an endomorphism of degree d > . Suppose that X admits a (flat?)fibration τ : X → Y whose general fibre is an elliptic curve and commutes with f .Then (up to base change) we have X is a product Y × E where E an elliptic curve.Proof. Some multiple bundle of the canonical bundle induces a fibration ϕ : X → C such that mK X ≃ ϕ ∗ L . By [NZ09, Thm. A] the endomorphism f induces anautomorphism g : C → C of finite order, so up to replacing f by some multiple wecan suppose that g = Id C . st case. The general ϕ -fibre has algebraic dimension one. By a statement due to Voisin (cf. [Cam06, Prop.3.10]), there exists a holomorphictwo-form on X whose restriction to the general fibre X c gives a non-zero holomor-phic two form. Thus by [CP00, Prop.4.2, Prop.6.7] the fibration ϕ is almost smoothand there exists a base change C ′ → C such that the normalisation of X × C C ′ gives an étale covering of X and is a principal bundle over C ′ . Arguing as in theproof of Theorem 5.3 we see that the endomorphism f lifts, so we can supposewithout loss of generality that ϕ : X → C is a principal bundle. By Corollary 4.8the induced endomorphism f : X c → X c has a non-empty finite fixed point set.Copying word by word the proof of [FN07, Thm.2.24], one sees that F ix ( f ) → C is finite and étale. Since X → C is a principal bundle, we see that after étale basechange X ≃ C × A . ϕ -fibre is algebraic. By Lemma 5.2 there exists a normal projective surface Z that admits an ellipticfibration ψ : Z → C and satisfies the following properties: • there exists an almost holomorphic fibration τ : X Z such that ϕ = ψ ◦ τ . • there exists an endomorphism f : Z → Z of degree d that commutes with ψ and such that f ◦ τ = τ ◦ f .By [Nak08, Thm.6.2.1] there exists a finite base change C ′ → C such that thenormalisation of Z × C C ′ is isomorphic to C ′ × E , where E is an elliptic curve. Notefurthermore that by [Nak08, Lemma 6.2.5] the base change is étale over the locuswhere the fibres of ψ are reduced, in fact the ramification order in c ∈ C equals themultiplicity of the fibre Z c . Since ϕ = ψ ◦ τ this shows that the ramification orderin c divides the multiplicity of the fibre X c . Thus by a classical argument the mapfrom the normalisation of X × C C ′ onto X is étale. As in the proof of Theorem5.3 we can use the universal property of the fibre product and a commutativediagram to see that (up to étale cover) we can suppose that Z ≃ C × E where E in elliptic curve. By [FN07, Lemma 2.25] we may suppose (up to making an étalebase change) that the endomorphism f is of the form id C × g E , where g E : E → E is an endomorphism of degree d .Since E is an elliptic curve, the meromorphic fibration p E ◦ τ : X E extends toa holomorphic map such that we have a commutative diagram X p E ◦ τ τ (cid:15) (cid:15) ✤✤✤ f / / X τ (cid:15) (cid:15) ✤✤✤ p E ◦ τ ~ ~ C × E p E (cid:15) (cid:15) f / / C × E p E (cid:15) (cid:15) E g E / / E .
Thus by Proposition 2.14 the fibration p E ◦ τ is smooth and by adjunction its fibres X e are surfaces with Kodaira dimension one that have a natural elliptic fibration ϕ | X e : X e → C . Since the general fibre of ϕ is algebraic, an easy application ofCampana’s theorem shows that a general fibre of X e has algebraic dimension one.Thus the relative algebraic reduction of X → E is holomorphic and identifies to τ . Thus τ is holomorphic and since f is étale of degree at least two, Proposition .14 implies that it is an equidimensional elliptic fibration whose singular locus isa disjoint union of c i × E .Let now X c be a very general ϕ -fibre. Then X c is covered by an algebraic torusand is an elliptic bundle τ c : X c → c × E ≃ E . Thus by Poincaré’s irreducibilitytheorem there exists an étale cover of X c by E ′ c × E . In particular we get a familyof elliptic curves in X c surjecting onto E . By countability of the components ofthe cycle space, we may suppose that the family of curves deforms with X c for c ∈ C very general. Let Y → C be the normalisation of the component of C ( X/C ) parametrising these curves, and denote by ˜ X the normalisation of the universalfamily. Denoting by p : ˜ X → X and q : ˜ X → Y the natural maps, we we havecommutative diagram ˜ X p / / q (cid:15) (cid:15) X ϕ (cid:15) (cid:15) p E ◦ τ / / EY / / C .
A general member of the family ˜ X y is an elliptic curve surjecting onto E . The étalebase change ˜ X y → E induces an étale covering of X × E ˜ X y → X , so up to replacing X by an étale cover we can suppose without loss of generality that a general ˜ X y isa p E ◦ τ -section and p is birational. We claim that p is an isomorphism. Assumingthis for the time being, let us show how to conclude: for simplicity of notation,identify X and ˜ X . Let X e be a general p E ◦ τ -fibre, then X e → Y is surjective andétale in codimension one. Thus the induced morphism X × T X e → X is étale incodimension one, so étale since X is smooth. Thus up to replacing X by an étalecover the p E ◦ τ -fibres X e are q -sections, in particular all the fibres are isomorphicto Y . Thus q × ( p E ◦ τ ) : X → Y × E is an isomorphism. Proof of the claim.
Since X is smooth and ˜ X is normal, it is sufficient that for every x ∈ X there exist at most finitely many ˜ X y passing through x . We will show thisproperty “fibrewise”: let first c ∈ C be a point that is not in the ϕ -singular locus.Then the fibre X c is an elliptic bundle over c × E and the ˜ X y are a one-dimensionalfamily of disjoint sections parametrised by Y c . Thus X c is a torus isomorphic to Y c × E and the ˜ X y form a family of line bundles such that the intersection productin X c equals zero.Let now c ∈ C be a point that is in the ϕ -singular locus. Then for every point y ∈ Y c there exists a small analytic neighbourhood D of c and a multisection S ⊂ Y over ∆ passing through y such that S ∩ Y c is a singleton. Let D := p ( q − ( S )) be the analytic subset of X covered by the curves parametrised by S . Then D is asurface in the threefold X , so it is locally principal. In particular for every c ∈ ∆ ,the intersection X c ∩ D is a Cartier divisor. Since for c ∈ D general the self-intersection is zero and the X c vary in a flat family, we see that the curve ˜ X y is aCartier divisor in X c with self-intersection zero. Thus if we show that every curve ˜ X y is irreducible, this implies that there are at most finitely many ˜ X y through agiven point and we are done.The irreducibility of ˜ X y can be seen as follows: by Kodaira’s list of singular fibres ofan elliptic fibration, the reduction of the fibre X c ≃ τ − ( c × E ) is either an abeliansurface (in this case the fibres over c × E are multiple elliptic curves) or the fibres of X c → E are cycles of (maybe singular) rational curves. We will deal with the second ase, the first case can be reduced to the preceding case by a local base change.Since f commutes with ϕ , we have an étale endomorphism f c : X c → X c of degree d and up to replacing f by some power f c maps every irreducible component of X c onto itself. Again by Kodaira’s list the normalisation ν : T → X c is a disjoint unionof P -bundles T i over the elliptic curve E and f c lifts to an étale endomorphism f ic : T i → T i of degree d . Since f ic is étale, one sees easily (cf. [Nak02, Ch.2]) that T i does not contain any curves with negative self-intersection. The pull-back ν ∗ ˜ X y isan effective divisor with self-intersection that surjects onto E . The claim followsby an easy intersection calculus on T i . (cid:3) Proof of Theorem 1.2.
By Proposition 3.8 there exists a finite sequence ofsmooth compact Kähler threefolds X = X µ = µ −→ X µ −→ . . . µ n −→ X n such that X n is described by Theorem 5.1 and X i − is the blow-up of X i along anelliptic curve C i with numerically trivial normal bundle such that f − i ( C i ) = C i .We will now use the list in Theorem 5.1 to see what blow-ups are actually possible. κ ( X ) = 0 . In this case X n is a torus or a product Y × E where Y is a K3surface and E is an elliptic curve. a) X n is a torus. We claim that if X = X n , then X n is isogenous to Y × E where Y is a torus and E an elliptic curve: by hypothesis X n contains the elliptic curve E := C n , so we have a quotient map ϕ : X n → X n /E . Since f − n ( C n ) = C n thereexists an étale endomorphism g : X n /E → X n /E such that g ◦ ϕ = ϕ ◦ f n . Since f − n ( C n ) = C n the fibre g − ( ϕ ( C n )) is a singleton, so g is an automorphism. Thus X n is decomposable by Proposition 4.7, i.e. a product of an elliptic curve and atwo-dimensional torus. In particular the algebraic dimension of X n is at least oneand if it equals one, the unique decomposition possible is Y × E . If the algebraicdimension of X n equals two, the quotient X n /E has algebraic dimension one andthe claim follows from Proposition 4.6.Thus X n ≃ Y × E and up to making an étale base change we can suppose that f is of the form ( g, h ) where g : Y → Y is an automorphism and h : E → E an endomorphism of degree d (cf. [FN07, Lemma 2.25] which does not use theprojectiveness assumption). Let × C n be the copy of E ⊂ X n that is the centerof the blow-up. Then the commutative diagram X n f / / p E (cid:15) (cid:15) X np E (cid:15) (cid:15) E h / / E and deg h > immediately implies that C n is not contained in some Y × e for e ∈ E . Thus C n surject onto E and up to replacing by an étale cover, it is of theform y × E for some y n ∈ Y . Thus the blow-up of X n along C n is isomorphic to Bl y Y × E . The restriction of f n − to Bl y Y is an automorphism, so we obtain thestatement by induction. b) X n ≃ Y × E with Y a K3. Since h ( Y, O Y ) = 0 , the projection onto E is theAlbanese map of X n . Since a K3 does not admit endomorphisms of degree at leasttwo, the induced endomorphism h : E → E has degree d . Since the automorphism roup of a K3 surface is discrete we see that the restriction of f n to Y × e does notdepend on e , so f n = ( g, h ) , where g : Y → Y is an automorphism. We can nowargue as in Case a). κ ( X ) = 1 . According to Theorem 5.1 we distinguish two cases. a) X n ≃ C × A with C a curve of general type and A a torus of algebraic dimensionat most one. We claim that in this case it is not possible to have X = X n and argueby contradiction. The projection onto C is the Iitaka fibration, so up to replacing f by some power we may suppose that the induced endomorphism on C is the identity.Moreover since the algebraic dimension of A is at most one, any morphism from C to A is constant. This implies that f n = ( Id C , g ) with g an endomorphism of A ofdegree d . The elliptic curve C n does not map surjectively onto C , so it is containedin some torus c × A . In particular A has algebraic dimension one and C n is afibre of the algebraic reduction A → T . Since g − ( C n ) = C n , the endomorphisminduced by g on T is an automorphism. Hence A is algebraic by Proposition 4.6, acontradiction. b) X n ≃ E × S where E is an elliptic curve and S a surface with a ( S ) = κ ( S ) = 1 . Let ψ : S → C be the algebraic reduction of S . Then p E × ψ : X → E × C is thealgebraic reduction of X , so there exists an induced endomorphism g : E × C → E × C . In general the endomorphism g does not preserve the projection onto E ,but by [FN07, Lemma 2.25] there exists an étale base change E ′ → E such that thisis the case. Since all the X i are fibre spaces over E we can suppose without loss ofgenerality that there exists an endomorphism g E : E → E such that g E ◦ p E = p E ◦ g .Thus we also have g E ◦ p E = p E ◦ f and the restriction of f to some e × S gives anendomorphism f S of S . Since S has algebraic dimension one, Theorem 1.1 impliesthat f S is an automorphism. Thus g E has degree d and we can conclude as in thefirst case. (cid:3) Uniruled manifolds
The aim of this section is to prove Theorem 1.3. We will start with the easy casewhere f is étale, for the case of ramified endomorphisms we make a case distinctionin terms of the algebraic dimension. With increasing algebraic dimension the proofsget more and more involved, this confirms our philosophy that non-algebraicity is anobstruction to the existence of endomorphisms. For the convenience of the readerwe will repeat at the start of every subsection the part of Theorem 1.3 that we areabout to prove.6.A. Proof of Theorem 1.3: étale endomorphisms.6.1. Theorem.
Let X be a non-algebraic compact Kähler threefold which is notuniruled, and let f : X → X be an étale endomorphism of degree d > . Then X is (up to étale cover) a projectivised bundle P ( E ) over a non-algebraic torus A and c ( E ) = 4 c ( E ) .Proof of Theorem 6.1. Since the endomorphism f is étale, we have χ ( X, O X ) = 0 .Since X is non-algebraic and uniruled, we have h ( X, O X ) = 0 and h ( X, O X ) ≥ .Therefore h ( X, O X ) ≥ , and we denote by α : X → Y the Stein factorisation ofthe Albanese map X → Alb ( X ) whose general fibre has dimension at least one. st case. Y is a surface. The general fibre of α is a rational curve, so Y is not alge-braic since otherwise X would be algebraic. By Proposition 2.9 the endomorphism f induces an endomorphism g on Y which has degree d since f is étale. ApplyingTheorem 1.1 we obtain κ ( Y ) = 0 , so Y → Alb ( X ) is surjective and étale. By theuniversal property of the Albanese torus, we obtain Y = Alb ( X ) .We will now show that α : X → Y = Alb ( X ) is a P -bundle. By Proposition 2.14the α -singular locus ∆ is empty or has pure dimension one and satisfies g − (∆) = ∆ .Since Y is a non-algebraic two-dimensional torus, the components of ∆ are ellipticcurves that are contracted by the algebraic reduction Alb ( X ) → E . The endo-morphism g induces an endomorphism h on E by Proposition 2.12. The condition g − (∆) = ∆ implies that h is an isomorphism which contradicts a ( Alb ( X )) ≤ byProposition 4.6.Up to making an étale base change, we can suppose that X is a projectivised bundle P ( E ) . The canonical bundle of a is trivial, so K X ≃ O P ( E ) ( − ⊗ ϕ ∗ det E . Since f is étale, we have K X = 0 . An elementary computation shows that c ( E ) = 4 c ( E ) . Y is a curve. We will prove that in this case X is necessarily algebraic.Since h ( Y, O Y ) = h ( X, O X ) ≥ , the endomorphism induced on Y is an auto-morphism of finite order, so up to replacing f by some power we may suppose that g = Id Y . Let S → C be the unique irreducible component of the relative cyclespace C ( X/C ) that parametrizes the rational curves in X . The push-forward ofcycles f ∗ acts on C ( X/C ) and since f induces a meromorphic endomorphism on thebase of the rationally connected quotient X S (Prop. 2.9). Thus we obtain anendomorphism f ∗ : S → S that commutes with S → C . Since f is étale and thegeneral fibre of X S is a P , we see that f ∗ has degree d . The general fibre of S → C is an elliptic curve since otherwise S is algebraic. By Lemma 4.2 we seethat S → C has a multisection, so S is algebraic. Therefore X is algebraic. (cid:3) Proof of Theorem 1.3: algebraic dimension zero.6.2. Theorem.
Let X be a compact Kähler threefold of algebraic dimension zerowhich is uniruled, and let f : X → X be a ramified endomorphism of degree d > .Then X is (up to étale cover) a projectivised bundle P ( E ) over a torus A of algebraicdimension zero, f induces an endomorphism on A of degree at least two, and E isa direct sum of line bundles. If the algebraic dimension is zero, we have the following elementary, but usefullemma at our disposition.
Let X be a compact Kähler threefold of algebraic dimension zeroand let f : X → X be an endomorphism of degree d > . Suppose that X admits afibration ϕ : X → S onto a normal surface S . Then there exists an endomorphism g : S → S such that g ◦ ϕ = ϕ ◦ g .Proof. By a theorem of Fischer and Forster [FF79, Thm.] there exist only finitelymany divisors on X . Since the images and preimages of divisors are divisors, wesee that f induces a bijective map on the set of divisors on X . Since this set isfinite, some iteration f k induces the identity.By the rigidity lemma it is sufficient to prove that any ϕ -fibre is mapped by f intoa ϕ -fibre. By what precedes we may suppose that this is true for the divisorial bre components. Since S contains only finitely many curves, it also holds for thegeneral ϕ -fibres. (cid:3) Proof of Theorem 6.2.
The rationally connected quotient is an almost holomorphicmap ϕ : X S onto a compact Kähler surface of algebraic dimension zero. ByLemma 2.18 we can replace S by a singular K3 surface or a torus without curves, sowe may suppose that ϕ is holomorphic. By Lemma 6.3 there exists a holomorphicmap g : S → S such that ϕ ◦ f = g ◦ ϕ . Since S does not contain any curves, the ϕ -singular locus is a finite union of points. g is an automorphism. Since the ϕ -singular locus is a finite union of points,the conditions of Proposition 2.16 are satisfied (if S is a singular K3 surface, we alsouse Corollary 2.19). Thus after finite base change X is bimeromorphic to S × P .In particular it has algebraic dimension at least one, a contradiction. g is not an automorphism. We claim that S is a torus: otherwise S would be a singular K3 surface and by Proposition 2.21 there exists a Galois coverby a torus ν : A → S that is étale in codimension one such that g lifts to anendomorphism g A : A → A of degree deg g . The following commutative diagramand the universal property of the fibre product show that f lifts to an endomorphism f ′ : X × S A → X × S A . X × S A ❍❍❍❍❍❍❍❍❍ ϕ ′ (cid:15) (cid:15) f ′ ∃ univ. prop. fibre prod. / / X × S A { { ✈✈✈✈✈✈✈✈✈ ϕ ′ (cid:15) (cid:15) X ϕ (cid:15) (cid:15) f / / X ϕ (cid:15) (cid:15) S g / / SA ν : : ✈✈✈✈✈✈✈✈✈✈ g A / / A ν d d ❍❍❍❍❍❍❍❍❍❍ Since the endomorphism g A is étale and A does not contain any curves, the mor-phism ϕ ′ is smooth by Proposition 2.14. Thus the reduction of every ϕ -fibre isisomorphic to P . Suppose that there exists a non-reduced fibre ϕ − ( s ) : then − K X · ϕ − ( s ) = 2 implies that the fibre is a double P and the reduced fibre l satisfies − K X · l = 1 . By a theorem of Ein-Kollár [Kol91, Thm.5.3] the rationalcurve l deforms in a one-dimensional family that covers a surface D in X . Since ϕ is equidimensional, ϕ ( D ) is a curve in S , a contradiction. Thus we see that ϕ is asmooth map and hence S is a smooth Kähler surface of algebraic dimension zerothat admits an endomorphism of degree at least two. This shows that S is a torus.Up to replacing X by an étale cover, we can suppose that X = P ( E ) where E is arank two vector bundle which splits by Lemma 6.5 below. (cid:3) The same proof shows the following statement.
Let X be a compact Kähler threefold, and let f : X → X bean ramified endomorphism of degree d > . Suppose that the rationally connectedquotient is a fibration ϕ : X → S onto a normal Kähler surface without curves. uppose furthermore that the induced endomorphism g : S → S has degree at leasttwo. Then S is a torus and X a P -bundle over S . Let A be a two-dimensional torus of algebraic dimension zero, and let ϕ : X = P ( E ) → A be the projectivisation of a rank two vector bundle E . Supposethat there exists a ramified endomorphism f : X → X and an étale endomorphism g : A → A of degree at least two such that g ◦ ϕ = ϕ ◦ f . If the algebraic dimensionof X is zero, then E is (up to finite étale base change) a direct sum of line bundles. Remark.
The statement of the lemma as well as the techniques in the proof aresimilar to [Ame03, Thm.2]. Nevertheless there is an important difference due tothe condition on A : our conclusion holds after étale base change. Proof.
Since X has algebraic dimension zero, there exist only finitely many divisorson X . Thus up to replacing f by some multiple we can suppose that f − ( D ) = D for every irreducible effective divisor on X . Since f is ramified, there exists aneffective divisor D ⊂ R and we denote by f D : D → D the restriction of f to D .The torus A does not contain any curve, so the holomorphic map ϕ D : D → A issurjective and not ramified. Since g ◦ ϕ D = ϕ D ◦ f D and g étale of degree at leasttwo, Proposition 2.14 shows that ϕ D does not have any higher-dimensional fibres.Thus ϕ D is étale and D is a torus of algebraic dimension zero. Making an étalebase change D → A , we can suppose that D is a ϕ -section. Moreover the followingcommutative diagram and the universal property of the fibre product show that f lifts to an endomorphism f ′ : X × A D → X × A D . X × A D $ $ ❍❍❍❍❍❍❍❍❍ (cid:15) (cid:15) f ′ ∃ univ. prop. fibre prod. / / X × A D z z ✈✈✈✈✈✈✈✈✈ (cid:15) (cid:15) X ϕ (cid:15) (cid:15) f / / X ϕ (cid:15) (cid:15) A g / / AD ϕ D : : ✉✉✉✉✉✉✉✉✉✉ f D / / D ϕ D d d ■■■■■■■■■■ Thus up to finite étale base change the ramification divisor R has one irreduciblecomponent D that is a ϕ -section. Since any endomorphism of P of degree atleast two ramifies in at least two points, this implies that R has at least anotherirreducible component D . Arguing as before we see that D is an étale cover of A .Moreover the intersection with D is empty since D does not contain any curves.Up to making a second étale base change D → A we can suppose that ϕ has twodisjoint sections. Thus E is a direct sum of line bundles. (cid:3) Proof of Theorem 1.3: algebraic dimension one.6.6. Theorem.
Let X be a compact Kähler threefold of algebraic dimension onewhich is uniruled, and let f : X → X be a ramified endomorphism of degree d > .Then (up to étale cover) one of the following holds: .) X is a product S × P , where S is a compact Kähler surface of algebraicdimension zero and f induces an automorphism on S .2.) X is a projectivised bundle P ( E ) over a torus A of algebraic dimension atmost one such that the induced morphism on A has degree at least two and E is a direct sum of line bundles.Proof of Theorem 6.6. The rationally connected quotient is an almost holomorphicmap ϕ : X S onto a compact Kähler surface of algebraic dimension at mostone. S has algebraic dimension one Up to replacing S by a bimeromorphic model, we may suppose that S is a relativelyminimal elliptic surface ψ : S → C . Since a ( S ) = 1 there does not exist any curvein S that surjects onto C . The general fibre of ψ ◦ ϕ : X C is uniruled, hencealgebraic, so ψ ◦ ϕ extends to a holomorphic map by Corollary 3.3. Furthermore weknow by [CP00, Cor.7.3] that the general ψ ◦ ϕ -fibre is isomorphic to P ( O E ⊕ L ) → E where E is an elliptic curve and L is a numerically trivial line bundle that is nottorsion.By Proposition 2.9 there exists a meromorphic endomorphism g : S S such that g ◦ ϕ = ϕ ◦ f . Thus by Proposition 2.12 there exists an endomorphism g C : C → C such that g C ◦ ψ = ψ ◦ g . Thus we get a commutative diagram X ψ ◦ ϕ ϕ (cid:15) (cid:15) ✤✤✤ f / / X ϕ (cid:15) (cid:15) ✤✤✤ ψ ◦ ϕ ~ ~ S ψ (cid:15) (cid:15) g / / ❴❴❴ S ψ (cid:15) (cid:15) C g C / / C and make another case distinction. a) g C is an automorphism. We will show that this case does not exist: the restric-tion of f to the general fibre X c ≃ P ( O E ⊕ L ) gives an endomorphism f c : X c → X c degree d > . Denote by π : X c → E the canonical projection, then there exists anendomorphism g E : E → E such that g E ◦ π = π ◦ f c . Note that g E can’t be anautomorphism: otherwise Amerik’s theorem [Ame03, Thm.1] would imply that X c is a product after étale cover, so L is a torsion line bundle. We will now use theMori program: since X is uniruled, it admits a contraction.Suppose first that X admits a fibre type contraction. Since X contains only onecovering family of rational curves and the rationally connected quotient is onlydefined up to birational equivalence, we can suppose that the contraction is therationally connected quotient ϕ : X → S . Since ϕ is flat, the rigidity lemmaimplies that g extends to a holomorphic endomorphism g : S → S . Since therestriction of g to a general ψ -fibre is g E , we see that g has degree at least two.Thus S is a torus [FN05] and admits an endomorphism g of degree at least twosuch that g C ◦ ψ = ψ ◦ g . Since g C is an automorphism, Proposition 4.6 impliesthat S is isogenous to a product of elliptic curves, a contradiction to a ( S ) = 1 . f X admits a contraction of birational type µ : X → X ′ , we argue using Proposition3.10. First observe that the composed map ψ ◦ ϕ : X → C is locally projective(blow up to make ϕ a (locally projective) morphism, then a priori ψ ◦ ϕ might onlylocally Moishezon, but [CP04, Thm.10.1] gives local projectivity). Then changepossibly µ to get local projectivity of the induced map τ ′ : X → C . Then we getinductively a sequence of a compact manifolds bimeromorphically equivalent to aKähler manifold X = X → X → . . . → X n such that X n admits a fibre type contraction and g C is an automorphism. But wehave just seen that such a X n does not exist (the Kähler property is not neededthere). b) g C is not an automorphism. In this case the curve C is elliptic or P . Wecan’t have C ≃ P , since otherwise S is algebraic by Proposition 2.22. Thus C iselliptic and the endomorphism g C is étale. It follows by Lemma 2.15 that ψ ◦ ϕ isa submersion, thus all the fibres are uniruled and have b ( X c ) = b ( X c ) = 2 . Thiseasily implies that all the fibres are P -bundles over elliptic curves. Thus ϕ extendsto a holomorphic map and X is a P -bundle over S which is an elliptic bundleover S . Since S is Kähler but not algebraic, it follows from [BHPVdV04, V.5.B)]that S is a torus of algebraic dimension one. Thus the meromorphic endomorphism g : S S also extends to a holomorphic map of degree at least deg g C . Up tomaking an étale base change, we can suppose X ≃ P ( E ) , where E is a rank twovector bundle which splits by Lemma 6.7 below. S has algebraic dimension zero In this case X is bimeromorphic to P × S where S is a compact Kähler surfaceof algebraic dimension zero [CP00, Cor.7.6]. The general fibre of the algebraicreduction is bimeromorphic to S , so by Proposition 3.4 we have a holomorphicalgebraic reduction r : X → P . On the other hand, we know by Lemma 2.18that, up to replacing S by some bimeromorphic normal model, we may supposethat S does not contain any curves. Thus we may suppose that the rationallyconnected quotient is a holomorphic map ϕ : X → S , so we get a holomorphic map r × ϕ : X → P × S of degree one. In particular we have r ∗ O P (1) · f = 1 , where f is a general ϕ -fibre. We know by Proposition 2.9 that there exists anendomorphism g S : S → S such that g S ◦ ϕ = ϕ ◦ f . Moreover by Proposition 2.12there exists an endomorphism g P : P → P such that g P ◦ r = r ◦ f . Togetherthey induce an endomorphism g P × g S : P × S → P × S such that we have acommutative diagram X r × ϕ (cid:15) (cid:15) f / / X r × ϕ (cid:15) (cid:15) P × S g P × g S / / P × S We distinguish two cases: a) g S is not an automorphism. By Proposition 6.4 we know that X → S is a P -bundle and S is a torus. Since r ∗ O P (1) · f = 1 the fibres of the algebraic reduction re ϕ -sections. Thus r × ϕ is an isomorphism onto P × S and we are in the secondCase of Theorem 6.6 (in this case E ≃ O ⊕ S ). b) g S is an automorphism. Since r × ϕ has degree one, this implies that g P hasdegree d . Let T ⊂ P × S be the set such that the r × ϕ -fibre has dimension two.Then T is finite and by Proposition 2.14, the endomorphism g P × g S is totallyramified in every point of T . Since g S is an automorphism, this shows that g P hasramification order d in every point of p P ( T ) .We will now use Mori theory to discuss the structure of X : since X is uniruled itadmits at least one contraction. b1) X admits a fibre type contraction ψ : X → S ′ . Then X is a P - or conic bundleover the smooth surface S ′ (Theorem 3.5). Since X contains only one coveringfamily of rational curves, the general ψ -fibre is a general ϕ -fibre. The fibration ψ is flat, so the rigidity lemma shows that we have a factorisation τ : S ′ → S which is actually a birational map. Since ρ ( X/S ′ ) = 1 and r ∗ O P (1) · f = 1 , wesee that ψ is not a conic bundle. The endomorphism g S is an automorphism, sothe endomorphism g S ′ induced by f on S ′ is an automorphism. We satisfy theconditions of Proposition 2.16, so there exists an étale cover of S ′ such that X becomes a product after base change. Thus we are in the first case of Theorem 6.6.Note furthermore that f = ( g S , h ) since the space of endomorphism of P is affine[Ame03, Lemma 1.2]. b2) X admits a birational contraction ψ : X → X ′ . Denote by E the exceptionallocus. Since S does not contain any curve, E is contained in a higher-dimensionalfibre of ϕ . Since there are only finitely many higher-dimensional fibres, we cansuppose (up to replacing f by f k ) that f − ( E ) = E . In particular we get anendomorphism f ′ : X ′ → X ′ of degree d such that f ′ ◦ ψ = ψ ◦ f . We claim that ψ is of type 3.) in Theorem 3.5, i.e. it contracts a divisor onto a smooth curve. Proof of the claim:
We argue by contradiction and suppose that ψ contracts a divi-sor E onto a point. Since all the curves contained in E are numerically equivalentand P × S does not contain any ruled surface (recall that S contains no curve),we see that r × ϕ maps E onto a point c contained in T . By what precedes, theendomorphism g P has ramification order d in p P ( c ) . Denote by X c = r ∗ c the r -fibre over c . Since g P ◦ r = r ◦ f and g ∗ P c = dc , we have dX c = f ∗ X c . Thus the ramification order of f along E ⊂ X c equals d and E = N E/X = 0 byTheorem 3.5. Since f − ( E ) = E , this contradicts Corollary 3.9. (cid:3) Thus X ′ is smooth and since S does not contain any curves, there exists a holo-morphic map ϕ ′ : X ′ → S such that ϕ = ϕ ′ ◦ ψ . Note that we also obtain amorphism t : X ′ → P × S. Instead of applying Proposition 3.10, which is tedious, we argue as follows. Weapply directly [Nak87] to the projective morphism r × ϕ : X → P × S nd obtain a possibly new birational morphism ν : X → X ′ and a factorisation X ′ → P × S which is again projective. Since P × S is Kähler, so is X ′ . Arguingby induction, we get a sequence of birational contractions X = X → X → . . . → X n such that X n is compact Kähler, admits an endomorphism f n : X n → X n of degree d and a fibre type contraction. By Case a) there exists an étale cover of X n by S n × P and f n = ( g n , h n ) where g n is an automorphism of S n and h n : P → P an endomorphism of degree d . Since π ( X ) ≃ π ( X ) ≃ . . . ≃ π ( X n ) we can suppose that X n = S n × P . The manifold X n − is obtained by blowing upa curve C n such that f − n ( C n ) = C n . It is elementary to see that such a curve isnecessarily of the form s n × P where s n ∈ S n is a point. Thus X n − ≃ Bl s n S n × P and we conclude inductively that we are in the first case of Theorem 6.6. (cid:3) Let A be a two-dimensional torus of algebraic dimension one, and let ϕ : X = P ( E ) → A be the projectivisation of a rank two vector bundle E . Supposethat there exists a ramified endomorphism f : X → X and an étale endomorphism g : A → A of degree at least two such that g ◦ ϕ = ϕ ◦ f . If the algebraic dimensionof X is one, then E is (up to étale cover) a direct sum of line bundles.Proof. Let ψ : A → E be the algebraic reduction of the torus A , and denoteby g E : E → E the endomorphism induced by g on E . Since X has algebraicdimension one, ψ ◦ ϕ is the algebraic reduction of X . Since f is ramified and g étaleany effective divisor D ⊂ R maps surjectively onto A , in particular it is not a polardivisor. Since there are only finitely many non-polar divisors [FF79, Thm.], we cansuppose (up to replacing f by some power) that f − ( D ) = D for every irreduciblecomponent of D ⊂ R effective divisor on X .Denote by f D : D → D the restriction of f to some D ⊂ R . Since g ◦ ϕ D = ϕ D ◦ f D and g is étale of degree at least two, Proposition 2.14 shows that ϕ D does nothave any higher-dimensional fibres. If ϕ D is ramified, the branch locus ∆ ⊂ A is a finite union of curves such that g − (∆) = ∆ . Yet this implies that g E isan automorphism, so A would be algebraic by Proposition 4.6. Thus ϕ D is étaleand D a torus. Arguing as in the proof of Lemma 6.5 we see that up to étale basechange, the ramification R has at least two irreducible components: one component D that is a ϕ -section and a second D that is an étale cover of A . Moreover theintersection with is empty since D ∩ D would be a bunch of curves such that f − D ( D ∩ D ) = D ∩ D . Once more Proposition 4.6 would then imply that D ≃ A is algebraic. Thus up to making a second étale base change D → A ,we can suppose that ϕ has two disjoint sections. Thus E is a direct sum of linebundles. (cid:3) Proof of Theorem 1.3: algebraic dimension two.6.8. Theorem.
Let X be a compact Kähler threefold of algebraic dimension twowhich is uniruled, and let f : X → X be a ramified endomorphism of degree d > .Then (up to étale cover) one of the following holds: .) X is a prodcut Y × P where Y is a surface of algebraic dimension one and f induces an automorphism on Y or2.) X is a projectivised bundle P ( E ) over a torus A of algebraic dimension oneand f induces an endomorphism g of degree at least two on A .
1. It is obvious that the list is effective: in the second case take X := A × P with A a torus of algebraic dimension one. The following exampleshows that in the second case X is in general not a product after étale cover.Let E be an elliptic curve without complex multiplication, and let L be a numer-ically trivial line bundle on E that is not torsion. By [Nak02, Prop.5] there existsa ramified endomorphism h : P ( O E ⊕ L ) → P ( O E ⊕ L ) preservering the fibrationsuch that P ( O E ⊕ L ) → E and inducing an endomorphism g E : E → E of degreeat least two. Since End ( E ) ≃ Z it is clear that g E is of the form x nx + e forsome e ∈ E . Let now ψ : A → E be a torus of algebraic dimension one, and let g : A → A be the endomorphism x nx + a where a ∈ ψ − ( e ) . Then we obviouslyhave g E ◦ ψ = ψ ◦ g .The fibre product X := P ( O E ⊕ L ) × E A is a P -bundle over A that is not aproduct, even after étale cover. An easy diagram chase shows that X admitsa ramified endomorphism f preserving the P -bundle structure and inducing theendomorphism g on A .2. One might also ask for a more precise classification of the vector bundle E inthe second case of the theorem: if A → B is the algebraic reduction of A , it is nothard to see E | A b ≃ L b ⊕ L b ⊗ T b , where T b is a torsion line bundle on A b . Since L b and T b might deform with b ∈ B , a further discussion is quite tedious and we leaveit as an exercise to the interested reader. Proof of Theorem 6.8.
The rationally connected quotient is an almost holomorphicmap ϕ : X S onto a compact Kähler surface of algebraic dimension one [CP00,Thm.9.1].Up to replacing S by a bimeromorphic model, we may suppose that S is a relativelyminimal elliptic surface ψ : S → C . Since a ( S ) = 1 there does not exist any curvein S that surjects onto C . The general fibre of ψ ◦ ϕ : X C is uniruled, hencealgebraic, so ψ ◦ ϕ extends to a holomorphic map by Corollary 3.3.By Proposition 2.9 there exists a meromorphic endomorphism g : S S such that g ◦ ϕ = ϕ ◦ f . Thus by Proposition 2.12 there exists an endomorphism g C : C → C such that g C ◦ ψ = ψ ◦ g . Thus we get a commutative diagram X ψ ◦ ϕ ϕ (cid:15) (cid:15) ✤✤✤ f / / X ϕ (cid:15) (cid:15) ✤✤✤ ψ ◦ ϕ ~ ~ S ψ (cid:15) (cid:15) g / / ❴❴❴ S ψ (cid:15) (cid:15) C g C / / C g C is not an automorphism. In this case the curve C is elliptic or P . Wecan’t have C ≃ P , since otherwise S is algebraic by Proposition 2.22. Thus C iselliptic and the endomorphism g C is étale. It follows by Lemma 2.15 that ψ ◦ ϕ is a ubmersion. The general fibre X c is uniruled and the base of its rationally connectedquotient is an elliptic curve S c = ψ − ( c ) , so b ( X c ) = 2 . Since ψ ◦ ϕ is smooth,this implies that the rationally connected connected quotient of all the fibres isa holomorphic fibration over an elliptic curve. Thus ϕ extends to a holomorphicmap and S is an elliptic bundle over the elliptic curve C . Since S is Kähler butnot algebraic, it follows from [BHPVdV04, V.5.B)] that S is a torus of algebraicdimension one. Thus the meromorphic endomorphism g : S S extends to aholomorphic étale map of degree at least deg g C > . Thus by Proposition 2.14the ϕ -singular locus ∆ is empty or a finite union of curves such that g − (∆) = ∆ .Since S has algebraic dimension one, the irreducible components of ∆ are ψ -fibres.Thus g − (∆) = ∆ implies g − C ( ψ (∆)) = ψ (∆) . Since g C is étale, we see that ψ (∆) is empty. Thus X is a P -bundle over the torus S and (up to another étale basechange) we are in the second case of Theorem 6.8. g C is an automorphism. We will use Mori theory to discuss the structureof X : since X is uniruled it admits at least one contraction. a) X admits a fibre type contraction. Since X contains only one covering fam-ily of rational curves and the rationally connected quotient is only defined up tobimeromorphic equivalence, we can suppose that the contraction is the rationallyconnected quotient ϕ : X → S . Then X is a P - or conic bundle over the smoothsurface S (Theorem 3.5) which is an elliptic surface over C . Since ϕ is flat, therigidity lemma implies that g extends to a holomorphic endomorphism g : S → S . a1) g is not an automorphism. In this case S is a torus and admits an endomor-phism g of degree at least two such that g C ◦ ψ = ψ ◦ g . Since g C is an automorphism,Proposition 4.6 implies that S is isogenous to a product of elliptic curves, a contra-diction to a ( S ) = 1 . a2) g is an automorphism. .We claim that in this case the variety X admits a holomorphic elliptic fibration r : X → Y such that a ( Y ) = 2 , i.e. the algebraicreduction of X can be taken holomorphic. Proof of the claim.
Let M ⊂ S be an irreducible curve. By Kodaira’s classificationof fibres of elliptic fibrations, the normalisation ˜ M → M is an elliptic or rationalcurve. Set X M := ϕ − ( M ) , then the general fibre of ϕ M : X M → M is a P or two P ’s meeting transversallyin a point. Moreover (up to replacing f by some power) the restriction of f to X M gives an endomorphism f M : X M → X M such that g | M ◦ ϕ M = ϕ M ◦ f M . Denote by ˜ X M → X M the normalisation and by ˜ ϕ M : ˜ X M → M the induced map, then f M lifts to an endomorphism ˜ f M : ˜ X M → ˜ X M such that g | ˜ M ◦ ˜ ϕ M = ˜ ϕ M ◦ ˜ f M . The general fibre of ˜ ϕ M is a P or a disjointunion of two P ’s and we denote by the same letter ˜ ϕ M : ˜ X M → ˜ M ′ its Steinfactorisation. It is well-known that this new fibration is a P -bundle and ˜ M ′ → ˜ M ramifies exactly in the points where the ϕ -fibre is a double line. Since such a pointgives a singular point of the discriminant locus ∆ ⊂ S and ∆ is necessarily containedin ψ -fibres, Kodaira’s classification shows that there are at most two ramificationpoints. Thus we see that ˜ M ′ is elliptic or rational. he P -bundle ˜ X M → ˜ M ′ admits an endomorphism of degree at least two suchthat the induced endomorphism on ˜ M ′ is an automorphism (here we use that g is an automorphism). Thus by Amerik’s theorem 2.16 there exists an étale cover E → ˜ M ′ such that the fibre product ˜ X M × ˜ M ′ E is a product E × P and there existsa finite group G acting diagonally on E × P such that ˜ X M = ( E × P ) /G . Thusthe projection on the second factor E × P → P induces a fibration r M : ˜ X M → P /G ≃ P . This fibration descends to a fibration X M → P : this is clear in thecomplement of the non-normal locus and the non-normal locus gives a section of ˜ X M → ˜ M ′ , so by construction it is contracted by r M .The construction of the algebraic reduction r is now obvious: let Y be the normal-isation of the unique irreducible component of C ( X ) parametrising a general fibreof the algebraic reduction (the algebraic reduction is almost holomorphic by Thm.3.2, so this is well-defined). Let Γ → Y be the universal family over Y and denoteby p : Γ → X the natural map. Then p is an isomorphism: it is sufficient to checkthat the fibre over every point x ∈ X is a singleton, but this is obvious since x iscontained in some X M → P which realises Γ → Y “locally”. This proves the claim. Since the endomorphism f preserves the algebraic reduction, we clearly have ameromorphic map g Y : Y Y . We have constructed the variety Y as the nor-malisation of some component of C ( X ) , thus we can identify g Y generically to thepush-forward of cycles f ∗ . Since f ∗ is a holomorphic map on the Chow scheme, wesee that g Y extends to a holomorphic map. Since g is an automorphism, the endo-morphism g Y has degree at least two. By [CP00], there exists a fibration τ : Y → C ,and it is easy to check that g C ◦ τ = τ ◦ g Y . Since g C is an automorphism and Y is uniruled, we deduce by [Nak08, Lemma 6.1.1,(2)] that Y is a P -bundle over C .Moreover another application of Amerik’s theorem shows that Y (after étale basechange) a product C × P Thus the fibre product Y × C S is a product S × P and theendomorphisms g and g Y induce an endomorphism f ′ : S × P → S × P of degreeat least two. Since the space of endomorphisms of P is affine, we have f ′ = ( g, h ) where h : P → P is an endomorphism of degree at least two. By the universalproperty of the fibre product there exists a holomorphic map µ : X → S × P . Weclaim that µ has degree one. Since b ( X ) = b ( S ) + 1 = b ( S × P ) it is obviously finite and therefore X ≃ S × P . Proof of the claim.
By construction we have a commutative diagram X µ / / f (cid:15) (cid:15) S × P f ′ (cid:15) (cid:15) X µ / / ϕ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ S × P p S | | ①①①①①①①①① S and we denote by M the branch locus of µ . A straightforward adaptation of Propo-sition 2.14 to the case of finite maps shows that f ′− ( M ) = M . Since f = ( g, h ) with h of degree at least two, the divisor M is of the form p − S ( D ) for some divisor on S . In particular if s ∈ S is general point, the restricted map µ | ϕ − ( s ) : ϕ − ( s ) → s × P is not ramified. Since P is simply connected and ϕ − ( s ) irreducible, the restriction µ | ϕ − ( s ) has degree one. This proves the claim . b) X admits a birational contraction ψ : X → X ′ . Denote by E the exceptionaldivisor. Since the fibers of X → C are algebraic, the divisor E must be containedin some fiber X c . Hence - at least after passing to some f k - we have f − ( E ) = E and E is not contained in the branch locus by Lemma 2.15. Thus by Corollary 3.9the contraction ψ is the blow-up of a smooth curve C ′ ⊂ X ′ and X ′ is a compactKähler manifold. Thus we can proceed inductively and get a sequence X = X → X → . . . → X n such that X n is compact Kähler, admits an endomorphism f n : X n → X n of degree d and a fibre type contraction. By Case a) we know that X n ≃ S n × P . We cannow argue in the proof of Theorem 6.6 to see that X ≃ S × P . (cid:3) References [AKP08] Marian Aprodu, Stefan Kebekus, and Thomas Peternell. Galois coverings and en-domorphisms of projective varieties.
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Andreas Höring, Université Paris 6, Institut de Mathématiques de Jussieu, Equipe deTopologie et Géométrie Algébrique, 175, rue du Chevaleret, 75013 Paris, France
E-mail address : [email protected] Thomas Peternell, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth,Germany
E-mail address : [email protected]@uni-bayreuth.de