Pluricanonical maps of varieties of Albanese fiber dimension two
aa r X i v : . [ m a t h . AG ] F e b PLURICANONICAL MAPS OF VARIETIES OF ALBANESEFIBER DIMENSION TWO
HAO SUN
Abstract.
In this paper we prove that for any smooth projective varietyof Albanese fiber dimension two and of general type, the 6-canonical map isbirational. And we also show that the 5-canonical map is birational for anysuch variety with some geometric restrictions. Introduction
Let X be a smooth complex projective irregular variety of general type, i.e.,variety of general type with q ( X ) >
0. We define the Albanese fiber dimensionof X to be e = dim X − dim a ( X ), where a : X → Alb( X ) is the Albanese map.Recently, the birationality of the n -th pluricanonical map ϕ | nK X | of X has attracteda lot of attention.When e = 0, i.e., X is of maximal Albanese dimension, it was shown by Chenand Hacon [3, 4] that ϕ | K X | is birational. This result was improved by Jiang,Lahoz and Tirabassi [8], showing that ϕ | K X | is birational. When e = 1 or 2, Chenand Hacon [4] proved that ϕ | ( e +5) K X | is a birational map. Recently, Jiang and theauthor [9] showed that ϕ | K X | is birational if e = 1. The main results of this paperare some improvements of the result of Chen and Hacon in the case of Albanesefiber dimension two: Theorem 1.1.
Let X be a smooth complex projective variety of Albanese fiberdimension two and general type. Then the -canonical map ϕ | K X | is birational. Theorem 1.2.
Let X be a smooth complex projective variety of Albanese fiberdimension two and general type. If the translates through of all components of V ( ω X ) generate Pic ( X ) , then ϕ | K X | is birational. By Theorem 1.1, we can immediately obtain a recent result of Chen, Chen andJiang:
Corollary 1.3. [2, Theorem 1.1]
Let V be a smooth complex projective irregular -fold of general type. Then ϕ | K X | is birational. Acknowledgments.
The author would like to thank Jungkai A. Chen, Zhi Jiangand Lei Zhang for various comments and useful discussions. This work was donewhile the author was visiting Emmy Noether Research Institute for Mathematics.He is grateful to this institution for hospitality.
Date : November 16, 2012.2000
Mathematics Subject Classification.
Key words and phrases.
Irregular variety, pluricanonical map, surface.This work is supported by NSFC and the Mathematical Tianyuan Foundation of China. Definitions and lemmas
In this section, we recall some notion and useful lemmas.
Definition 2.1.
Let F be a coherent sheaf on a smooth projective variety Y .(1) The i -th cohomological support loci of F is V i ( F ) = { α ∈ Pic ( Y ) | h i ( F ⊗ α ) > } . (2) We say F is IT if H i ( F ⊗ α ) = 0 for all i ≥ α ∈ Pic ( Y ).(3) F is called M -regular if codim V i ( F ) > i for every i > Y is an abelianvariety.(4) We say F is continuously globally generated at y ∈ Y (in brief CGG at y )if the nature map M α ∈ U H ( F ⊗ α ) ⊗ α ∨ → F ⊗ C ( y )is surjective for any non-empty open subset U ⊂ Pic ( Y ).(5) F is said to have no essential base point at y ∈ Y if for any surjective map F → O y , there is a non-empty open subset U ⊂ Pic ( Y ) such that for all α ∈ U , the induced map H ( F ⊗ α ) → H ( O y ⊗ α ) is surjective. Lemma 2.2.
Let π : X → Y be a double covering branched along a reduced divisor B ∈ | L | , where X is a projective variety, Y is a smooth projective variety and L is a divisor on Y . Let D be a divisor on Y . Then | π ∗ D | induces a birational mapif and only if | D | induces a birational map of Y and H ( Y, O Y ( D − L )) = 0 .Proof. We know that π ∗ π ∗ O Y ( D ) = O Y ( D ) ⊕ O Y ( D − L ). Hence we have H ( X, O X ( π ∗ D )) = H ( Y, O Y ( D )) ⊕ H ( Y, O Y ( D − L )) . The only-if-direction follows immediately from the above isomorphism.To prove the if-direction, we take an open affine subset U ⊂ Y . Suppose that U = Spec R for some ring R . We know that π − ( U ) = Spec R [ z ] / ( z − f ), where f ∈ R is the local defining equation of B . From the isomorphism of R -modules R [ z ] / ( z − f ) ∼ = R ⊕ Rz, it follows that we can choose s , . . . , s k , t , . . . , t k ∈ R such that 1 , s , . . . , s k is abasis of H ( D ) and t z, . . . , t n z is a basis of H ( D − L ). Let y ∈ Y be a generalpoint. Since H ( D − L ) = 0, we can assume that f ( y ) = 0 and t ( y ) = 0.Let { x , x } be the preimage of y , where x = ( y, p f ( y )) and x = ( y, − p f ( y )).Since | π ∗ D | separates two general points on two distinct general fibers of π , we canfind a section s + t z ∈ H ( X, O X ( π ∗ D )) vanishes along y , where s ∈ H ( D ) and t z ∈ H ( D − L ). Hence section s + ( t ( y ) − t ( y )) p f ( y ) + t z ∈ H ( π ∗ D )vanishes along x but does not vanish along x . It follows that | π ∗ D | separates twopoints on a general fiber of π . Therefore | π ∗ D | induces a birational map. (cid:3) Lemma 2.3.
Let f : X → W be a morphism between smooth projective varietieswith a general fiber F . Suppose that κ ( W ) ≥ and K X is W -big, i.e., sK X ≥ f ∗ L for some ample divisor L on W and some integer s ≫ . Suppose further that LURICANONICAL MAPS OF VARIETIES OF ALBANESE FIBER DIMENSION TWO 3 h ( mK F ) > for some m ≥ . Then after replacing X by an appropriate birationalmodel, there exist positive integers b , c and there is a normal crossing divisor B ∈ | bc ( m − K X − f ∗ bM | such that ⌊ Bbc ⌋| F ≤ B m,F , ⌊ Bbc ⌋ ≤ B m,α , for all α ∈ Pic ( W ) . Here B m,F (resp. B m,α ) is the fixed part of | mK F | (resp. | mK X + f ∗ α | ), M is a given nef and bigdivisor on W and b , c are sufficiently large integers depending on M and K X .Proof. See [5, Lemma 2.5]. (cid:3)
Lemma 2.4. If F is a coherent sheaf on a smooth projective variety Y and y is apoint on Y , then F is CGG at y if and only if F has no essential base point at y .Proof. Firstly, we assume that F is CGG at y . For any surjective map F → O y ,the induced map F ⊗ C ( y ) → O y is also surjective. The definition of CGG impliesthat the composition M α ∈ U H ( F ⊗ α ) ⊗ α ∨ → F ⊗ C ( y ) → O y is surjective for any non-empty open subset U ⊂ Pic ( Y ). It follows that for anynon-empty open subset U ⊂ Pic ( Y ), there is an α ∈ U such that the induced map H ( F ⊗ α ) → H ( O y ⊗ α ) is surjective. By semi-continuity, one sees that for ageneral α ∈ Pic ( Y ), the induced map H ( F ⊗ α ) → H ( O y ⊗ α ) is surjective.Thus F has no essential base point at y .Conversely, suppose that F has no essential base point at y . One can write F ⊗ C ( y ) = L ki =1 V i , where V i ∼ = C ( y ), i = 1 , , . . . , k . Let p i : F ⊗ C ( y ) → V i be the canonical projection and ϕ i : F → F ⊗ C ( y ) → V i be the composition.Since F has no essential base point at y , we know that there exists non-emptyopen subsets U i (1 ≤ i ≤ k ) of Pic ( Y ) such that for any α ∈ U i the induced map˜ ϕ i : H ( F ⊗ α ) → V i ⊗ α is surjective. For any non-empty open subset U ⊂ Pic ( Y ),since ∩ ki =0 U i is also a non-empty open subset, the map M α ∈ U H ( F ⊗ α ) ⊗ α ∨ −→ k M i =1 V i = F ⊗ C ( y )is surjective. It follows that F is CGG at y . (cid:3) Lemma 2.5. If F is a non-zero M -regular sheaf on a complex abelian variety A ,then F has no essential base point at any y ∈ A .Proof. The conclusion follows from [11, Proposition 2.13] and Lemma 2.4. (cid:3)
Lemma 2.6.
Let D an effective divisor on an abelian variety A . Take T , . . . , T k subtori of A such that they generate A as an abstract group and let γ i , i = 1 , . . . , k some points of A . Then D ∩ ( T i + γ i ) = ∅ for at least one i .Proof. See [12, Lemma 2]. (cid:3)
HAO SUN Proof of the main theorems
From now on, we let X be a smooth complex projective variety of Albanese fiberdimension two and general type. Denote a : X → Z = a ( X ) ⊂ A the Albanesemap of X , where A is the the Albanese variety of X . Let ν : W → Z be adesingularization of the Stein factorization over Z . Replacing X by an appropriatebirational model, we may assume that there is a morphism f : X → W whosegeneral fiber is a connected smooth surface S . Then we obtain a commutativediagram X f / / a ❇❇❇❇❇❇❇❇ W ν (cid:15) (cid:15) Z We prove the following statement, which is more general than Theorem 1.1.
Theorem 3.1. | K X + α | induces a birational map for any α ∈ Pic ( X ) .Proof. By [5, Corollary 2.4], we know that ϕ | K X + α | is birational for any α ∈ Pic ( X ) if ϕ | K S | is birational. Hence we can assume that ϕ | K S | is not birational.It follows from [1, Main Theorem] that K S = 1 and p g ( S ) = 2, where S is theminimal model of S . The theorem also tells us that | K S | is base point free, and ϕ | K S | is a generically double covering onto its image. Thus the natural morphism f ∗ f ∗ ω ⊗ X → ω ⊗ X defines a rational map X Y ⊂ P ( f ∗ ω ⊗ X ) over W , where Y isthe closure of the image.Let Y ′ → Y be a resolution of singularities of Y , and let h : X ′ → Y ′ bea resolution of indeterminacies of the corresponding rational map X Y ′ . Weknow that h is a generically double covering branched along a reduced divisor B .Let µ : e Y → Y ′ be a log resolution of B , such that B := µ ∗ B − ⌊ µ ∗ B ⌋ is smooth(see [13, Lemma 1.3.1]). We assume that B ∈ | L | for some divisor L on e Y . Let e X → e Y be the double covering branched along B . One sees that e X is smooth.Thus we obtain K e X = π ∗ ( K e Y + L ). Now we have the commutative diagram amongsmooth projective varieties e X π / / e f ❆❆❆❆❆❆❆❆ e Y p (cid:15) (cid:15) W. We only need to show that ϕ | K f X + α | is birational for all α ∈ Pic ( e X ).Let H (resp. F ) be the fiber of e f (resp. p ) over a general point w ∈ W .From the construction, we know that π | H : H → F is a double covering betweensmooth surfaces branched along a smooth divisor B | F ∈ |O F (2 L ) | . Hence we have K H = ( π | H ) ∗ ( K F + L | F ).By Lemma 2.3, for some m ≥ σ : b X → e X there exists positive integers b , c and there is a normal crossing divisor B m ∈ | bc ( m − K b X − g ∗ bM | such that ⌊ B m bc ⌋| H ′ ≤ B m,H ′ , ⌊ B m bc ⌋ ≤ B m,α , for all α ∈ Pic ( b X ). Here M is a givennef and big divisor on W , g is the composite map e f ◦ σ , H ′ is the general fiber of LURICANONICAL MAPS OF VARIETIES OF ALBANESE FIBER DIMENSION TWO 5 g and b , c are sufficiently large integers depending on M and K b X . Thus we obtain( m − K b X − ⌊ B m bc ⌋ ≡ c g ∗ M + { B m bc } . By [10, Theorem 10.15], we know that H i ( A, ν ∗ g ∗ O b X ( mK b X − ⌊ B m bc ⌋ + α )) = H i ( W, g ∗ O b X ( mK b X − ⌊ B m bc ⌋ + α )) = 0for all α ∈ Pic ( b X ) and all i >
0. It follows from σ ∗ O b X ( mK b X/ e X ) = O e X that J m := σ ∗ O b X ( mK b X/ e X − ⌊ B m bc ⌋ )is an ideal sheaf of O e X . One sees that J m ⊃ I m,α , J m | H ⊃ I m,H and H i ( A, ν ∗ e f ∗ ( O e X ( mK e X + α ) ⊗ J m )) = 0for all α ∈ Pic ( e X ) and all i >
0, where I m,α (resp. I m,H ) is the base ideal of | mK e X + α | (resp. | mK H | ). In particular, ν ∗ e f ∗ ( O e X ( mK e X ) ⊗ J m ) is IT .Since π ∗ O e X (6 K e X ) = O e Y (6 K e Y + 6 L ) ⊕ O e Y (6 K e Y + 5 L ) , we conclude that there exists ideal sheaves J and J on e Y such that π ∗ ( O e X (6 K e X ⊗ J ) = ( O e Y (6 K e Y + 6 L ) ⊗ J ) ⊕ ( O e Y (6 K e Y + 5 L ) ⊗ J ) . Hence ν ∗ p ∗ ( O e Y (6 K e Y + 5 L ) ⊗ J ) is IT .On the other hand, we know that H ( H, O H (6 K H )) = H ( H, O H (6 K H ) ⊗ J | H )= H ( F, O F (6 K F + 6 L | F ) ⊗ J | F ) ⊕ H ( F, O F (6 K F + 5 L | F ) ⊗ J | F ) . This implies H ( F, O F (6 K F + 6 L | F )) = H ( F, O F (6 K F + 6 L | F ) ⊗ J | F )and H ( F, O F (6 K F + 5 L | F )) = H ( F, O F (6 K F + 5 L | F ) ⊗ J | F ) . Since | K H | is birational for a surface H , we obtain H (6 K F +5 L | F ) = 0 by Lemma2.2. Thus we have H ( F, O F (6 K F + 5 L | F ) ⊗ J | F ) = H ( F, O F (6 K e Y + 5 L ) ⊗ J | F ) = 0 . This implies ν ∗ p ∗ ( O e Y (6 K e Y + 5 L ) ⊗ J )is nonzero. Therefore we conclude that ν ∗ p ∗ ( O e Y (6 K e Y + 5 L ) ⊗ J )is a nonzero IT sheaf on A .Hence for any α ∈ Pic ( A ), H ( e Y , O e Y (6 K e Y + 5 L + α )) = 0. By [5, Theorem2.8], one sees that ϕ | K f X + α | separates two general points on two distinct generalfibers of π . Therefore ϕ | K f X + α | is birational by Lemma 2.2. (cid:3) HAO SUN
Next we study the 5-canonical map of X . We take B ∈ | bc ( m − K X − f ∗ bM | as in Lemma 2.3 and L m := ( m − K X − ⌊ Bbc ⌋ ≡ c f ∗ M + { Bbc } ,m ≥
2. One sees that H (( K X + L m ) | S ) ∼ = H ( mK S ) and H ( K X + L m + α ) ∼ = H ( mK X + α ) , for all α ∈ Pic ( X ). By [10, Theorem 10.15], we have H i ( A, a ∗ O X ( K X + L m ) ⊗ α ) = 0 , for all α ∈ Pic ( A ) and all i >
0. In particular, a ∗ O X ( K X + L m ) is IT . Lemma 3.2.
Let x ∈ X be a general point. Then K X + L m has no essential basepoint at x .Proof. Let F = X a ( x ) . By x being general, we mean that a ∗ O X ( K X + L m ) ⊗ C ( a ( x )) ∼ = H (( K X + L m ) | F ) ∼ = H ( mK F ) ,F is smooth and x is not a base point of | ( K X + L m ) | F | . Hence pushing forwardthe standard exact sequence0 → I x ( K X + L m ) → O X ( K X + L m ) → O x → A , we obtain0 → a ∗ ( I x ( K X + L m )) → a ∗ O X ( K X + L m ) → O a ( x ) → . Since a ∗ O X ( K X + L m ) is IT , by Lemma 2.5, a ∗ O X ( K X + L m ) has no essentialbase point at a ( x ). It follows that K X + L m has no essential base point at x . (cid:3) We denote U m , the open subset of Pic ( X ) where h ( mK X + α ) has minimalvalue. Let D m be the closure of the divisorial part of S = { ( x, α ) ∈ X × U m | x is a base point of | mK X + α |} in X × Pic ( X ). By Lemma 3.2, we know that dim S < dim( X × Pic ( X )). For ageneral x ∈ X , the fiber of the projection D m → X is a divisor, that we call D m,x .Now we prove the following theorem, which is more general than Theorem 1.2. Theorem 3.3.
If, for any effective divisor D of X × Pic ( X ) which dominates X ,the intersection D ∩ ( X × V ( ω X )) still dominates X , then | K X + α | is birationalfor any α ∈ Pic ( X ) .Proof. This theorem will be proved by three steps.
Step 1.
Let y ∈ X be a general point. Then a ∗ ( I y ( K X + L )) has no essentialbase point at any point of Z . This argument has essentially been proved by Jiang in the proof of [7, Theorem4.1], but we still give a proof for readers’ convenience.Because of Lemma 2.5, we want to show that a ∗ ( I y ( K X + L )) is M -regular.We have the exact sequence0 → a ∗ ( I y ( K X + L )) → a ∗ O X ( K X + L ) → O a ( y ) → . Considering the long exact sequence obtained from the above short exact sequence,we conclude that H i ( a ∗ ( I y ( K X + L )) ⊗ α ) = 0 for all α ∈ Pic ( A ) and i ≥
2. Hence, by y being general, y is a base point of | K X + L + α | if and only if LURICANONICAL MAPS OF VARIETIES OF ALBANESE FIBER DIMENSION TWO 7 α ∈ V ( a ∗ ( I y ( K X + L )). This implies y is a base point of | K X + α | if andonly if α ∈ V ( a ∗ ( I y ( K X + L )). One sees that D ,y is the divisorial part of V ( a ∗ ( I y ( K X + L )). Thus, by definition of M -regular, we know that a ∗ ( I y ( K X + L ) is M -regular if D = 0.Now we can assume that D = 0 and pr X : D ∩ ( X × V ( ω X )) → X is dominant.By Lemma 3.2, we know that D is dominant on X and Pic ( X ) via the naturalprojections. Since the base locus of | K X + α | is a proper closed subscheme of X forany α ∈ Pic ( X ), one sees that T x ∈ X D ,x = ∅ . This implies X × V ( ω X ) * D .Because pr X : D ∩ ( X × V ( ω X )) → X is dominant, there exists a component C of V ( ω X ) such that C * D ,y and C ∩ D ,y is not empty. By [6, Theorem 0.1], wecan write C = α + T , where T is a subtorus of b A := Pic ( X ) and α is a pointof b A .Let p : b A → b A/T be the quotient map. Sincedim p ( D ,y − α ) = dim( D ,y − α ) − dim T ∩ ( D ,y − α ) = dim( b A/T ) , we know that p ( D ,y − α ) = b A/T . This implies that D ,y − α − T = b A , i.e., D ,y − C = b A .Since y ∈ X is a general point, we can choose a non-empty open subset U ⊂ C such that y is not a base point of | K X + β | , for any β ∈ U . By considering themap H ( K X + α ) ⊗ H (2 K X + β ) → H (3 K X + α + β ) , we conclude that y is a base point of | K X + β | for any β ∈ V ( a ∗ ( I y ( K X + L ))) − U . By y being general, we know that y is also a base point of | K X + L + β | forany β ∈ V ( a ∗ ( I y ( K X + L ))) − U .It follows from D ,y − C = b A that V ( a ∗ ( I y ( K X + L ))) − U contains a non-empty open subset of Pic ( X ). This contradicts that K X + L has no essential basepoint at y . Step 2. If | K S | is birational, then | K X + α | induces a birational map for any α ∈ Pic ( X ) . We need the following:
Claim 3.4.
Let x , x ∈ X be general points. Then | K X + L + α | separates x , x for general α ∈ Pic ( X ) . We follow the idea in the proof of [5, Corollary 2.4]. If x and x are on a general F of a . Since H (( K X + L ) | F ) ∼ = H (3 K F ), we know that | ( K X + L ) | F | separates x , x . This implies a ∗ ( I x ,x ( K X + L )) = a ∗ ( I x ( K X + L )). Hence we have anexact sequence0 → a ∗ ( I x ,x ( K X + L )) → a ∗ ( I x ( K X + L )) → O a ( x ) → . If a ( x ) = a ( x ), we obtain a ∗ ( I x ( K X + L )) ⊗ C ( a ( x )) ∼ = a ∗ ( K X + L ) ⊗ C ( a ( x )).Thus we still obtain0 → a ∗ ( I x ,x ( K X + L )) → a ∗ ( I x ( K X + L )) → O a ( x ) → . By Step 1, a ∗ ( I x ( K X + L )) has no essential base point at a ( x ). Thus forgeneral α ∈ Pic ( X ) h ( a ∗ ( I x ,x ( K X + L )) ⊗ α ) = h ( a ∗ ( I x ( K X + L )) ⊗ α ) − . HAO SUN
Therefore for general α ∈ Pic ( X ) h ( I x ,x ( K X + L ) ⊗ α ) = h ( I x ( K X + L ) ⊗ α ) − h ( O X ( K X + L ) ⊗ α ) − α ∈ Pic ( X ), we choose general β ∈ Pic ( X ) and consider the map H (2 K X + α − β ) ⊗ H (3 K X + β ) → H (5 K X + α ) . We conclude that | K X + α | induces a birational map for any fixed α ∈ Pic ( X ). Step 3. If | K S | is not birational, then | K X + α | induces a birational map forany α ∈ Pic ( X ) . Since | K S | is not birational, it is well known that S satisfies ( K S , p g ) = (1 , , S is the minimal model of S . Thus ϕ | K S | is a generically doublecovering onto its image (cf. [1]). The natural morphism f ∗ f ∗ ω ⊗ X → ω ⊗ X define arational map g : X Y ⊂ P ( f ∗ ω ⊗ X ) over W , where Y is the closure of the image.As the construction in the proof of Theorem 3.1, we have the commutative diagramamong smooth projective varieties e X π / / e f (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ e Y p (cid:15) (cid:15) W, here the double cover π is a birational modification of g . The arguments in Step 2show that ϕ | K f X + α | separates two general points on two distinct general fibers of π for any α ∈ Pic ( e X ). By the same method in the proof of Theorem 3.1, we canobtain Step 3. We leave the details to the interested reader. (cid:3) Now we can easily prove Theorem 1.2.
Proof of Theorem 1.2. If D = 0, one sees that | K X | is birational by Theorem3.3. Hence we can assume that D = 0. This implies D ,x is an effective divisor onPic ( X ) for general x ∈ X .By [6, Theorem 0.1], we can write V ( ω X ) = k [ i =1 ( T i + γ i ) , where T i ’s are subtori of Pic ( X ) and γ i ’s are some points of Pic ( X ). It followsfrom Lemma 2.6 that D ,x ∩ V ( ω X ) = ∅ , for general x ∈ X . Thus the projection pr X : D ∩ ( X × V ( ω X )) → X is dominant. By Theorem 3.3, we obtain ourconclusion. (cid:3) References
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