On the Iitaka fibration of varieties of maximal Albanese dimension

We prove that the tetracanonical map of a variety $X$ of maximal Albanese dimension induces the Iitaka fibration. Moreover, if $X$ is of general type, then the tricanonical map is birational.


Introduction
In the study of smooth complex algebraic varieties, the natural maps defined by differential forms are of special importance. In particular pluricanonical maps have been extensively studied.
When X is of maximal Albanese dimension, the study was started by Chen and Hacon [CH2,CH3]. Their results were improved by Jiang [J2], who showed that the 5-th canonical map induces the Iitaka fibration. Recently, Tirabassi [Ti], showed that if X is of general type then the 4-th pluricanonical is birational.
Combining Theorems 4.1 and 5.1, we obtain the following statement.
Theorem A. Let X be a smooth projective variety of maximal Albanese dimension. Then, (1) the linear system |4K X | induces the Iitaka fibration of X, (2) if X is of general type, then the linear system |3K X | induces a birational map.
These bounds are optimal for varieties of maximal Albanese dimension. On the one hand, if X is a curve of genus 2, then |2K X | is not birational (varieties of general type whose bicanonical map is not birational have been studied in [BLNP, L]). Furthermore, we produce varieties of dimension at least 4, whose tricanonical map does not induce the Iitaka fibration (see Example 5.6).
We observe that when χ(X, ω X ) > 0, the birationality of the tricanonical map was proved by Chen and Hacon [CH3,Thm. 5.4]. Hence, we restrict ourselves to the case χ(X, ω X ) = 0. In this situation we have a special fibration where the m-th pluricanonical linear system restricts surjectively to a general fibre for m ≥ 3 (see Lemma 3.5). On the base of this fibration, we construct two positive line bundles (see Lemmas 3.3 and 3.4). One of them induces a birational map ϕ on the base. The other one is used to prove a non-vanishing result which shows that the fibration followed by ϕ factors through the pluricanonical map. We use these line bundles to apply Lemma 2.1, which allows us to proceed by induction on the dimension of X. When X is of general type, it is enough to consider the tricanonical map to get the non-vanishing mentioned above. The lack of this non-vanishing is what forces us to consider the tetracanonical map for non-general type varieties (as we note in Remark 5.5).
Notation. In the sequel, X will always be a smooth complex projective variety of maximal Albanese dimension. We denote by a X : X → A X the Albanese morphism. Given an abelian variety A, let A be its dual. For a morphism t : X → A to an abelian variety and a coherent sheaf F on X, we denote by V i (F , t) the i-th cohomological support loci: {P ∈ Pic 0 (A) | H i (X, F ⊗ t * P ) = 0}. We will use the terms M-regular, I.T.-index j (IT j for short) and continuously globally generated (cgg for short) as they are defined in [PP1]. We will say that F is a GV -sheaf if codim Pic 0 (X) V i (F , a X ) ≥ i for all i ≥ 0. Given two divisors E, F , we will write E F if F − E is effective.

Preliminaries
We begin with some easy lemmas.
fibrations between smooth projective varieties. Let L be an line bundle on X. If the following two conditions hold: 1) The image of H 0 (X, L) → H 0 (X y , L| Xy ) induces a map birationally equivalent to g| Xy : X y → Z y for a general fibre X y of f ; 2) There are line bundles is effective and the multiple evaluation map is birational. Then, the linear system |L| induces a map birationally equivalent to g : X → Z.
Proof. Since the L − f * H i are effective we have a map π that induces the following diagram Condition 2) guarantees that ϕ |L| separates generic fibres of f and condition 1) shows that the map ϕ |L| factorizes as ϕ Z • g and a general fibre of h is mapped birationally via ϕ Z .
We will need the following lemma to ensure the birationality of ϕ Y in the previous lemma.
Lemma 2.2. Let π : Y → Y be a generically abelian Galois covering between smooth projective varieties of maximal Albanese dimensions. Denote by G the Galois group of π. We denote where H χ is the torsion-free rank-one sheaf corresponding to the character χ ∈ G * . Then, there exists H χ 0 such that the multiple evaluation map If π is an isomorphism, then Lemma 2.2 is contained in the proof of [CH3,Thm. 4.4].
where L 0 = ω Y and all L i are torsion-free rank-one sheaves on Y .
Since V 0 (ω Y , b Y ) = Pic 0 (Y ), we conclude that there exists χ 1 such that V 0 (L χ 1 , a Y ) = Pic 0 (Y ). Denote by Z the closed subset where L χ 1 is not locally free. Then, for any P ∈ Pic 0 (Y ), the line bundle Since there is the natural G-map of torsion-free sheaves on Y , Hence, there is an open dense subset U of Y such that the sheaf H χ 0 | U is locally free and for any P ∈ Pic 0 (Y ), H χ 0 ⊗ P is globally generated over U, i.e. ϕ P | U is a morphism.
Therefore, H ′ χ 0 ⊗ P is globally generated and locally free on the open subset U and Supp(H χ 0 /H ′ χ 0 ) is contained in Y U. Now, let V = U Exc(a Y ) and for any point y ∈ V , from the exact sequence is a M-regular sheaf, so it is cgg (see [PP1,Prop. 2.13]). Hence for any z ∈ V different from y, there exists P ∈ Pic 0 (Y ) such that I y ⊗ H ′ χ 0 ⊗ P is globally generated on z. This shows that for any two different points y, z ∈ V there exists P ∈ Pic 0 (Y ) and a divisor D P in |H χ 0 ⊗ P | such that y ∈ D P but z / ∈ D P . Therefore ϕ P (y) = ϕ P (z). We take P 1 , . . . , P M such that ϕ P 1 ···P M becomes stable, namely ϕ P 1 ···P M is birational equivalent to ϕ P 1 ···P M P for any P ∈ Pic 0 (Y ). Then, ϕ P 1 ···P M is birational.
Lemma 2.3. Let f : X → Y be a surjective morphism between smooth projective varieties. Assume that X is of maximal Albanese dimension. Then, K X/Y is effective.
We then consider the exact sequence X is generically globally generated and hence so is Q.
Hence det(Q) is also an effective divisor.
The following lemma is used in [J1,Lem. 2.3] under the assumption that the plurigenera are positive (since X is of maximal Albanese dimension, this condition is automatically satisfied). We recall it here for easy reference.
Lemma 2.4. Suppose that f : X → Y is an algebraic fibre space between smooth projective varieties. Assume that Y is of general type. Then, the Iitaka model of (X, 3. Positive bundles on the base and surjectivity of the restriction map to a fibre We will use the following definition that it is strongly related to Hypotheses 4.7 in [BLNP].
Definition 3.1. Let F be a coherent sheaf on an abelian variety A. We Let X be a smooth projective variety of dimension n and maximal Albanese dimension. We know that the pushforward of the canonical bundle a X * ω X is a GV -sheaf but it often fails to be M-regular, which makes the tricanonical map difficult to study. Hence we consider the set which measures how far a X * ω X is from being almost M-regular.
Setting 3.2. Assume that S X is not empty. We denote by k the maximal number of S X and Q + B ⊆ V k (ω X , a X ) a codimension-k component, where Q is zero or a torsion element of Pic 0 (X) B. Let the following commutative diagram be a suitable birational modification of the Stein factorization of the composition pr •a X , such that Y is smooth.
Proof. We know by [GL,Theorem 0.1] that the dimension of a general fibre of f is k.
Moreover, the pull-back by pr of any codimension-j component of [K2,Thm. 3.1]). Hence by the maximality of k, we know that We may choose Q such that G ∩ B = 0, where G := Q is the subgroup generated by Q.
Claim. Up to a birational modification of X and Y , there exists a diagram where all varieties are smooth and the vertical morphisms are fibrations (in particular, Y is a modification of the Stein factorization of f • π). Moreover π : X → X is birational to G-cover, π : Y → Y is a generically G-cover, and f is a G-equivariant morphism.We also have Proof of the claim. We first consider theétale cover π : X → X induced by G. Let Y be the Stein factorization of f • π. By [PP2,Thm. 5.8 and Prop. 3.14], If we denote by X y a general fibre of f , then we know that Q| Xy is O Xy . Hence, a general fibre of f is isomorphic via π to a general fibre of f . So G is also the Galois group of the field extension k( Y )/k(Y ). After modifications of f : X → Y , we may assume that G acts also on Y , f is a G-equivariant morphism, and π : Y → Y is a generically G-cover of smooth projective varieties. Hence where L 0 = ω Y and all L i are torsion-free rank-one sheaves on Y and we conclude the proof of the Claim.
The same arguments as in the case where Q is trivial show that where all varieties are smooth, all slanted arrows are birational modifications, and π ′ is birational equivalent to theétale cover induced by G.
Since we have π * L i ֒→ ω Y , we conclude that ω Y ′ ⊗ π ′ * L ′−1 i has a non-trivial section. We know K X ′ / Y ′ is an effective divisor by Lemma 2.3. Hence, So there exists j such that ω X ′ ⊗ η * Q j ⊗ f ′ * L ′−1 i is effective. Thus in 2), we can take f ′ : X ′ → Y ′ to be the birational model of f and take L to be any L ′ i for i = 0. Lemma 3.4. Assume S X is not empty, so we are in Setting 3.2. Consider the birational model obtained in Lemma 3.3. Then, after modifying f by blowing-up Y , there exists a line bundle H on Y and i ∈ Z such that O X (2K X + iQ) ⊗ f * H −1 has a non-trivial section. Moreover we can take P i ∈ Pic 0 (Y ), 1 ≤ i ≤ M such that the multiple evaluation map Proof. If Q is trivial, we just take H to be ω 2 Y . By Lemma 2.2, we conclude.
If Q ∈ Pic 0 (X) B, then we use the same notations as in the proof of Lemma 3.3. Notice that we can consider diagram (1) obtained in the Claim. Now, we apply Lemma 2.2 to π and we take H to be the direct summand H χ 0 of π * ω 2 Y .
As in diagram (4), we can take an appropriate model of f and assume that H is a line bundle. Since π * H ֒→ ω 2 Y , as in the proof of Lemma 3.3, there exists an integer i ∈ Z such that O X (2K X + iQ) ⊗ f * H −1 is effective.
Lemma 3.5. Assume S X is not empty, so we are in Setting 3.2. Let L be the line bundle obtained in Lemma 3.3. Then, for y a general point of Y , the restriction map is surjective, for any m ≥ 2 and P ∈ V 0 (ω m X , a X ). Proof. We just prove the statement for P = O X , the same argument works for any P ∈ V 0 (ω m X , a X ). There are two distinguished cases, whether Q is trivial or not, which we address with slightly different techniques.
Case A. Assume that Q is trivial.
We have seen in the proof of Lemma 3.3 that χ(Y, ω Y ) > 0, so Y is of general type. By Lemma 2.4, the Iitaka model of (X, K X + (m − 2)K X/Y ) dominates Y and by [J1,Lem. 2.1] there exists an asymptotic multiplier ideal sheaf I : [PP1,Prop. 2.13], is cgg outside the exceptional locus of a Y .
We conclude, similarly to [BLNP,Prop. 4.4 and Cor. 4.11] that, F ⊗ ω Y is globally generated in an open dense subset of Y . Indeed, we first notice that |ω Y ⊗ P | is not empty for all P ∈ B. Take P i , 1 ≤ i ≤ N such that the evaluation map Moreover, by [J1,Lem. 3.5], F is a non-zero sheaf on Y of rank P m (X y ). Hence, over a general point y ∈ Y , F ⊗ k(y) is isomorphic to H 0 (X y , O Xy (mK Xy )). Since and they have the same rank P m (X y ), we conclude the proof of the lemma when Q is trivial.
Case B. If Q is non-trivial, we use the same notation as in the proof of Lemma 3.3. After the modification performed in diagram (4), we can assume that we have a birational model of f : X → Y such that there exists a diagram where all varieties are smooth, the vertical morphisms are fibrations, and π is birational to anétale cover. Moreover, if π * ω Y = ⊕ i L i , then the line bundle L constructed in Lemma 3.3 is a birational modification of L i for i = 0.
We claim that the Iitaka model of (X, (m where E is some π-exceptional divisor. Since π is birational to anétale cover, the claim is clear.
Then, by [J1, Lem. 2.1] there is an ideal I of X such that is an IT 0 sheaf and by [J1,Proof of Lem. 3.9] f * O X (mK X ) ⊗ I ⊗ L −(m−2) has rank P m (X y ). We conclude as before, that for i = 0, f * (O X (mK X − (m − 3)f * L )) is globally generated over an open dense subset of Y .
Remark 3.6. Observe that we can consider a birational model f : X → Y such that both L and H are line bundles in Y that fulfil the desired properties listed in lemmas 3.3, 3.4 and 3.5. In particular, when in Setting 3.2 Q is non-trivial, we also have the following diagram: where all varieties are smooth, the vertical morphisms are fibrations, and π is birational to anétale cover. Moreover, ω Y ⊗ π * L −1 and ω 2 Y ⊗ π * H −1 have non-trivial sections.

General type case
In this section X will be a variety of general type.
Theorem 4.1. Let X be a smooth projective variety, of maximal Albanese dimension and general type. Then, the linear system |3K X + P | induces a birational map, for any P ∈ Pic 0 (X).
Proof. We reason by induction on the dimension of X, that we will denote by n. Note that for n = 1 the result is well known. So we assume that for any P Y ∈ Pic 0 (Y ), |3K Y + P Y | induces birational map for any smooth projective variety Y of maximal Albanese dimension, general type, and dim Y ≤ n − 1.
Observe that S X is empty if and only if a X * ω X is almost M-regular. Since X is of general type, if S X is empty, then χ(X, ω X ) > 0 (e.g. [BLNP,Prop. 4.10]). We notice that Chen and Hacon have proved that |3K X + P | induces a birational map in this situation (see [CH3,Thm. 5.4]).
From now on, we will assume S X is not empty. As in the last section, we are in Setting 3.2: Consider an appropriate birational model of f : X → Y in the sense of Remark 3.6.
Let y ∈ Y be a general point and denote by X y a general fibre of f . By Lemma 3.5, the restriction map is surjective for any P ∈ Pic 0 (X) and, by the induction hypothesis, |3K Xy + P | Xy | induces a birational map. We have also produced interesting line bundles on Y in Lemma 3.3 and Lemma 3.4. Let H be the line bundle on Y constructed in Lemma 3.4. According to Lemma 2.1, in order to conclude the proof of the theorem, we just need to prove the following claim.
Claim ‡. For every P ∈ A X and every P ′ ∈ B, the line bundle has a non-trivial section.
where N is an integer large enough and H is an ample divisor on Y . For any P ∈ A X , we define Observe that to conclude the proof of the claim it is enough to see that For any ample divisor H ′ on Y , we have that for any i > 0. We postpone the proof of ( †) to the end of the proof of this theorem. From ( †) we deduce that for any i > 0 (see e.g. [La,Lemma 4.3.10]). Therefore, is constant for P ∈ A X . By Lemma 3.3 and Lemma 3.4, there exist integers i and j and effective divisors D 1 ∈ |K X + iQ − f * L | and D 2 ∈ |2K X + jQ − f * H |.
by [La,Thm. 11.1.8], we have On the other hand, we see by [J1,Lem. 2.5] that F P is a GV -sheaf for any P ∈ A X . Therefore, for P ′ ∈ B general and any P ∈ A X , Hence, by semicontinuity, for any P ∈ A X , V 0 (F P , a Y ) = B.
Proof of ( †). We use the same notation as in the proof of Lemma 3.3 and Lemma 3.4, which is summarized in Remark 3.6. Notice is a consequence of Nadel vanishing theorem (see [La,Thm. 11.2.12]).

Iitaka fibration
In this section X will not necessarily be a variety of general type.
Theorem 5.1. Let X be a smooth projective variety, of maximal Albanese dimension. Then, the linear system |4K X + P | induces a model of the Iitaka fibration of X, for any P ∈ V 0 (ω 2 X , a X ). Before starting the proof the Theorem 5.1, which is parallel to the proof of Theorem 4.1, let us fix the notation.
Setting 5.2. Consider the following diagram: where g : X → Z is a model of the Iitaka fibration of X such that Z is smooth. Let K be the kernel of pr Z . We denote by X z a general fibre of g, which is birational to its Albanese variety K, and the natural map K → K is an isogeny. We know that pr * Z A Z is an irreducible component of K := ker( A X → Pic 0 (X z )) and denote by Q := K / pr * Z A Z . Observe that Q can be also identified with ker( K → K).
Remark 5.3. The group Q is often non-trivial and this is exactly the reason why the tricanonical map can not always induce the Iitaka fibration. In some specific cases, given information about Q, we can prove that the tricanonical map or some twisted tricanonical map (the maps induced by |3K X + P | for some P ∈ Pic 0 (X)) will induce the Iitaka fibration (see Remark 5.5).
Nevertheless we will construct a variety of maximal Albanese dimension (see Example 5.6), where NONE of the twisted tricanonical maps is birationally equivalent to the Iitaka fibration.
Before proving the theorem, we start with an easy well-known observation. We add its proof for the convenience of reader.
Lemma 5.4. The kernel K defined in Setting 5.2, satisfies Proof. It is clear that V 0 (ω m X , a X ) ⊆ K . If P ∈ K , then g * (ω m X ⊗P ) is a nontrivial torsion-free sheaf. By [J1,Lem. 2 . Proof of Theorem 5.1. We will prove the theorem by induction on the dimension of X. We suppose the statement is true in dimension ≤ n−1 and assume dim X = n. If X is of general type, then we are back to Theorem 4.1. Hence we can assume κ(X) = dim Z = n − l, for some number l > 0. In particular, S X is not empty.
Hence, we are in Setting 3.2. Let k be the maximal number of S X and let Q + B be an irreducible component of V k (ω X , a X ) of codimension k. Since V k (ω X , a X ) ⊆ V 0 (ω X , a X ), by [CH2,Lem. 2.2], B ֒→ pr * Z A Z (recall Setting 5.2). Hence, Setting 3.2 and 5.2 combine in the following commutative diagram where we choose an appropriate birational model of f : X → Y in the sense of Remark 3.6. Let y ∈ Y be a general point and denote by X y and Z y general fibres of f and h. By Easy Addition Formula (e.g. [Ii,Thm. 10.4]), dim Y + κ(X y ) ≥ κ(X) = dim Z. Hence κ(X y ) ≥ dim Z y and thus g| Xy : X y → Z y is the Iitaka fibration of X y . By Lemma 3.5 and 5.4, the restriction map is surjective, for any P ∈ K . Notice that P | Xy ∈ V 0 (ω 2 Xy , a Xy ), so by induction hypothesis, |4K Xy + P | Xy | induces the Iitaka fibration g| Xy : X y → Z y . Let H be the line bundle on Y constructed in Lemma 3.4. Then, by Lemma 2.1, we just need to prove the following claim to complete the proof of the theorem.
Claim. For every P ∈ K and every P ′ ∈ B, has a non-trivial section.
where N is an integer large enough and H is an ample divisor on Y . For any P ∈ K , we define To conclude the proof of the claim it is enough to see that V 0 (h * G P , a Y ) = B. By [J1, Lemma 2.1], we have for any i ≥ 1, any ample divisor H ′ on Y , and any Q ′′ ∈ A Z . Hence, for any i > 0 (see e.g. [La,Lemma 4.3.10]). Therefore, is constant for Q ′′ ∈ A Z . By Lemma 3.3 and Lemma 3.4, we know there exists m ∈ Z such that Observe that P − mQ is not necessarily in pr * Z A Z . But, since P − mQ ∈ K , we have that a Z * g * O X (K X + P − mQ) is a non-trivial GVsheaf. In particular, V 0 (g * O X (K X + P − mQ), a Z ) = ∅. Hence there exists Q 0 ∈ pr * Z A Z such that P − mQ + Q 0 ∈ V 0 (ω X , a X ). Therefore is the sum of two effective divisors. By the same argument as in (6), We know that V 0 (L , a Y ) = B (see Lemma 3.3) and h * (G P ⊗ Q ′ ) is a GV -sheaf for any P ∈ K and any Q ′ ∈ A Z (see [J1,Lem. 2.5]). Hence, for P ′ ∈ B general, By semicontinuity, V 0 (h * G P , a Y ) = B for any P ∈ K . Now, we can make more precise Remark 5.3.
Remark 5.5. In the previous proof, observe that if P − mQ lies in pr * Z A Z , then 3K X + P + Q 0 − f * H − f * L is effective for some Q 0 ∈ pr * A Z . So, we could have improved the result to the tricanonical map (assuming the induction hypothesis). In particular, if Q := K / pr * Z A Z is trivial for X and the successive fibres of the induction process, then the tricanonical map twisted by an element in K induces the Iitaka fibration.
Moreover, if for some P ∈ K , P + pr * Z A Z is an irreducible component of V 0 (ω X , a X ), then we can again prove that the tricanonical map twisted by an element in K induces the Iitaka fibration. This shows that varieties of maximal Albanese dimension, where none of the twisted tricanonical map is birational equivalent to the Iitaka fibration, are closely related to varieties of maximal Albanese dimension, of general type with vanishing holomorphic Euler characteristic.
We finish with an example of maximal Albanese dimension, whose tricanonical map does not induce the Iitaka fibration. This example is based on the famous Ein-Lazarsfeld threefold, which is constructed in [EL,Ex. 1.13] and further investigated in [CDJ].
Example 5.6. We take three bielliptic curves C i of genus 2, i = 1, 2, 3. Let ρ i : C i → E i be the double cover over an elliptic curve E i and denote by τ i the involution of fibres of ρ i . We write Let Y be the threefold (C 1 × C 2 × C 3 )/(τ 1 , τ 2 , τ 3 ), which has only rational singularities. We know that a Y : Y → E 1 × E 2 × E 3 is a (Z/2Z × Z/2Z)-cover.
We then take an abelian variety A and a (Z/2Z × Z/2Z)-étale cover A → A. Set {O A , P 1 , P 2 , P 3 } to be the kernel A → A.
Denote H = (Z/2Z × Z/2Z) and let X ′ be the variety (Y × A)/H, where H acts diagonally on Y × A. Notice that X ′ has only rational singularities and let X be a resolution of singularities of X ′ . The Albanese morphism is birationally a (Z/2Z × Z/2Z)-cover.
After permutation of {P i , i = 1, 2, 3}, we have It is easy to check that for any P ∈ Pic 0 (X), the linear series |3K X + P | can not induce the Iitaka fibration X → E 1 × E 2 × E 3 .
Acknowledgements. The first-named author enjoyed the hospitality of the Max Planck Institut for Mathematics in Bonn during the preparation of this paper. The second-named author was supported by the SFB/TR 45 'Periods, moduli spaces, and arithmetic of algebraic varieties' and partially by the Proyecto de Investigación MTM2009-14163-C02-01.
The third-named author thanks her advisor, Prof. G. Pareschi, for having introduced her to the subject of pluricanonical maps and for many suggestions. She is also deeply indebted to C. Ciliberto for many mathematical conversations. Finally she is grateful toward the department of Mathematics of Università di Roma "Tor Vergata" for its kind hospitality.
Finally, we want to thank the referees for carefully reading the paper and for many suggestions that improved the exposition.