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On the Boundedness of Canonical Models

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On the Boundedness of Canonical Models ON THE BOUNDEDNESS OF CANONICAL MODELS JUNPENG JIAO Abstract. It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We con- firm this conjecture when the general fibers of the corresponding Iitaka fibration are in a fixed bounded family of polarized Calabi-Yau pairs. As a consequence, we prove that in this case, the fibration is birationally bounded, and when it has terminal singularities, the corresponding min- imal model is bounded in codimension 1. 1. Introduction Throughout this paper, we work with varieties defined over the complex numbers. By analogy with the definition of volumes of divisors, the Iitaka volume of a Q-divisor is defined as follows. Let X be a normal projective variety and D be a Q-Cartier divisor. When the Iitaka dimension κ(D) of D is nonnegative, then the Iitaka volume of D is defined to be κ(D)!h0(X, O (⌊mD⌋)) (1.1) Ivol(D) := lim sup X m→∞ mκ(D) Theorem 1.1. Fix C a log bounded class of polarized log Calabi-Yau pairs and V,v > 0 two positive rational numbers. Let (X, ∆) be a klt pair, L a divisor on X and f : X → Z an algebraic contraction which is birationally equivalent to the Iitaka fibration of KX +∆. If the general fiber (Xg, ∆g,Lg) of f is in C , then (1) Ivol(X, KX + ∆) is in a DCC set; (2) If Ivol(X, KX + ∆) ≤ V , then X is birationally bounded; arXiv:2103.13609v1 [math.AG] 25 Mar 2021 (3) If Ivol(X, KX +∆)= v is a constant, then ∞ 0 Proj ⊕m=0 H (X, OX (mKX + ⌊m∆⌋)) is in a bounded family As an application, we prove the following boundedness result. Theorem 1.2. Fix C a bounded class of polarized Calabi-Yau varieties and V > 0 a positive rational number. Then there is a bounded family of varieties Xm → T satisfying the following. Assume X is a variety as follows: (1) X is projective normal with terminal singularities, (2) there is an algebraic contraction f : X → Z which is birationally equivalent to the Iitaka fibration of KX , 1 2 JUNPENG JIAO (3) the general fiber (Xg,Lg) of f is in C , and (4) Ivol(X, KX ) ≤ V . Then there is a closed point t ∈ T , the fiber Xm,t is a good minimal model of X. Theorem (1.1) is a special case of the following conjecture. Conjecture 1.3. Let n be a positive integer, v a non-negative rational num- ber, and I ⊂ [0, 1] a DCC set of rational numbers. We let D(n,v, I) be the set of varieties Z satisfying the following: (1) (X, ∆) is a klt pair of dimension n, with coeff(∆) ⊂I. (2) Ivol(KX +∆) = v is a constant, and (3) f : X 99K Z is the Iitaka fibration associated with KX +∆, where ∞ 0 Z = Proj ⊕m=0 H (X, OX (m(KX + ∆))). Then D(n,v, I) is in a bounded family. ∞ 0 Note that by [BCHM06], R(X, KX +∆) := ⊕m=0H (X, OX (m(KX +∆))) is finitely generated and in particular Z is well defined normal projective variety and v is a positive rational number. When KX +∆ is big, Conjecture 1.3 is proved by [HMX14], when the general fiber of f is ǫ-lc Fano type, it is proved by [Li20]. Usually, boundedness of varieties is connected with the DCC property of volumes, see [HMX13]. The following conjecture is closely related with Conjecture 1.3. Conjecture 1.4. Let n ∈ N be a fixed number, and I ⊂ [0, 1]∩Q be a DCC set. Then the set of Iitaka volumes {Ivol(KX + ∆)|(X, ∆) is klt, dimX = n, and coefficients of ∆ are in I} is a DCC set. Our main result in this direction is to prove the DCC of Iitaka volumes and the boundedness of the canonical models when the locus of singular fibers in the corresponding Iitaka fibrations is ”bounded”. To be precise, we are interested in the following set of log pairs and the corresponding Iitaka volumes. Definition 1.5. Fix a DCC set I ⊂ [0, 1] ∩ Q, integers n,l,r > 0 , define DI,n,r,l to be be the set of log pairs (X, ∆), such that (1) (X, ∆) is a klt pair of dimension n, with coeff∆ ⊂ I. (2) f : X → Z is the canonical model of (X, ∆). (3) KX +∆ ∼Q F for an effective Q-divisor F and there is a reduced divisor D such that (X, Supp(∆ − F )) is log smooth over Z \ D and ∗ Ivol(KX +∆+ f D) ≤ rIvol(KX + ∆). ONTHEBOUNDEDNESSOFCANONICALMODELS 3 (4) By the canonical bundle formula, there is a nef Q-divisor M and a Q-divisor B on a birational model W of Z such that lM is Cartier and 0 0 H (X, OX (m(KX + ∆))) =∼ H (W, OZ (m(KW + B + M))) for m sufficiently divisible. Theorem 1.6. Fix a DCC set I ⊂ [0, 1] ∩ Q, integers n,l,r > 0, then the set {Ivol(KX + ∆) | (X, ∆) ∈ DI,n,r,l} satisfies the DCC. As an application, we prove the following boundedness result Theorem 1.7. Fix a DCC set I ⊂ [0, 1] ∩ Q, integers n,l,r > 0 and a positive number C > 0. Then the set {ProjR(X, KX + ∆) | (X, ∆) ∈ DI,n,r,l, Ivol(KX +∆)= C} is bounded. The main idea is to prove that we can choose a SNC model (see definition 2.8) of (X, ∆ − F ) → Z belonging to a bounded family, this is where we use condition (3) in Definition 1.5. We believe that the corresponding integer r naturally arises from the moduli space of the general fiber of f. Theorem 1.1 is an application of Theorems 1.6 and 1.7 that is based on this idea. Acknowledgement. I would like to thank my advisor, Professor Christo- pher Hacon, for many useful suggestions, discussions, and his generosity, I would also like to thank Stefano Filipazzi, Zhan Li, Yupeng Wang, Jingjun Han and Jihao Liu for many helpful conversations. The author was partially supported by NSF research grant no: DMS-1952522 and by a grant from the Simons Foundation; Award Number: 256202. 2. Prelimiary 2.1. Notation and conventions. Let I ⊂ R be a subset, we say I satisfies the DCC if there is no strictly decreasing subsequence in I. For a birational −1 morphism f : Y → X and a divisor B on X, f∗ (B) denotes the strict transform of B on Y , and Exc(f) denotes the sum of reduced exceptional divisors of f. A fibration means a projective and surjective morphism with connected fibers. For an R-divisor D, a map defined by the linear system |D| means a map defined by |⌊D⌋|. Given two R-Cartier R-divisors A, B, A ∼Q B means that there is an integer m> 0 such that m(A − B) ∼ 0. A pair (X, ∆) consists of a normal variety X over C and an Q-divisor ∆ on X such that KX +∆ is Q-Cartier. If g : Y → X is a birational morphism and E is a divisor on Y , the discrepancy a(E,X, ∆) is −coeffE(∆Y ) where KY + ∗ ∆Y := g (KX + ∆). A pair (X, ∆) is call subklt (respectively sublc) if for every birational morphism Y → X as above, a(E,X, ∆) > −1 (respectively 4 JUNPENG JIAO ≥−1) for every divisor E on Y . A pair (X, ∆) is called klt (respectively lc) if (X, ∆) is subklt (respectively sublc) and ∆ is effective. A generalised pair (X, ∆+ M) consists of a normal variety X equipped f with a birational morphism X′ −→ X where X is normal, a Q-boundary ∆, ′ ′ and an Q-Cartier nef divisor M on X such that KX +∆+ M is Q-Cartier, ′ ′ ′ ′ where M = f∗M . Let ∆ be the Q-divisor such that KX′ +∆ + M = ∗ f (KX +∆+M), we call (X, ∆+M) a generalised klt (respectively lc) pair, if (X′, ∆′) is a subklt (respectively sublc) pair. Let X → Z be an algebraic contraction and R a divisor on X, we write R = Rv + Rh, where Rv is the vertical part and Rh is the horizontal part. Definition 2.1. For a klt pair (X, ∆) such that KX +∆ is Q-Cartier and κ(X, ∆) ≥ 0, by [BCHM06], the canonical ring 0 R(X, KX +∆) := ⊕m≥0H (X, OX (m(KX + ∆))) is finitely generated. Therefore if κ(X, KX +∆) ≥ 0, we define the canonical model of (X, ∆) to be ProjR(X, KX + ∆). Next we state some well known results that we will use in what follows. Theorem 2.2 ([BZ16, Theorem 1.3]). Let d, r be two positive integers and I ⊂ [0, 1] a DCC set of real numbers. Then there is a positive number m0 depending only on d, r and I satisfying the following. Assume that (1) (Z,B) is a projective lc pair of dimension d, (2) B ∈ I, (3) rM is a nef Cartier divisor, and (4) KZ + B + M is big, then the linear system |m(KZ + B + M)| defines a birational map for every positive integer m such that m0 | m. Theorem 2.3 ([BZ16, Theorem 8.1]). Let I be a DCC set of nonnegative real numbers and d a natural number. Then there is a real number e ∈ (0, 1) depending only on I,n such that if (1) (Z,B) is projective lc of dimension d, (2) M = µjMj where Mj are nef Cartier divisors, (3) the coefficients of B and the µj are in I, and P (4) KZ + B + M is a big divisor, then KZ + eB + eM is a big divisor.
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