aa r X i v : . [ m a t h . AG ] N ov UNIFORM BOUNDS FOR THE IITAKA FIBRATION
GABRIELE DI CERBOAbstract.
We give effective bounds for the uniformity of the Iitaka fibration.These bounds follow from an effective theorem on the birationality of someadjoint linear series. In particular we derive an effective version of the maintheorem in [17]. Introduction
Hacon and M c Kernan in [8] proposed the following conjecture.
Conjecture 1.1.
There is a positive integer m n,κ such that for any m ≥ m n,κ sufficiently divible, φ | mK X | is birationally equivalent to the Iitaka fibration of X forall smooth projective varieties X of dimension n and Kodaira dimension κ . They proved in [7] the case when κ ( X ) = n . For different proofs see also [19] and[22]. Their result is not effective and it remains a difficult question to find boundsfor these numbers. When κ ( X ) < n the standard approach to this problem is to usethe canonical bundle formula of Fujino and Mori [6]. Roughly speaking, it says thatfor the Iitaka fibration f : X Y the question is equivalent to finding a uniformbound for the birationality of linear systems on Y of type K Y + M Y + B Y where( Y, B Y ) is a klt pair and M Y is a nef Q -divisor. With this approach m n,κ dependsalso on some other numerical parameters which appear in the formula, see Section4 for details. Under these new assumptions there are some partial results. Fujinoand Mori proved the case κ ( X ) = 1. Some years later Viehweg and Zhang in [23]proved the case κ ( X ) = 2. If dim( X ) = 3 Rigler gave a different proof in [18]. Forlow dimensional varieties, i.e. dim( X ) ≤
4, the log version of Conjecture 1.1 hasbeen studied in [20] and [21]. The first result for arbitrary Kodaira dimension isdue to Pacienza in [17] but he needs to assume that K Y is pseudo-effective and M Y is big. Recently Jiang in [10] proved the case where M Y is numerically trivial byreducing the problem to a result on log pluricanonical maps in [9]. In summary, thecanonical bundle formula suggests that we need some general theorems for adjointlinear systems in order to prove Conjecture 1.1. In fact, thanks to the followingresult, Pacienza derived his theorem on the uniformity of the Iitaka fibration. Date : November 28, 2011. NIFORM BOUNDS FOR THE IITAKA FIBRATION Theorem 1.2 (Pacienza [17]) . For any positive integers n and ν , there exists aninteger m n,ν such that for any smooth complex projective variety X of dimension n with pseudo-effective canonical divisor, and any big and nef Q -divisor M on X such that νM is a Z -divisor, the pluriadjoint map φ m ( K X + M ) : X P H ( X, O X ( m ( K X + M ))) is birational for all m ≥ m n,ν divisible by ν . His proof relies on some techniques developed by Takayama in [19] and Debarre[4]. There are some deep results involved, like Takayama’s extension theorem andthe weak positivity theorem of Campana [3]. Unfortunately all these theorems allowus only to derive non-effective statements. On the other hand Koll´ar’s proof in [12]of Angehrn-Siu’s theorem is effective but it only deals with big and nef divisors. Inthis note we explain how to use the method of Koll´ar to derive an effective version ofPacienza’s theorem. Furthermore our proof relies on more elementary techniques.Our main result is:
Theorem 1.3.
Let ( X, ∆) be a klt pair. Let M be a big and nef Z -divisor on X and E a pseudo-effective Q -divisor on X . Then for any m > (cid:18) n + 22 (cid:19) the map induced by | ⌈ K X + ∆ + E + mM ⌉ | is birational. Taking X smooth, ∆ = 0 and E = ( m − K X we get an effective version ofTheorem 1.2.We now show how Theorem 1.3 gives a uniform result for the Iitaka fibration. Let f : X Y be the Iitaka fibration of X . As we will see in Section 4 there are twopositive integers b and N , depending only on the general fiber of f , such that bN M Y is an integral divisor. For the definition of b and N see Definition 4.1. Furthermorewe will see in Proposition 4.2 that | mK X | gives a map birationally equivalent tothe Iitaka fibration if and only if the linear series | ⌊ m ( K Y + B Y + M Y ) ⌋ | gives abirational map. Then Theorem 1.3 implies the following. Theorem 1.4.
Let f : X Y be the Iitaka fibration of X , where X is a smoothprojective variety of Kodaira dimension κ . Suppose K Y + B Y is pseudo-effectiveand M Y is big. Then for any m > bN (cid:18) κ + 22 (cid:19) divisible by bN , the pluricanonical map φ | mK X | is birationally equivalent to theIitaka fibration. NIFORM BOUNDS FOR THE IITAKA FIBRATION One can give conditions only on X and the generic fiber of f such that Theorem1.4 applies, see for example Corollary 4.3. Of course we would like to prove a similarstatement without the assumption K Y + B Y pseudo-effective. In Section 3 we studythe pseudo-effective threshold of ( X, ∆) with respect to a big and nef divisor M . Inparticular we obtain a similar result if we assume that the pseudo-effective thresholdis bounded away from one. 2. Pluriadjoint maps
We follow the notation and terminology of [12] and [13]. However we state heresome definitions we will need later.
Definition 2.1.
A pair ( X, ∆) consists of a normal variety X and a Q -Weil divisor ∆ ≥ such that K X + ∆ is Q -Cartier. The multiplier ideal of a divisor D on a normal variety X is denoted by J ( X, D ).We refer to [15] for the definition.
Definition 2.2.
The non-klt locus
Nklt( X, ∆) of a pair ( X, ∆) is Nklt( X, ∆) := { x ∈ X | ( X, ∆) in not klt at x } . We will use the following relationNklt( X, ∆) = Supp( O X / J ( X, ∆)) red . See [15] Section 9.3.B for the proof.
Proposition 2.3.
Let ( X, ∆) be a pair. Let M be a big and nef Cartier divisor on X and N be a Cartier divisor on X such that N − K X − ∆ is pseudo-effective. Let x and x be two general points in X . Suppose there are t > and an effective Q -divisor D such that (1) D ∼ Q t M ; (2) x , x ∈ Nklt( X, ∆ + D ) ; (3) x is an isolated point in Nklt( X, ∆ + D ) .Then for any m > t the linear system | N + mM | separates x and x .Proof. Let E a pseudo-effective Q -divisor such that N = K X + ∆ + E . Fix m > t and write D := D + E . Note that D is equivalent to an effective Q -divisor. Let V := Nklt( X, ∆ + D ). Let x and x be two general points not contained inSupp( E ), then we have that x , x ∈ V and x is isolated in V . In order to getseparation of points we want the following map to be surjective H ( X, O X ( N + mM )) → H ( V, O X ( N + mM ) | V ) . NIFORM BOUNDS FOR THE IITAKA FIBRATION It fits in the long exact sequence given by0 → O X ( N + mM ) ⊗ J ( X, ∆ + D ) → O X ( N + mM ) → O X ( N + mM ) | V → , then it is enough to prove that H ( X, O X ( N + mM ) ⊗ J ( X, ∆ + D )) = 0 . Since N + mM − ( K X + ∆ + D ) ∼ Q ( m − t ) M is big and nef, the above vanishing follows from Nadel vanishing on singular vari-eties, see Theorem 2.16 in [12] or Theorem 9.4.17 in [15]. (cid:3) Remark 2.4.
In Proposition 2.3 we work with D ∼ Q t M instead of working with D ∼ Q t ( M + E ) as Pacienza does in his Lemma 6.3. This is a crucial differencebetween our approach and that of Pacienza. We recall a result of Koll´ar in [12], Theorem 6.5.
Theorem 2.5 (Koll´ar) . Let ( X, ∆) be a projective klt pair and M a big and nef Q -Cartier Q -divisor on X . Let x and x be closed points in X and c ( k ) positivenumbers such that if Z ⊂ X is an irreducible subvariety with x ∈ Z or x ∈ Z then ( M dim Z · Z ) > c (dim Z ) dim Z . Assume also that n X k =1 k √ kc ( k ) ≤ . Then there is an effective Q -divisor D ∼ Q M such that: (1) x , x ∈ Nklt( X, ∆ + D ) ; (2) x is an isolated point in Nklt( X, ∆ + D ) . We recall the definition of the augmented base locus B + ( M ), see Definiton 10.3.2in [15]. Definition 2.6.
The stable base locus of a divisor M is B ( M ) := \ m ≥ Bs( | mM | ) , where Bs( | M | ) is the base locus of M .The augmented base locus of a divisor M is the Zariski-closed set B + ( M ) := B ( M − ǫA ) , for any ample A and sufficiently small ǫ > . Theorem 2.5 easily implies the following useful result.
NIFORM BOUNDS FOR THE IITAKA FIBRATION Corollary 2.7.
Let ( X, ∆) be a klt pair. Let M be a big and nef Cartier divisoron X . Then for any x , x / ∈ B + ( M ) there exists an effective Q -divisor D with (1) D ∼ Q (cid:0) n +22 (cid:1) M ; (2) x , x ∈ Nklt(
X, D + ∆) ; (3) x is an isolated point in Nklt(
X, D + ∆) .Proof. Recall that B + ( M ) is a proper subset of X if and only if M is big, seeExample 1.7 in [5]. Furthermore B + ( M ) = \ M = A + E Supp( E ) , where the intersection is taken over all decomposition M = A + E , where A is ampleand E effective, see Remark 1.3 in [5]. Then for any variety Z through x , x / ∈ B + ( M ) we have that M dim( Z ) · Z >
0. Since M is integral then ( M dim Z · Z ) ≥ n X k =1 k √ k < n X k =1 (cid:18) k (cid:19) k = (cid:18) n + 22 (cid:19) − , we see that the divisor (cid:0) n +22 (cid:1) M satifies the conditions of Theorem 2.5 with c ( k ) = (cid:0) n +22 (cid:1) . (cid:3) Now our main theorem is a consequence of the above results.
Proof of Theorem 1.3.
By Corollary 1.4.3 in [1] we can assume that ( X, ∆) is a Q -factorial klt pair, see also Corollary 4.4 in [16] for a more detailed proof.We can write ⌈ K X + ∆ + E + mM ⌉ = K X + ∆ + E ′ + mM, where E ′ := E + ⌈ K X + ∆ + E ⌉ − ( K X + ∆ + E )is a pseudo-effective Q -divisor. Then Proposition 2.3 and Corollary 2.7 imply that | ⌈ K X + ∆ + E + mM ⌉ | separates any two points x and x not in B + ( M ). Since X − B + ( M ) is a dense open subset of X , the result follows. (cid:3) Corollary 2.8.
Let ( X, ∆) be a klt pair with K X + ∆ pseudo-effective. Let M be abig and nef Q -divisor on X . Let ν be an integer such that νM is a Z -divisor, thenfor any m > ν (cid:18) n + 22 (cid:19) divisible by ν , the map induced by | ⌈ m ( K X + ∆ + M ) ⌉ | is a birational map.Proof. Let m be as in the statement and set E := ( m − K X +∆). Then Theorem1.3 gives the result. (cid:3) NIFORM BOUNDS FOR THE IITAKA FIBRATION Note that if we take X smooth and ∆ = 0, we get an effective version of Theorem1.2.Of course one can ask a weaker question about the non-vanishing of the cohomologygroup H ( X, O X ( ⌈ m ( K X + ∆ + M ) ⌉ )). Using a similar version of Proposition 2.3and Theorem 6.4 in [12] we get: Theorem 2.9.
Let ( X, ∆) be a klt pair with K X + ∆ pseudo-effective. Let M be abig and nef Q -divisor on X . Let ν be an integer such that νM is a Z -divisor, thenfor any m > ν (cid:18) n + 12 (cid:19) divisible by ν , | ⌈ m ( K X + ∆ + M ) ⌉ | 6 = ∅ . In the application to the Iitaka fibration we need to study the round down ofthese linear series instead of the round up.
Definition 2.10.
The index of a variety X is the smallest natural number a ( X ) such that a ( X ) K X is a Cartier divisor. Corollary 2.11.
Let ( X, ∆) be a klt pair such that K X + ∆ is pseudo-effective.Let M be a big and nef Q -divisor on X and let ν be an integer such that νM isa Z -divisor. Suppose ⌊ k ∆ ⌋ ≥ ( k − for any k ∈ Z > divisible by ν and a ( X ) .Then for any m > ν (cid:18) n + 22 (cid:19) divisible by ν and a ( X ) the map induced by | ⌊ m ( K X + ∆ + M ) ⌋ | is birational.Proof. Let m be as in the statement, then we can write ⌊ m ( K X + ∆ + M ) ⌋ = K X + ( m − K X + ∆) + ⌊ m ∆ ⌋ − ( m − mM. Let E := ( m − K X + ∆) + ⌊ m ∆ ⌋ − ( m − K X + E + mM isan Z -divisor. Then the result follows from Theorem 1.3. (cid:3) Pseudo-effective Threshold
In this section we deal with the case where K X + ∆ is not pseudo-effective.Following [1] and [23] we define the pseudo-effective threshold. Definition 3.1.
Let ( X, ∆) be a pair such that K X + ∆ is not pseudo-effective.Let M be a big divisor on X . We define the pseudo-effective threshold e ( X, ∆ , M ) of ( X, ∆) with respect to M as e ( X, ∆ , M ) := inf { e ′ ∈ R | K X + ∆ + e ′ M is pseudo-effective } . If there is no risk of confusion we denote it only by e ( M ). NIFORM BOUNDS FOR THE IITAKA FIBRATION Proposition 3.2.
Let ( X, ∆) be a klt pair such that K X +∆ is not pseudo-effective.Let M be a big and nef Z -divisor on X such that K X + ∆ + M is big. Let e ( M ) bethe pseudo-effective threshold of ( X, ∆) with respect to M . Then for any m > − e ( M ) (cid:18) n + 22 (cid:19) the map induced by the linear system |⌈ m ( K X + ∆ + M ) ⌉| is birational.Proof. Let m be as in the statement. Since K X + ∆ + M is big, by Corollary 2.2.24in [14], we know that e ( M ) <
1. Then we can find a rational number e ′ such that e ( M ) ≤ e ′ < m > − e ′ (cid:18) n + 22 (cid:19) . We can write m ( K X + ∆ + M ) = K X + ∆ + ( m − K X + ∆ + e ′ M ) + ( m (1 − e ′ ) + e ′ ) M. In particular it is enough to prove that the map induced by round up of the linearsystem K X + ∆ + ( m − K X + ∆ + e ′ M ) + m (1 − e ′ ) M is a birational map. Then the result follows from Theorem 1.3. (cid:3) We have an analogue statement for the round down.
Proposition 3.3.
Let ( X, ∆) , M and e ( M ) as in Proposition 3.2. Furthermoreassume that ⌊ k ∆ ⌋ ≥ ( k − for any k ∈ Z > divisible by a ( X ) . Then for any m > − e ( M ) (cid:18) n + 22 (cid:19) divisible by a ( X ) , the map induced by the linear system |⌊ m ( K X + ∆ + M ) ⌋| isbirational.Proof. The argument is the same as in Proposition 3.2 and Corollary 2.11. (cid:3) Iitaka Fibration
We now show how the previous results gives the uniformity of the Iitaka fibrationunder some extra conditions. For the definition and basic properties of the Iitakafibration we refer to [14]. We recall the canonical bundle formula and some of hisproperties, see [6] for details. Let f : X Y be the Iitaka fibration of X with Y nonsingular and general fiber F . Blowing up X we may assume that f is amorphism. Then the canonical bundle formula says that there are Q -divisors B Y and M Y such that K X ∼ Q f ∗ ( K Y + B Y + M Y ) . NIFORM BOUNDS FOR THE IITAKA FIBRATION B Y is called the boundary divisor and it is an effective divisor such that ( Y, B Y ) isa klt pair. M Y is called the moduli part and it is a nef Q -divisor. We now definetwo numbers which play a key role in the canonical bundle formula. Definition 4.1.
Let b := min { b ′ > | | b ′ K F | 6 = ∅} . Let B be the ( n − κ ( X )) -th Betti number of a non-singular model of the cover E → F associated to the unique element of | bK F | . We define N = N ( B ) := lcm { m ∈ Z > | ϕ ( M ) ≤ B } , where ϕ is the Euler function. We list some properties we will need later.
Proposition 4.2.
The following hold true: (1) bN M Y is a Z -divisor; (2) for any m ∈ Z > divisible by b , we have H ( X, O X ( mK X )) ∼ = H ( Y, O Y ( ⌊ m ( K Y + M Y + B Y ) ⌋ ));(3) for any m ∈ Z > divisible by bN , ⌊ mB Y ⌋ − ( m − B Y is effective; (4) K Y + M Y + B Y is a big Q -divisor; (5) if F has a good minimal model and V ar ( f ) is maximal then M Y is big.Proof. For (1), (2) and (3) see [6]. (4) follows from (2). Finally (5) follows from atheorem of Kawamata in [11]. See also Corollary 3.1 in [17]. (cid:3)
In particular (2) implies that | mK X | is birational to the Iitaka fibration if andonly if | ⌊ m ( K Y + M Y + B Y ) ⌋ | gives a birational map. Proof of Theorem 1.4.
It is just Proposition 4.2 and Corollary 2.11 with a ( Y ) = 1because Y is smooth. (cid:3) We can now prove an effective version of Theorem 1.2 in [17].
Corollary 4.3.
Let X be a smooth projective variety of Kodaira dimension κ and f : X Y be the Iitaka fibration. Assume that (1) Y is not uniruled; (2) f has maximal variation; (3) the generic fiber F has a good minimal model.Then for any m > bN (cid:18) κ + 22 (cid:19) divisible by bN , the pluricanonical map φ | mK X | is birationally equivalent to f . NIFORM BOUNDS FOR THE IITAKA FIBRATION Proof.
The main result in [2] implies that if Y is not uniruled then K Y is pseudo-effective. By Proposition 4.2 we know that M Y is big. Then Theorem 1.4 applies. (cid:3) If we use Theorem 2.9 instead of Theorem 1.3 we can prove the following.
Theorem 4.4.
Let X as in Theorem 1.4. Then for any m > bN (cid:18) κ + 12 (cid:19) divisible by bN , the cohomology group H ( X, O X ( mK X )) is non-zero. In particular if κ ( X ) = n − b = 12 and N = 22.If K Y + B Y is not pseudo-effective we have a similar result but the bound dependson the pseudo-effective threshold e ( M Y ). Theorem 4.5.
Let f : X Y the Iitaka fibration of X with general fiber F .Assume that (1) f has maximal variation; (2) the generic fiber F has a good minimal model.Then for any sufficiently divisible m > bN − e ( M Y ) (cid:18) κ + 22 (cid:19) the map associated to | mK X | is birational to the Iitaka fibration.Proof. Apply Proposition 4.2 and Proposition 3.3 with a ( Y ) = 1. (cid:3) It is now natural to ask the following.
Question 4.6.
Is possible to find a universal bound e < which depends only onthe dimension of Y , b and N such that e ( Y, B Y , M Y ) ≤ e for any Y , B Y and M Y as in Theorem 4.5? Viehweg and Zhang gave an affirmative answer to Question 4.6 in the casedim( Y ) = 2, see Lemma 2.10 and Lemma 2.11 in [23]. Acknowledgments
First I would like to express my gratitude to Professor J´anos Koll´ar for hisconstant support and many enlightening discussions. I also would like to thankProfessor Gianluca Pacienza for constructive comments on the paper.
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