POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION

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1 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION JUNGKAI ALFRED CHEN AND ZHI JIANG Abstract. Given a generically finite morphism f from a smooth projective variety X to an abelian variety A, we show that f ω X is sufficiently positive on A. As an application, we prove that ω 2 X is globally generated away from the exceptional locus of f and the global sections of ω 2 X define a generically finite map of X. We also study the structure of X when X is moreover of general type and satisfies χ(x, ω X) = 0. We formulate a conjectural characterization of such X and prove the conjecture when A has only three simple factors. Résumé en français: Étant donné un morphisme f génériquement fini d une variété projective lisse X sur une variété abélienne A, nous montrons que f ω X est suffisamment positif sur A. Comme application, nous montrons que ωx 2 est engendré par ses sections globales, hors du lieu exceptionnel de f, et les sections globales de ωx 2 définissent une application rationnelle génériquement finie sur son image. Nous étudions également la structure de X lorsque X est en outre de type général et χ(x, ω X ) = 0. Nous formulons une caractérisation conjecturale de X et démontrer la conjecture lorsque A n a que trois facteurs simples. 1. Introduction A smooth projective variety X is said to be a variety of maximal Albanese dimension if there exists a generically finite morphism f : X A to an abelian variety. In this article, we study birational geometry of varieties of maximal Albanese dimension. Our purpose is twofold. First of all, we study the sheaf f ω X. It is nown that the birational geometry of X is very much governed by the positivity of f ω X and the sheaf f ω X is nown to be a GV-sheaf but is not necessary M-regular (cf. [PP1] and [PP2]). Our first main result is a general decomposition theorem for f ω X (see Theorem 3.4 and Theorem 3.5 for details), which implies that f ω X is not far from being M-regular Mathematics Subject Classification. 14J10, 14F17, 14E05. Key words and phrases. Generic vanishing, cohomological support loci, varieties of general type, Albanese dimension, Albanese variety, Euler characteristic. 1

2 2 J. CHEN AND Z. JIANG Theorem 1.1. Let f : X A be a generically finite morphism to an abelian variety. Then, we have f ω X (p i F i P i ), i where p i : A A i are quotients of abelian varieties, F i are M-regular sheaves on A i, and P i are torsion line bundles on A. Remar 1.2. In the above formulation, we allow p i : A A i to be trivial quotients, namely p i could be an isomorphism or a fibration to Spec C. This decomposition theorem and its variant can be applied to prove globally generated properties for canonical or pluricanonical bundles. For instance, combining Theorem 3.5 with properties of M-regular sheaves, we have Theorem 1.3. Let f : X A be a generically finite morphism to an abelian variety. Then ωx 2 is globally generated away from the exceptional locus of f. Moreover, the bicanonical map of X is generically finite onto its image. Recall that for a polarized abelian variety (A, H), 2H is globally generated and 3H is very ample. Theorem 1.3 and [JLT, Theorem A] are the birational analogues for canonical bundles of varieties of maximal Albanese dimension. The second purpose of this article is to study the structure of smooth projective varieties X of general type and of maximal Albanese dimension with χ(x, ω X ) = 0. In recent years, these varieties have attracted considerable attention. Green and Lazarsfeld showed in [GL1] that a variety of maximal Albanese dimension satisfies χ(x, ω X ) 0. It was conjectured by Kollár [K3, ] that a variety of general type and maximal Albanese dimension would satisfy χ(x, ω X ) > 0. A couple years later, Ein-Lazarsfeld disprove the conjecture by providing an example of threefold of general type and maximal Albanese dimension with χ(x, ω X ) = 0. In fact, in the recent studies on the structure of the pluricanonical maps and of the Iitaa map, it has been realized that the case χ(x, ω X ) = 0 is usually the hardest case. For example, it was shown in [CH1] that the tricanonical map is birational for varieties of general type and maximal Albanese dimension with χ(x, ω X ) > 0. However, if we assume χ(x, ω X ) = 0 instead, then it is more difficult to prove that the tricanonical map is birational (see [JLT]). It is thus natural and important to characterize or classify varieties of general type and maximal Albanese dimension with χ(x, ω X ) = 0. Our previous joint wor [CDJ] with Olivier Debarre was the starting point toward this direction, in which we prove that the Albanese variety of X has at least three simple factors and the example of Ein and Lazarsfeld is the only possible variety in dimension three. Note that the characterizing properties of X are preserved under birational maps and finite étale maps, the classification of X would be up to birational maps and étale covers.

3 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 3 Inspired by the results in [CDJ], we formulate a conjectural characterization (see Conjecture 6.6) of X. The method in [CDJ] relies heavily on special properties of surfaces and curves and thus is rather difficult to be generalized to higher dimensions. We here propose a different approach. First, as an application of Theorem 1.1, we prove some birational criteria for morphisms between varieties of maximal Albanese dimension. Then, we show that these criteria surprisingly give strong restraints on the structure of X. In particular, when that A X has only three simple factors, we are able to prove Conjecture 6.6. Theorem 1.4. Let X be a variety of general type and of maximal Albanese dimension and assume that A X has only three simple factors. If χ(x, ω X ) = 0, there exist simple abelian varieties K 1, K 2, K 3, double coverings from normal varieties F i K i with associated involution τ i, and an isogeny η : K 1 K 2 K 3 A X, such that the base change X is birational to (F 1 F 2 F 3 )/ σ where σ = τ 1 τ 2 τ 3 is the diagonal involution. following commutative diagram: That is, we have the X (F 1 F 2 F 3 )/ σ ε a X K 1 K 2 K 3 where ε is a desingularization. X a X A X. It is worth mentioning that the arguments in the proof of Theorem 1.4 are quite general and should be useful to attac Conjecture 6.6. The paper is organized as follows. In section 2 we introduce definitions and prove some basic results on Fourier-Muai transform of GV sheaves. Section 3 is devoted to prove the decomposition theorem and section 4 contains several applications of the decomposition theorem on pluricanonical systems. In section 5 we provide several birational criteria of morphisms between varieties of maximal Albanese dimension. In section 6 we study the general structure of varieties X of general type and of maximal Albanese dimension with χ(x, ω X ) = 0. Finally, in section 7, we restrict ourselves to the case when A X has only three simple factors and prove Theorem 1.4. Acnowledgements. This wor started during the second author s visit to NCTS (Mathematics Division, Taipei Office). The second author thans NCTS for their warm hospitality and the excellent research atmosphere. The authors than Olivier Debarre for numerous conversations on this subject. η

4 4 J. CHEN AND Z. JIANG 2. Notation and Preliminaries For any smooth projective variety X, we will denote by a X : X A X the Albanese morphism of X and ÂX = Pic 0 (X) the dual of the Albanese variety. We will denote by D(X) the bounded derived category of coherent sheaves on X. Following [PP2], for any object E D(X), we write R (E ) := RH om(e, ω X ). For an abelian variety A and its dual Â, we denote by P A the normalized Poincaré line bundle on A Â. For α Â, we denote by P α the line bundle that represents α. By [Mu], the following functors give equivalence between D(A) and D(Â): RΦ PA : D(A) D(Â), RΦ P A ( ) = Rp (p A( ) P A ), RΨ PA : D(Â) D(A), RΨ P A ( ) = Rp A (p  ( ) P A). For any coherent sheaf F on X and any morphism f : X A to an abelian variety, we define the i-th cohomological locus V i (F, f) := {α  Hi (X, F P α ) 0}. If f = a X is the Albanese morphism, we will simply denote by V i (F ) the i-th cohomological locus. For an abelian variety A and its dual Â, we always use the notation to denote an abelian subvariety of Â, and then = is the natural quotient of A. We recall the definition of GV-sheaves and M-regular sheaves on abelian varieties (see [PP1] and [PP2]). Definition 2.1. Let F be a coherent sheaf on an abelian variety A. Then F is a GV-sheaf if codimâ Supp R i Φ PA (F ) i for all i 0; F is a M-regular sheaf if for all i > 0. codimâ Supp R i Φ PA (F ) > i Remar 2.2. The following properties of M-regular sheaves and GV-sheaves are quite useful: 1) if F is a GV-sheaf (resp. M-regular sheaf) on A, then R (F ) := RΦ PA (R (F ))[g] is a coherent (resp. torsion-free) sheaf of  supported on V 0 (F ) ([H1, Theorem 1.2], [PP3, Proposition 2.8]); 2) if F is a GV-sheaf, then E xt i ( R (F ), OÂ) ( 1Â) R i Φ PA (F ) (see [PP2, Remar 3.13]);

5 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 5 3) F is a GV-sheaf (resp. M-regular sheaf) if and only if codimâ V i (F ) i (resp. > i) for all i 1 ([PP2, Lemma 3.6]). The following proposition is a generalization of [BL, Proposition (b)] Proposition 2.3. Let f : A B be a quotient of abelian varieties. Assume that dim A = g and dim B = g 1. Then (1) we have the natural isomorphism of functors from D(B) to D(Â): RΦ PA f [g] f RΦ PB [g 1 ]; (2) and the natural isomorphism of functors from D( B) to D(A): f RΨ PB RΨ PA f Remar 2.4. Note that the direct image f needs not to be derived for a closed embedding and similarly, f needs not to be derived for a smooth morphism. Proof. We note that (2) is equivalent to (1) by Fourier-Muai equivalence. It suffices to prove (1). For brevity, we will abuse the notation of pull-bac and push-forward with its derived functors. We consider the following commutative diagram: f p A A A  pâ fâ π πâ B B B   p f B f B B B p B B, where each map is either the natural projection, dual map, or base change. We then have RΦ PA (f F ) = pâ (p Af F P A ) pâ (f  π BF P A ) πâ fâ (f  π BF P A ) πâ (π BF fâ P A ) (projection formula). Similarly, f RΦ PB (F )) = f p B (p BF P B ) πâ fb ( f Bπ BF P B ) πâ (π BF f B P B ) (projection formula). Thus if f has connected fibers, we conclude the proof by Lemma 2.5 below.

6 6 J. CHEN AND Z. JIANG If f has disconnected fibers, we consider the Stein factorization of f: A g B π B, where π is an isogeny between abelian varieties and g is a fibration. Let F D(B). Since π is an isogeny, by [BL, Proposition (b)], we have RΦ PB (π F ) π RΦ PB (F ). Since g is a fibration, we also have Thus we have This completes the proof. ĝ RΦ PB (π F )[g 1 ] RΦ PA (g π F )[g]. f RΦ PB (F )[g 1 ] ĝ π RΦ PB (F )[g 1 ] ĝ RΦ PB (π F )[g 1 ] RΦ PA (g π F )[g] RΦ PA (f F )[g]. Lemma 2.5. Let f : A B be a quotient of abelian varieties with connected fibers. Keeping the notation as in Proposition 2.3, we have Rf P A [g] f B P B [g 1 ] D(B Â). Proof. Let K be the ernel of f. Step 1. We first assume that A = B K and f : A B is the natural projection. Then P A = p 13 P B p 24 P K on A  = B K B K and f : B B K is the closed embedding x (x, 0 K) for x B. We consider A  = B K B K fâ=(p 1,p 3,p 4 ) B  = B B K p 24 q K K K K p K Therefore, f B p B B B fâ P A = fâ (p 13P B p 24P K ) = fâ (f  p BP B p 24P K ) = p BP B fâ p 24(P K ) = p BP B q Kp K (P K ) = p BP B q KC 0 K [g 1 g] ( f B P B )[g 1 g].

7 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 7 Step 2. We assume that f : A B has connected fibers. By Poincaré s reducibility theorem, we can tae π B : B B to be an isogeny such that the fiber product f : à := A B B B is isomorphic to the projection B K B. We then have the following commutative diagram: à à f à B à f B B B π 1 π 2 π 3 à  h A B  h B B B τ 1 τ 2 τ 3 A  fâ B  f B B B. By [BL, (14.2)], one sees that π 1 P à = τ 1 P A and π 3 P B = τ 3 P B. Together with base change and Step 1, it is easy to verify that τ 2 fâ P A [g] π 2 f à PÃ[g] π 2 f B P B[g 1 ] τ 2 f B P B [g 1 ]. Thus fâ P A [g g 1 ] is a coherent sheaf supported on the image of f B and of ran 1 on each point of its support. Therefore, we have fâ P A ( fb P B L ) [g 1 g] for some L Im(Pic 0 (B) Pic 0 (B Â)). Rp P A = C 0Â[ g], we have L = O B Â. Finally, by the fact that Proposition 2.6. Let f : A B be a quotient of abelian varieties and let F be a GV-sheaf on B. Then R (f F ) := RΦ PA (R (f F ))[g] is a coherent sheaf supported on f( B). Moreover, if F is M-regular on B, then R (f F ) is a pure sheaf supported on f( B). Proof. Since F is GV-sheaf, R (F ) := RΦPB (R (F ))[g 1 ] is a coherent sheaf on B by Remar By Lemma 2.3, it follows that RΦ PA (R (f F ))[g] = R f ( R (F )) = f ( R (F )) is also a coherent sheaf supported on f( B). If F is M-regular on B, then R (F ) is torsion-free by Remar Hence R (f F ) is a pure sheaf supported on f( B). Corollary 2.7. Let f : A B be a quotient of abelian varieties. Let F 1 be a M-regular sheaf on B and let F 2 be a GV-sheaf on A. Assume that f( B) is not contained in V 0 (F 2 ), then Hom A (f F 1, F 2 ) = 0.

8 8 J. CHEN AND Z. JIANG Proof. We have Hom A (f F 1, F 2 ) = Hom D(A) (f F 1, F 2 ) Hom D(A) (R (F 2 ), R (f F 1 )) Hom D( Â) ( R (F 2 ), R (f F 1 )) by [Mu, Corollary 2.5] = HomÂ( R (F 2 ), R (f F 1 )). By Proposition 2.6 and Remar 2.2, R (f F 1 ) is a pure sheaf supported on f( B) and R (F 2 ) is supported on V 0 (F 2 ). Since f( B) is not contained in V 0 (F 2 ), we have We conclude the proof. HomÂ( R (F 2 ), R (f F 1 )) = Decomposition theorems for f ω X Let f : X A be a generically finite morphism onto its image and A is an abelian variety. In this section we study the sheaf f ω X. By the wor of Green-Lazarsfeld ([GL2]) and Simpson [S], we now that V i (ω X, f) = V i (f ω X ) is a union of torsion translated abelian subvarieties of Â. Moreover, each irreducible component of V i (ω X, f) in  is of codimension i ([GL1]). In particular f ω X is a GV-sheaf on A. However f ω X often fails to be M-regular, namely V i (f ω X ) often has an irreducible component of codimension i in Â. In particular, if f is generically finite onto A, then O A is a direct summand of f ω X. Definition 3.1. Let f : X A be a generically finite morphism to an abelian variety A. When f is surjective onto A, then we define W X/A or simply W to be the direct summand of f ω X so that f ω X = O A W X/A. When f is not surjective onto A, we define W X/A to be f ω X. Instead, we are interesting in the M-regularity of W. This is actually the main technical difficulty to study birational geometry of varieties of maximal Albanese dimension. Hence we consider the following set which measures how far W X/A is from being M-regular. Definition 3.2. Let F be a GV-sheaf on A, of which cohomological support loci consists of torsion translated subtori. We define S i (F ) to be the set of torsion translated subtori of  consisting of irreducible components of V i (A, F ) of codimension i in Â. We define S(F ) := n i=0 Si (F ) and S(F ) := n i=1 Si (F ). For brevity, We use SX i (resp. S X, SX ) to denote S i (W X/A ) (resp. S(W X/A ), S (W X/A ). Therefore, by definition, S(F ) is empty if and only if F is M-regular. Note that V i (W X/A ) = V i (f ω X ) unless q(x) = dim X and i = dim X, and hence every irreducible component of S X has dimension > 0 by [EL, Theorem 3].

9 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 9 Convention. For each component T = P +  SX i, we have the dual abelian variety A together with a surjection p : A A. We consider a modification of the Stein factorization: (1) X f A q p h Y A where q is a fibration, Y is a smooth projective variety. We denote by M the image of h. We now that R i q ω X = ω Y and, if T passes through the origin, by [CDJ, Theorem 3.1],V 0 (h ω Y ) = Â, where i = codim(â, Â). Lemma 3.3. For each  SX i j 0, we have passing through the origin of  and each S j Y = { Si+j X  Â}. Proof. By Kollár s results [K1, Proposition 7.6] and [K2, Theorem 3.1], we have R i q ω X = ω Y, and for P Â, we have i H i+j (X, ω X q h P ) H i+j t (Y, R t q ω X h P ) t=0 i H i+j t (A, h R t q ω X P ). t=0 Hence S j Y { Si+j X  Â}. On the other hand, by Hacon s generic vanishing theorem [H1, Corollary 4.2], h R t q ω X is a GV-sheaf on A for each t 0. Thus S j Y { Si+j X  Â}. Theorem 3.4. Assume that each component of S X passes through the origin of Â. Then for each  S X, there exists a nontrivial M-regular sheaf F on A, which is a direct summand of h ω Y. Moreover we have an isomorphism W X/A p F. Proof. For each  SX i, we use the notation in the diagram (1) and define Z := Y A A. Then Z Y is a smooth abelian fibration and Z is a smooth projective variety. We recall that the dimension of a general fiber of q is i (see for instance the proof of [EL, Theorem 3]). Hence we have the natural morphism g : X Z,

10 10 J. CHEN AND Z. JIANG which is generically finite and surjective. Considering the natural morphisms f (2) we have X Z q g f r A p Y h A, (3) f ω X = f g ω X = f (ω Z Q ) = p (h ω Y ) f Q, where the last equality holds because the right part of diagram (12) is Cartesian. We now let 0 < d N < d N 1 < < d 2 < d 1 < n be the positive numbers such that S d i X is not empty. Note that Theorem 3.4 holds in dimension 1. We argue by induction on dim X. Thus we suppose that Theorem 3.4 holds for varieties of dimension < dim X. Step 1. Define F i on A i for each Âi SX. Suppose Âm S dr X, we now by Lemma 3.3 that, S Ym = S 0 Y m S d r 1 d r Y m... S d 2 d r Y m S d 1 d r Y m = {Âm} { j<rs d j X  Âm}. By induction, we now that there exists a nontrivial M-regular sheaf F m on A m such that h m ω Ym is a direct sum of F m with sheaves pulled bac from the dual of elements of SY m. Therefore, for P Âm general, we have (4) dim H 0 (Y m, ω Ym h mp ) = dim H 0 (A m, F m P ), and moreover, by (3), p mf m is a direct summand of W X/A. Step 2. Derive the decomposition. We start with S d 1 X. By Step 1, for each  S d 1 X, there exists a coherent sheaf W on A such that we have the decomposition: W W/A = p F W. For distinct components  1,  2 in S d 1 X, we now consider the decomposition of identity (5) Id : p 2 F 2 W X/A = p 1 F 1 f 1 W 1 p 2 F 2. By Corollary 2.7, we now that Hom A (p 2 F 2, p 1 F 1 ) = 0.

11 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 11 Hence p 2 F 2 is a direct summand of W 1 and thus p 1 F 1 p 2 F 2 is a direct summand of W X/A. Continuing in this way, we finally get a decomposition: W X/A =  S d 1 X p F Wd1. Now we apply induction on d i. Suppose that we have the decomposition W X/A =  j r 1 S d j X p F Wdr 1, where r N. Then for Âm S dr X, we consider Id : p mf m  j r 1 S d j X p F Wdr 1 p mf m, we again conclude by (2.7) that Hom A (p mf m,  j r 1 S d j X p F ) = 0. It follows that p mf m is indeed a direct summand of W dr 1. As before, we then get the decomposition for each element of S dr X : W X/A =  j r S d j X p F Wdr. By induction, we end up with the decomposition (6) W X/A =  S X p F WdN. Step 3. Show that W dn is either M-regular on A if V 0 (f ω X ) =  or trivial otherwise. As a direct summand of W X/A, W dn is a GV-sheaf (possibly trivial). It suffices to show that S (W dn ) =. Assume to the contrary that S (W dn ) SX is not empty. We then pic  m S dr (W dn ) for some d r.

12 12 J. CHEN AND Z. JIANG For P Âm general, we have h dr (A, f ω X p mp ) = h dr (A, W X/A p mp ) = h dr (A, p F p mp ) + h dr (A, W dn p mp ) Â S X h dr (A, p m(f m P )) + h dr (A, W dn p mp ) > h dr (A, p m(f m P )) = i+j=d r h i (A m, R j p m p m(f m P )) = h 0 (A m, R dr p m p m(f m P )) = h 0 (A m, F m P ), where the second equality holds because of (6) and the last two equalities holds because for each j 0, R j p m p mf m is a direct sum of copies of F m and hence is M-regular on A m, and in particular R dr p m p m(f m ) = F m. On the other hand, since R j p m f ω X is a GV-sheaf on A m, we have h dr (A, f ω X p mp ) = h 0 (A m, R dr p m f ω X P ) = h 0 (A m, h m R dr q m ω X P ) = h 0 (A m, h m ω Ym P ). Combining all the (in)equalities, we get that, for P Âm general, h 0 (A m, h m ω Ym P ) > h 0 (A m, F m P ), which is a contradiction to (4). Therefore, S (W dn ) =. Theorem 3.5. Let f : X A be a generically finite morphism. (1) For each T = P + Â S X, there exists a nontrivial M-regular sheaf F T on A supported on M = h (Y ) such that we have an isomorphism W X/A p F T. T S X P 1 (2) Moreover, if T S X, then T S X and, as coherent sheaves supported on M, we have rf T = rf T. Problem 3.6. Let f : X E be a surjective morphism from a smooth projective variety (or even a Kähler manifold) to an elliptic curve. Fujita ([F, Theorem 3.1]) showed that f ω X is a direct sum of torsion line bundles with an ample vector bundle. It would be very nice to have a generalization of both Theorem 3.5 and Fujita s result. Let f : X A be a morphism from a smooth projective variety to an abelian variety, f ω X is still a GV-sheaf. Do we have a similar decomposition result for f ω X?

13 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 13 Proof. We first prove (1), which is a direct corollary of Theorem 3.4. We tae an étale cover π : A A such that, for the induced morphism X := X A A A, every element of S X contains the origin of Â. Note that W X /A π W X/A and hence R W X /A π R W X/A. We denote respectively by M X and M X the sheaves R (WX /A ) and R (WX/A ). Then, by Proposition 2.3, M X π M X. By Theorem 3.4 and Remar 2.2, we now that M X is a direct sum of pure sheaves supported on abelian subvarieties of Â. Since π is an isogeny between abelian varieties, M X is a direct summand of π π M X = π M X. Hence M X is also a direct sum of pure sheaves supported on torsion translates of abelian subvarieties of Â: M X = τ P ι M, where τ P is the translation by a torsion element P Â, ι :   is the embedding of an abelian subvariety of Â, and M is a torsion-free sheaf on Â. Note that, by Fourier-Muai duality, we have the formula: (7) ( 1 A ) W X/A R RΨ PA (M X ) R RΨ PA ( τp ι M ). Since τ P is a translation and by Proposition 2.3, we have RΨ PA (τ P ι M ) P 1 RΨ PA (ι M ) P 1 p RΨ P A M. Since p is a smooth morphism, by [Hu, (3.17)], we have R (P 1 p RΨ P A M ) R (p RΨ P A M ) P p ( R (RΨPA M ) ) P. We define F := R (RΨ PA (M )). Since p F P is a direct summand of the coherent sheaf W X/A, it follows that F is a sheaf. Moreover, RΦ PA R (F )[dim A ] ( 1) A M is a pure sheaf on A, hence F is M-regular. This concludes the proof of the first part of Theorem 3.5. In order to prove (2), we first begin with a technical lemma. Let B  be an abelian subvariety and let p : A B be the corresponding quotient. For each  B, we denote by p : B A the dual quotient. Lemma 3.7. Let m = dim A dim B. Then for Q Â, we have R m p (W X/A Q) R m p (p F T P 1 Q) T Q+ B T Q+ B ( p F T Q P 1 ) ( dim A dim A m ), where, in the last equality, we notice that Q P B by the condition T Q + B.

14 14 J. CHEN AND Z. JIANG Proof. For T S X, we first show that R m p (p F P 1 Q) 0 only when T Q + B. For each A, we denote by B the neutral component of B  in  and denote by B the abelian subvariety of  generated by B and Â. We then have the commutative diagram: A ε p p B α A δ γ B B, β where the right part is a Cartesian. We denote by j = dim A dim B = dim B dim B. Then we have R m p (p F T P 1 Since P 1 Q) = R j δ R m j ε (ε α F T P 1 Q). Q is a torsion line bundle, R m j ε (ε α F P 1 Q) 0 only if Q P B and in this case we write Q P = Q 1 + Q 2, with Q 1 B and Q 2  two torsion points. Then we have R m j ε (ε α F T P 1 Q) = α (F T Q 2 ) δ Q 1. By flat base-change, we have R j δ (α (F T Q 2 ) δ Q 1 ) = R j δ (α (F T Q 2 )) Q 1 = β R j γ (F T Q 2 ) Q 1. We claim that R j γ (F T Q 2 ) 0 only if j = 0, namely B =  B = B. Actually, F T is a direct summand of h R dim A dim A p (ω X P ). If j > 0 and R j γ (F T Q 2 ) 0, we can repeat the argument in the proof of Lemma 3.3 to show that S(F T Q 2 ) contains a sub-torus of B, which is absurd since F T Q 2 is M-regular. Overall, we see that R m p (p F T P 1 Q) 0 implies that  B and Q P B and hence T Q + B. Considering A p B p A, p then R m p (p F T P 1 Q) = ( p F T (P 1 Q) ) ( dim A dim A m ). We are now ready to prove (2). Let 0 < d N < d N 1 < < d 2 < d 1 < n be the positive numbers such that S d i X is not empty. We again argue by decreasing d i.

15 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 15 Let T S d 1 X and we may assume that [T ] Â/Â has order > 0, namely P / Â. By Lemma 3.7, F T = h R d 1 q (ω X P ) is M-regular. Then for y Y general, H d 1 (X y, ω Xy P ) 0. Thus P Xy = O Xy. Then R d 1 q (ω X P 1 R d 1 p (W X/A P 1 ) is a non-trivial GV-sheaf on A. Moreover, since P / Â, ) = h R d 1 q (ω X P 1 ) is non-trivial. We apply again Lemma 3.7 and note that A is minimal among inclusion, hence T S d 1 X and rf T = deg(y /M ) = rf T. We then assume the claim in (2) holds for each T j s 1 S d j X. Now assume that T S ds X and [T ] Â/Â has order > 0. As before, R ds p (W X/A P 1 ) is non-trivial and has the same ran as R ds p (W X/A P ). By Lemma 3.7, we have R ds p (W X/A P ) = ( p j F Tj P j P 1 ) ( dim A dim A j m ) and T j T = F T R ds p (W X/A P 1 ) = T j T T j T ( p j F Tj P j P 1 ) ( dim A dim A j m ), ( p j F Tj P j P 1 ) ( dim A dim A j m ). Note that T j T if and only if T j T. Moreover, by induction, rf Tj = rf Tj. Thus the following two sheaves on M, ( p j F Tj P j P 1 ) ( dim A dim A j m ) and, T j T T j T ( p j F Tj P j P 1 ) ( dim A dim A j m ) have the same ran. Thus rf T = rf T > 0 and hence F T 0. This finishes the proof of (2). The following corollary is clear from the above theorem and will be used to prove a birational criterion for morphisms between varieties of maximal Albanese dimension in Section 5. Corollary 3.8. Let f : X A be a generically finite morphism. Assume that Q is a direct summand of f ω X. Then we have the decomposition for the sheaf R (Q): R (Q) M, P +Â S(Q) where M is a pure sheaf supported on P + Â.

16 16 J. CHEN AND Z. JIANG 4. Pluricanonical systems Theorem 3.5 can also be used to get information about pluricanonical systems of X. Theorem 4.1. Let f : X A be a generically finite morphism. Then there exists an abelian Galois étale cover π A : à A with the base change f : X := X A à à such that K is globally generated away from the X exceptional locus of f. Proof. By Theorem 3.5, we have W X/A P + S X P 1 p F, where F is M-regular on A. By [D, Proposition 3.1], there exists an abelian Galois étale cover π A : à A such that πa 1 (P p F ) is globally generated. Considering the base change: X f à X π f we have W X/ à = π A W X/A. Therefore, W X/ is globally generated and so is à f ω X. This implies that K X is globally generated away from the exceptional locus of f. Theorem 4.2. Let f : X A be a generically finite morphism. Then 2K X is generated by its global sections, away from the exceptional locus of f. Proof. The proof uses Theorem 3.5 and Pareschi-Popa s argument ([PP1, Proposition 2.12]). For simplicity, we may assume that W X/A = f ω X. By [PP1, Proposition 2.13], M-regular sheaves are continuously globally generated. Hence there exists N > 0, for each T S X and P 1,..., P N  general, the sum of twisted evaluation map H 0 (p F T P 1 P i ) P P 1 i F T 1 i N T S X 1 i N A, is surjective. Thus the sum of twisted evaluation map H 0 (X, ω X P 1 P i ) P P 1 i π A ω X is surjective away from the exceptional locus of f. Note that since P 1,..., P N T are general then P 1N,..., P N T are also general. We then consider the commutative diagram,i H0 (ω X P P 1 i ) H0 (ω X P 1 P i ) O X H 0 (ω 2 X ) O X,i H0 (ω X P 1 P i ) ω X P P 1 i ω 2 X.

17 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 17 Hence H 0 (ωx 2 ) O X ωx 2 of f. is surjective, away from the exceptional locus Theorem 4.3. Let f : X A be a generically finite morphism and assume that X is of general type. Then the bicanonical map of X is generically finite onto its image. Proof. Let ϕ : X P N be the bicanonical map of X. Taing an appropriate birational model of X, we may assume that ϕ is a morphism. Let F be an irreducible component of a general fiber of ϕ. Then we have dim Im(H 0 (X, 2K X ) H 0 (F, 2K F )) = 1. We need to prove that dim F = 0. Assume that dim F > 0. We note that f F : F A is generically finite onto its image. Chen and Hacon proved that the translates through the origin of the irreducible components of V 0 (ω X, f) generates  ([CH1, Theorem 1]). Hence we can tae T = P + B a positive dimensional component of V 0 (ω X, f) such that the composition of morphisms g : F f F A B is nonconstant. Note that each positive dimensional irreducible component of V 0 (ω X, f) is an element of S X. Thus T S X and by Theorem 3.5, T S X. In particular, for each Q T, H 0 (X, K X +Q) 0 and H 0 (X, K X Q) 0. Since g is nonconstant, we can tae a curve C T such that the restriction map C Pic 0 (F ) is generically finite. For Q C, considering the commutative diagram K X + Q K X Q 2K X restriction K F + Q F K F Q F 2K F, restriction we see that the image of the restriction map H 0 (X, 2K X ) H 0 (F, 2K F ) has dimension > 1, which is absurd. 5. Criteria of birationality We consider a surjective and generically finite morphism between smooth projective varieties t : X Y. We are interested to now when t is birational. When there exists a generically finite morphism p : Y A over an abelian variety A, there are several cohomological criterion about the birationality of t, see for instance [HP, Theorem 3.1], and [CDJ, Lemma 5.4]. In this section, we consider the case when X t Y g A are generically finite over the abelian variety A. We denote by f = g t. We will always assume that (8) V 0 (ω X, f) = N i=1âi,

18 18 J. CHEN AND Z. JIANG where Âi is a proper abelian subvariety of Â, and in particular, we have χ(x, ω X ) = 0. We may write (9) t ω X = ω Y Q. Then t is birational if and only if Q = 0. Since f ω X is a GV-sheaf, so is g Q. Hence Q = 0 if and only if V 0 (Q, g) =. For any irreducible component Âi V 0 (ω X, f), we consider the Stein factorizations (10) and the set and its complementary set X t Y h Xi h Yi g A X i t i Y i g i A i Σ b := {1 j N t j is birational}, Σ nb := {1 j N t j is not birational}. Proposition 5.1. Under the above assumptions, it follows that 1) for any T S(g Q), T j Σnb  j, and in particular, T  j ; j Σ nb T S(g Q) 2) if all t i are birational, t is also birational. Proof. We first prove 1). Assume that S(g Q) and tae T S(g Q) S(f ω X ). By Simpson s theorem [S], we can write T = P + K, where P is a torsion point and K is a abelian subvariety of Â. Assume P + K Â1, it suffices to prove that 1 Σ nb. Since P Â1, we may tae an étale cover A 1 A 1 such that the pullbac of P is trivial. After base change by A 1 A 1, diagram (10) now reads: X t X 1 h X Y t 1 Y 1 h Y π i g A g 1 A 1 where all h X, h Y are fibrations and t 1 is birational if and only if t 1 is birational. We now that P + K is an irreducible component of V (Q, g). Then we consider the composition of morphisms X t Y g A A K and tae π

19 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 19 smooth models of the Stein factorization of X K and Y K. We get X t Y g A µ X µ Y Z X t K Z Y g K K We write t ω X = ω Y Q, where Q is the pull-bac of Q on Y and moreover, since the pull bac of P is trivial on A, π K K Â is an irreducible component of V (Q, g ). We now that R i µ Y Q is a GV-sheaf on Z Y for each i 0 by Hacon s version of generic vanishing theorem (see [H1, Corollary 4.2]), we conclude that V 0 (R µ Y Q, g K ) = K. We then observe that t K ω ZX = t K R µ X ω X by [K1, Proposition 7.6], = R µ Y t ω X = ω ZY R µ Y Q. Therefore, h 0 (Z X, ω ZX ) > h 0 (Z Y, ω ZY ), we conclude that deg t K > 1. Notice that we have X t Y π K X 1 t 1 Y 1 Z X t K Z Y where all the vertical morphisms are fibrations. Hence deg t K > 1 implies that deg t 1 > 1. Therefore, t 1 is not birational. This implies that 1 Σ nb. For 2), we note that it suffices to prove that V 0 (Q, g) =. Assume that V 0 (Q, g) is not empty, then it contains an irreducible component T of codimension dim X. By [PP2, Proposition 3.15], T is also an irreducible component of V (Q, g). In particular, T S(g Q). On the other hand, all t i are birational by assumption, thus Σ nb =. Therefore, we have by 1) that S(g Q) =, which is a contradiction. It will be useful to consider birationality when X dominates more than one varieties. Let s consider the following commutative diagram: f 1 X Y 1 q f 2 a 2 Y 2 A where all the morphisms are generically finite and surjective. We then write f i ω X = ω Yi Q i for i = 1, 2. a 1

20 20 J. CHEN AND Z. JIANG Proposition 5.2. Suppose that S(a 1 Q 1 ) S(a 2 Q 2 ) =, then either f 1 or f 2 is birational. Proof. Let s denote by r i the degree of f i and denote by s i the degree of a i. Certainly, r 1 s 1 = r 2 s 2 = deg q. We may assume that f 1 is not birational. Then Q 1 is a sheaf of ran r 1 1 > 0. We note that a i Q 1 is a direct summand of q ω X = a i ω Y2 a i Q 2, for i = 1, 2. Thus R (ai Q i ) is a direct summand of Corollary 3.8, we have the decomposition for R (ai Q i ): R (q ω X ). By (11) R (a i Q i ) P +Â S(a i Q i ) F, where F is a pure sheaf supported on P + Â. We now consider the morphisms: We notice that Id : a 1 Q 1 Ψ=(Ψ 1,Ψ 2 ) a 2 ω Y2 a 2 Q 2 a 1 Q 1 Hom D(A) (a 1 Q 1, a 2 ω Y2 a 2 Q 2 ) ( Hom D(A) R (a2 ω Y2 ) R (a 2 Q 2 ), R (a 1 Q 1 ) ) ( Hom D( R (a Â) 2 ω Y2 ) R (a 2 Q 2 ), R (a1 Q 1 ) ). We denote by Φ i the image of Ψ i under the above natural transformation. By assumption S(a 1 Q 1 ) S(a 2 Q 2 ) =, thus by (11), we have Φ 2 = 0 and hence Ψ 2 = 0. Therefore, a 1 Q 1 is a direct summand a 2 ω Y2. We note that h n (A, a 2 ω Y2 ) = 1 but h n (A, a 1 Q 1 ) = h n (A, q ω X ) h n (A, a 1 ω Y1 ) = 0. Hence ran a 2 ω Y2 > ran a 1 Q 1. We then compare the ran of these two sheaves: s 2 > s 1 (r 1 1). Hence r 1 s 1 = r 2 s 2 > r 2 s 1 (r 1 1) and r 2 < r 1 r 1 1. Therefore r 2 = 1 and f 2 is birational. Here is an application of Proposition 5.2 and will be used in the last section. Corollary 5.3. Under the setting of Proposition 5.1, suppose that all t i but possible one are birational, then t is birational. Proof. Assume that t is not birational, then Σ nb. We may assume that Σ nb = {1}. We define Z := X 1 A1 A. Then the induced morphism s : X Z is generically finite and surjective.

21 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 21 We now consider the commutative diagram: t X Y q a 2 Z A. s Write t ω X = ω Y Q Y and s ω X = ω Z Q Z. We note by Proposition 5.1 that all elements of S(a 1 Q Y ) are contained in Â1. On the other hand, by the construction of Z, any element of S(a 2 Q Z ) is not contained in Â1. Hence S(a 1 Q Y ) S(a 2 Q Z ) =. We now apply Proposition 5.2 to conclude that either t or s is birational. However, s cannot be birational because Z is not of general type. This is the desired contradiction. 6. Primitive varieties of χ = 0 We are now ready to apply the results proved in previous sections to study the structure of a smooth projective variety X of maximal Albanese dimension, of general type, and χ(x, ω X ) = 0. In this section, we will formulate a conjecture about the structure of such varieties. Inspired by the main theorem in [CDJ], we find the following definition useful. Definition 6.1. (1) Let X be a smooth projective variety of maximal Albanese dimension, of general type, and χ(x, ω X ) = 0. We say that X is primitive of χ = 0 if for any non-trivial fibration h : X Y to a normal projective variety with a general fiber F, we have χ(f, ω F ) > 0. (2) Let h : X Y be a fibration to a normal projective variety. We call h an irregular fibration if there exists a finite morphism from Y to an abelian variety. For a primitive variety of χ = 0, we now that q(x) = dim X = n, the Albanese morphism a X : X A X is generically finite and surjective, and A X has at least 3 simple factors (see [CDJ, Lemma 4.6 and Corollary 3.5]). Moreover, we see from the definition that primitive varieties of χ = 0 are the building blocs of varieties of maximal Albanese dimension, of general type, and of χ(x, ω X ) = 0. But more precisely, we have the following structural result. Proposition 6.2. Let X be a variety of general type, of maximal Albanese dimension, and χ(x, ω X ) = 0. Then either X is primitive of χ = 0 or there exists an irregular fibration f : X Z with a general fiber F primitive of χ = 0. Proof. If X is not primitive of χ = 0, we tae f : X Y to be a fibration with a general fiber F such that χ(ω F ) = 0 and assume that dim F is minimal among all such fibrations. a 1

22 22 J. CHEN AND Z. JIANG By Lemma 6.3 below and the minimality of dim F, we see that f is an irregular fibration and a X (F ) is a translate of an abelian variety K of A X. We claim that F is primitive of χ = 0. Otherwise, there exists a fibration of F, whose general fiber has χ = 0. Considering the generically finite morphism a X F : F K, we conclude again by Lemma 6.3 that there exists an abelian subvariety K of K such that an irreducible component F of a general fiber of F K K/K has χ(ω F ) = 0. Then considering the Stein factorization X g Z A X /K, F is a general fiber of g, which is again a contradiction to the minimality of the dimension of F. Lemma 6.3. Let α : X A be a generically finite morphism from a smooth projective variety of general type to an abelian variety. Assume that we have a fibration f : X Y with a general fiber F such that χ(ω F ) = 0. Then there exists a quotient of abelian varieties A B such that f factors birationally through the Stein factorization g : X Z of the induced morphism X A B. Moreover, we have χ(ω F ) = 0 for a general fiber F of g. Proof. We consider the morphism α F : F A. By [CH2, Theorem 4.2], there exists an abelian subvariety K of A such that α F (F ) is fibred by K and moreover, an irreducible component F of a general fiber F A A/K has χ(ω F ) = 0. We then tae B to be A/K and let X g Z B be the Stein factorization of X B. It is easy to chec that g is the irregular fibration we are looing for. Here we have some basic properties for primitive varieties of χ = 0. Lemma 6.4. Let X be a primitive variety of χ = 0, then 1) for any simple abelian sub-variety K ÂX, there exists an irreducible component  i of V 0 (ω X, a X ) such that the composition of morphisms Âi ÂX ÂX/ K is an isogeny; 2) for any simple abelian variety K ÂX, the induced morphism X A X K is a fibration. Proof. For 1), we consider the surjective morphism g : X A X K. Since X is primitive, for an irreducible component F of a general fiber of g, we have χ(f, ω F ) > 0. Hence the natural morphism V 0 (ω X, a X ) ÂX ÂX/ K is surjective. Therefore there exists an irreducible component Âi of V 0 (ω X, a X ) such that the natural morphism Âi ÂX/ K is an isogeny. We argue by contradiction to prove 2). Assume that X A is not a fibration, we tae a modification of Stein factorization X M K such that M is a smooth projective variety. Then the morphism M K is not an étale morphism and is of degree 1. Since K is simple, we have χ(m, ω M ) > 0. On the other hand, by 1), there is an irreducible component T i of V 0 (ω X ) such that T i K ÂX is an isogeny. After taing an étale

23 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 23 cover of X, we may assume that T i = Âi passes through the origin of ÂX. Hence X M Y i is generically finite and surjective. However, we then have χ(x, ω X ) χ(m, ω M )χ(y i, ω Yi ) > 0, which is a contradiction. Corollary 6.5. Let X be primitive of χ = 0. Then, (1) if A X has m simple factors, then V 0 (ω X ) has at least m irreducible components; (2) if A X has 3 simple factors, then each irreducible component Âi of V 0 (ω X ) has 2 simple factors. We end this section with a conjectural characterization of primitive varieties of χ = 0. Conjecture 6.6. Let X be a smooth projective variety. Then X is a primitive variety of χ = 0 if and only if there exist simple abelian varieties K 1, K 2,..., K 2+1, double coverings from normal projective varieties of general type F i K i with involutions σ i, and an isogeny K 1 K 2+1 A X such that the base change X is birational to (F 1 F 2+1 )/ σ 1 σ 2+1. Together with Proposition 6.2, Conjecture 6.6 gives all possible structures for varieties of general type, of maximal Albanese dimension, and of χ(ω X ) = 0. We note one direction of Conjecture 6.6 is fairly standard but the other direction seems rather difficult. In next section, we will prove Conjecture 6.6 when A X has only 3 factors. 7. Three simple factors In this section we assume that X is of general type, of maximal Albanese dimension, with χ(x, ω X ) = 0, and A X has only three simple factors. In particular, X is primitive of χ = 0 (see [CDJ, Proposition 4.5]). We are interested in the structure (up to étale covers and birational modifications) of such varieties. We are free to tae étale covers of X and hence we always assume that each component of V 0 (ω X ) passes through the origin of ÂX in this section. We then write V 0 (ω X ) = N i=1âi. We now that the complementary of  i ÂX is a simple abelian variety and N 3. The main theorem of this section is the following generalization of [CDJ, Theorem 5.1]. Theorem 7.1. Let X be a variety of general type, of maximal Albanese dimension, with χ(x, ω X ) = 0, and assume that A X has only three simple factors. There exist simple abelian varieties K 1, K 2, K 3, double coverings from normal varieties F i K i with associated involution τ i, and an isogeny η : K 1 K 2 K 3 A X, such that the base change X is birational to (F 1 F 2 F 3 )/ σ

24 24 J. CHEN AND Z. JIANG where σ = τ 1 τ 2 τ 3 is the diagonal involution. following commutative diagram: That is, we have the X (F 1 F 2 F 3 )/ σ ε a X K 1 K 2 K 3 where ε is a desingularization. X a X A X. t i Let X f i Y i Ai be a modification of the Stein factorization of the morphism X a X p i A X Ai such that Y i is smooth projective, and denote by a X ω X = O AX W X/AX and t i ω Yi = O Ai F i, where F i is M-regular by Lemma 6.4. Since A X has only three simple factors and X is primitive, by Lemma 6.4 2), we now that S X = {Âi 1 i N}. By Theorem 3.4, we have the following proposition, which is our starting point. Proposition 7.2. We have W X/AX N i=1 p i F i. In particular, N deg a X 1 = (deg t i 1). i= Characterization of special primitive varieties. In this subsection, we are going to prove Theorem 7.1 for special primitive varieties of χ = 0 satisfying : ( 1) A X = K 1 K 2 K 3. ( 2) V 0 (ω X ) contains three components Â1 = {0} K 2 K 3, Â 2 = K 1 {0} K 3, and Â3 = K 1 K 2 {0}, and ( 3) the induced morphism X Z := Y i K Y j is birational, for {i, j, } = {1, 2, 3}. Recall that we have generically finite morphism Y i A i and the induced fibrations h ij : Y i K j. These fit into the following diagram (12) Z 1 Z 2 g 32 Z 3 g 13 g 23 g 12 g31 g 21 Y 2 Y 3 h 13 Y 1 h 21 h 31 h 23 h 12 h 32 K 3 K 1 K 2, η

25 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 25 where Z i is the main component of the fiber product over K i. We note that, by [CH1, Theorem 2.3], Z i is of general type. We denote a i = deg t i = deg(y i /A i ) for i = 1, 2, 3. Then by assumption ( 3), deg a X = a i a j for any 1 i j 3. Hence a := a 1 = a 2 = a 3. Lemma 7.3. There are smooth varieties of general type F 1, F 2, F 3 generically finite over K 1, K 2, K 3 respectively such that the fibration h ij : Y i K j is isotrivial with a general fiber F, for {i, j, } = {1, 2, 3}. Proof. We will show that h 13 : Y 1 K 3 is isotrivial and the general fiber of h 13 is birational to a general fiber of h 31 : Y 3 K 1. The same argument wors for other fibrations. Since X Z 1 is birational, there exists a dominant map τ : Z 1 X Y 1 fits into the following commutative diagram: X Z 1 τ f 1 (f 12,f 13 ) K 2 K 3, (h 12,h 13) Y 1 where f 12 = h 32 g 13 and f 13 = h 23 g 12. Fix a general point t K 1, and denote respectively by F 2t and F 3t the fibers of h 31 and h 21 over t. Then the fiber of Z 1 K 1 is isomorphic to F 3t F 2t with the commutative diagram (13) h 32 F 2t F 3t τ F 2t Y 1 3t F h 12 h K 13 2 K 3 K 2 K 3 Since h 21 is a fibration, we now that h 23 deg(f 3t /K 3 ) = deg(y 2 /A 2 ) = a. Let F 2 be a general fiber of h 13 : Y 1 K 3. We also have that deg(f 2 /K 2 ) = deg(y 1 /K 2 K 3 ) = a. Let W be the main component of Y 1 K3 F 3t. From the right part of diagram (13), we see that there is a induced rational dominant map τ : F 2t F 3t W over Y 1. Since { deg(f2t F 3t /Y 1 ) = a 2 / deg(y 1 /K 2 K 3 ) = a, deg(w/y 1 ) = deg(f 3t /K 3 ) = a. It follows that τ is birational. Thus h 13 : Y 1 K 3 is isotrivial with a general fiber F 2 birational to F 2t for any general t.

26 26 J. CHEN AND Z. JIANG Lemma 7.4. V 0 (ω X, a X ) has exactly three irreducible components Â1, Â2, and Â3. Proof. Let F ij := F i F j and F 123 := F 1 F 2 F 3. Since there are dominant generically finite rational map F 12 Y 3 (resp. F 13 Y 2, F 23 Y 1 ), it is straightforward to see that there exists dominant generically finite rational map F 123 Z for = 1, 2, 3. In particular, they induce a dominant generically finite rational map F 123 X. Resolve the indeterminancy via ν : X F 123, we have a generically finite morphism ρ: X X. We claim that V 1 (X, a X ρ) consists of finite unions of translation of Â1, Â 2, and Â3. Since χ(ω X ) = 0, V 0 (ω X, a X ) = V 1 (ω X, a X ) V 1 (ω X, a X ρ) and each component of V 0 (ω X, a X ) is a non-simple abelian subvariety passing through the origin, we conclude the Lemma. It remains to prove the claim. Let p = (α, β, γ) : F 123 A X = K 1 K 2 K 3 be the given surjective morphism. Clearly, ν is birational and p ν = a X ρ. Hence V 1 (ω X, a X ρ) = V 1 (ω X, p ν) = V 1 (ω F123, p). Since K 1, K 2 and K 3 are simple abelian varieties, V 1 (ω F1, α), V 1 (ω F2, β) and V 1 (ω F3, γ) are finite union of isolated points. By Künneth formula, V 1 (ω F123, p) = ( V 1 (ω F1, α) K 2 K ) ( 3 K1 V 1 (ω F2, β) K ) 3 ( K1 K 2 V 1 (ω F3, γ) ). This verifies the claim. By Lemma 7.4, V 0 (ω X ) = Â1 Â2 Â3. Thus we apply Proposition 7.2 and get a 2 1 = 3(a 1). Hence a = 2 and deg(y i /A i ) = deg(f i /K i ) = 2, for i = 1, 2, 3. Replace F i by its Stein factorization over K i, we may and do assume that F i is normal and double cover over K i. Let τ i be the corresponding involution. The covering F 2 F 3 K 2 K 3 = A 1 is Galois with Galois group Z 2 Z 2. The function field K(Y 1 ) is an intermediate field of the extension K(F 2 F 3 )/K(A 1 ) and of degree 2 over K(A 1 ). Together with the fact that Y 1 is of general type, it follows that Y 1 is birational to (F 1 F 2 )/ τ 1 τ 2 by exhausting all intermediate fields. It is clear to see that Y 2, Y 3 has the same structure. In fact, the similar argument also shows that X is birational to (F 1 F 2 F 3 )/ τ 1 τ 2 τ 3. We thus conclude this subsection that Proposition 7.5. Under the hypothesis, there exist simple abelian varieties K 1, K 2, K 3, double coverings from normal varieties F i K i with

27 POSITIVITY IN VARIETIES OF MAXIMAL ALBANESE DIMENSION 27 associated involution τ i, such that X is birational to (F 1 F 2 F 3 )/ σ where σ = τ 1 τ 2 τ 3 is the diagonal involution The general case. We prove the main theorem in this subsection. To start with, it is convenient to introduce the following notions. Definition 7.6. A primitive variety of χ = 0 is said to be special if it is birational to (F 1 F 2 F 3 )/ τ 1 τ 2 τ 3 as in Proposition 7.5. A primitive variety X of χ = 0 is said to be quasi-special if there is an étale base change X X so that X is special. Given a variety X primitive of χ = 0, we said that X is minimal if X is minimal among smooth projective varieties of general type sitting between X and A X (up to birational equivalent). More precisely, let X 0 be a variety of general type sit between X and A X, then X X 0 is birational. Fix a primitive variety X with χ = 0. For any component Âi, we have an map t i : Y i A i. Let d i := deg(t i ). For any two distinct components Âi, Âj, let K ij be the neutral component of Âi Âj. We consider Z ij a desingularization of an irreducible component of the main component of (Y i Kij Y j ) Ai Kij A j A X. Replacing X by its higher model, we may assume that there exists induced maps ρ ij : X Z ij and a ij : Z ij A X. It is easy to see the following properties of Z ij : 1) Z ij is of general type. 2) deg(z ij /A X ) d i d j. 3) V 0 (ω Zij, a ij ) Âi, Âj and we have the natural morphisms Z ij Y i A i and Z ij Y j A j. In particular, Z ij is not quasi-special if d i 3 or d j 3. Lemma 7.7. Pic any three irreducible components, say Â1, Â2, and Â3 of V 0 (ω X ). Assume that all ρ 12, ρ 13, ρ 23 are birational. Then after an abelian étale base change X X, one may assume that X satisfying 1, 2, 3. Hence X is quasi-special. Proof. By [CDJ, Proposition 4.3], we now that after an abelian étale cover à X A X and mae the base change: X a X à X π X a X A X, we may assume that ÃX = K 1 K 2 K 3 and à 1 = π(â1) = {0} K 2 K 3, à 2 = π(â2) = K 1 {0} K 3, and à 3 = π(â3) = K 1 K 2 {0}. As before, for i = 1, 2, 3, we denote by X g i Ỹi t i à i a modification of the

28 28 J. CHEN AND Z. JIANG Stein factorization of the natural morphism X Ãi such that Ỹi is smooth projective. We claim that the induced morphism X Ỹi K Ỹ j is birational for {i, j, } = {1, 2, 3}. To see this, note that since ρ ij is birational, we have deg a X (d i d j ), for any i, j {1, 2, 3}. Thus deg a X = deg a X (d i d j ). We also now that t i and t i are respectively the Albanese morphisms of Y i and Ỹi. Thus deg t i d i. Thus deg ( (Ỹi K Ỹ j )/ÃX) di d j for {i, j, } = {1, 2, 3}. Hence the induced morphism X Ỹ i K Ỹ j is birational. Proposition 7.8. We have the following criterion for quasi-special primitive varieties with χ = 0: (1) if X is minimal, then X is quasi-special. (2) if V 0 (ω X, a X ) has three components, then X is quasi-special. (3) if deg(x/a X ) = 4, then X is quasi-special. Proof. (1). Suppose that X is minimal. We pic three components Â1, Â 2, and Â3. For (i, j) = (1, 2), (1, 3) and (2, 3), Z ij of general type sitting between X and A X and hence is birational to X by minimality of X. Therefore X is quasi-special by Lemma 7.7. (2). For each (i, j) = (1, 2), (1, 3) and (2, 3), we have the commutative diagram: X ρ ij Z ij A X Y t Y t A t, where t = i or j. Since V 0 (ω X ) consists of three components Â1, Â2, and Â3, by Corollary 5.3, ρ ij is birational. Thus by Lemma 7.7, X is quasi-special (3). Again let X 0 be a minimal primitive variety dominated by X. Since X 0 is quasi-special, deg(x 0 /A X ) = 4. It follows immediately that deg(x/x 0 ) = 1 and hence X is quasi-special. Proposition 7.9. It X is primitive of χ = 0 such that A X has three simple factors, then X is quasi-special. Proof. Suppose on the contrary that X is not quasi-special. Since minimal primitive varieties are quasi-special, there exists X sitting between X and A X such that X is not quasi-special but any other general type variety dominated by X is quasi-special. Replace X by X, we may and do assume that X is not quasi-special, but any variety of general type dominated by X and not birational to X is quasi-special. For X f i t Y i i Ai, we denote by d i the degree of t i, for each A i S X. We may assume that d 1 d N 2. Since X is not quasi-special, by