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Effective Semi-Ampleness of Hodge Line Bundles on Curves I

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Effective Semi-Ampleness of Hodge Line Bundles on Curves I EFFECTIVE SEMI-AMPLENESS OF HODGE LINE BUNDLES ON CURVES I CHUYU ZHOU Abstract. In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type. Contents 1. Introduction 1 Acknowledgement 2 2. Preliminaries 2 2.1. K-stability 3 2.2. CM-line bundle and Hodge line bundle 3 3. K-moduli space with a Hodge line bundle 4 4. Positivity of the Hodge line bundle 5 5. Proof of the main result 6 6. Further exploration: Semi-ampleness of Hodge line bundles 8 6.1. A more general setting 8 6.2. A criterion for semi-ampleness 8 References 9 Throughout, we work over C. 1. Introduction Let f :(X; ∆) ! C be a klt (resp. lc)-trivial fibration over a smooth projective curve, i.e. f is surjective with connected fibers and (X; ∆) is a klt (resp. lc) log pair such that ∗ KX + ∆ ∼Q f D for some Q-divisor D on C. It is well known that there exist two Q- arXiv:2105.11986v1 [math.AG] 25 May 2021 divisors on C, which are called discriminant part denoted by BC and moduli part denoted by MC such that D ∼Q KC + BC + MC . As the names suggest, the divisor BC measures the singularities of fibers while the divisor MC measures the variation of fibers. Suppose C is an Ambro model (see [Amb04,Amb05]), the effective semi-ampleness conjecture proposed by Prokhorov and Shokurov ([PS09, Conjecture 7.13]) says that, there should be a positive integer m(d; r) depending only on the dimension d and the torsion index r of general fibers such that mMC is base point free. We note here that the semi-ampleness of MC is addressed by [Amb05, Flo14]. The effective semi-ampleness is harder and only a few cases are known. For example, O.Fujino [Fuj03, Theorem 1.2] shows that if general fibers are K3 surfaces or abelian varieties of dimension d, then one can take m = 19k and m = k(d + 1) respectively 1 2 CHUYU ZHOU to achieve base point freeness, where k is a weight associated to the Baily-Borel-Satake compactification of the period domain. It is worth mentioning that the proof heavily depends on the existence of moduli space of fibers. The idea is that MY appears due to the variation of fibers, it is apparent that one will have a good understanding of it once the moduli space of fibers exists. Recently, a projective separated good moduli space has been constructed for Fano varieties with fixed invariants and K-stability due to works [Jia20, BLX19, Xu20, BX19, ABHLX20, BHLLX20, XZ20, LXZ21]. It is the so called K-moduli space. Thus one can apply the idea to the klt-trivial fibrations whose fibers are parametrized by a K-moduli space. For a klt (resp. lc) trivial fibration (X; ∆) ! C, we say that it is Q-Gorenstein if KX=C is Q-Cartier. The following theorem is the main result of this paper. Theorem 1.1. Fix positive integers d and r > 1. Let f :(X; ∆) ! C be a Q-Gorenstein klt-trivial fibration over a smooth projective curve C such that (1) for any closed point t 2 C, the fiber Xt is a Q-Fano variety of dimension d, i.e. Xt admits klt singularities and −KXt is ample, (2) r∆t is a Weil divisor on Xt and no component of ∆ is contained in a fiber, (3) for any closed point t 2 C, the fiber (Xt; ∆t) is a klt log Calabi-Yau pair, i.e. KXt + ∆t ∼Q 0, ∗ Write KX + ∆ = f (KC + ΛC ). Then there exists a positive integer m(d; r) depending only on d and r such that mΛC is base point free. Here ΛC is the moduli part of the klt-trivial fibration, also called Hodge line bundle (which is in fact a Q-line bundle) in this case. We roughly talk here about the idea of the proof which will be treated more carefully later. As (Xt; ∆t) is a klt log Calabi-Yau with −KXt being ample, the pair (Xt; (1 − )∆t) is a K-stable log Fano pair for any 0 < 1. Via K-stability, one can construct a K-moduli space M parametrizing the fibers with a Hodge line bundle ΛHodge on M. For any klt-trivial fibration as in Theorem 1.1 there is a morphism C ! M such that ΛC is the pullback of the Hodge line bundle on M. Thus one can read the information of ΛC from ΛHodge. Note that in Theorem 1.1, we require that every fiber is a klt log Calabi-Yau pair, which is a key point in the proof. In the last section, we will discuss about a more general setting. Acknowledgement. The author would like to thank Chen Jiang, Junpeng Jiao, Yuchen Liu, Zhan Li, Zsolt Patakfalvi, Javier Carvajal Rojas and Roberto Svaldi for helpful discus- sions. The author is grateful to Ziquan Zhuang for answering his questions on the paper [XZ20]. The author is supported by grant European Research Council (ERC-804334). 2. Preliminaries In this section, we give a quick introduction of the concepts of K-stability, CM-line bundle, and Hodge line bundle. We say (X; ∆) is a log pair if X is a projective normal variety and ∆ is an effective Q-divisor on X such that KX + ∆ is Q-Cartier. The log pair (X; ∆) is called log Fano if it admits klt singularities and −(KX + ∆) is ample; if ∆ = 0, we just say X is a Q-Fano variety. For the concepts of klt and lc singularities, please refer to [KM98, Kol13]. EFFECTIVE SEMI-AMPLENESS OF HODGE LINE BUNDLES ON CURVES I 3 2.1. K-stability. Let (X; ∆) be a log pair. Suppose f : Y ! X is a proper birational mophism between normal varieties and E is a prime divisor on Y , we say that E is a prime divisor over X and we define the following invariant ∗ AX;∆(E) := ordE(KY − f (KX + ∆)); which is called the log discrepancy of E associated to the pair (X; ∆). If (X; ∆) is a log Fano pair, we define the following invariant Z 1 1 ∗ SX;∆(E) := vol(−f (KX + ∆) − xE)dx: vol(−KX − ∆) 0 Denote βX;∆(E) := AX;∆(E) − SX;∆(E). By the works [Fuj19, Li17, BJ20], one can define K-stability of a log Fano pair by the following way. Definition 2.1. Let (X; ∆) be a log Fano pair. (1) We say that (X; ∆) is K-semistable if βX;∆(E) ≥ 0 for any prime divisor E over X. (2) We say that (X; ∆) is K-polystable if it is K-semistable and any prime divisor E over X satisfying βX;∆(E) = 0 is of product type (see [Fuj17, Section 3.2]). (3) We say that (X; ∆) is uniformly K-stable if there is a positive rational number 0 < 1 such that βX;∆(E) ≥ SX;∆(E) for any prime divisor E over X. 2.2. CM-line bundle and Hodge line bundle. In this section, we fix an integer r > 1 1 and a family π :(X ; r D) ! B of relative dimension d satisfying the following conditions: (1) π and πjD are flat projective morphisms, and D is a Weil divisor on X , (2) −KX =B is a relatively ample Q-Cartier divisor on X and D ∼Q;B −rKX =B, 1 (3)( X ; r D + Xt) is slc for every closed point t 2 B. Let L be a relatively ample Q-line bundle on X . We have the following Knudsen-Mumford expansion (see [KM76]): k k k (d+1) (d) (1) det(π∗(kL)) = λd+1 ⊗ λd ⊗ ::: ⊗ λ1 ⊗ λ0; k ~(d) ~ det(πjD∗(kLjD)) = λd ⊗ ::: ⊗ λ0; ~ where λi; i = 0; 1; :::; d + 1; and λj; j = 0; 1; :::; d; are Q-line bundles on B. By the flatness 0 0 of π and πjD, we know that h (Xt; kLt) and h (Dt; kLtjDt ) are independent of t. We write 0 d d−1 d−1 0 d−1 d−1 h (Xt; kLt) = a0k + a1k + o(k ) and h (Dt; kLtjDt ) =a ~0k + o(k ): It is not hard to see Ld −K Ld−1 Ld−1D a = t ; a = Xt t ; a~ = t t : 0 d! 1 2(d − 1)! 0 (d − 1)! Definition 2.2. We define a Q-line bundle λCM;c (i.e. CM-line bundle) on B with respect to the polarization L as follows: 2a1 +d(d+1) a~0 a0 −2(d+1) a0 ~−(d+1) M1 := λd+1 ⊗ λd and M2 := λd+1 ⊗ λd ; c λ := M − M : CM;c 1 r 2 4 CHUYU ZHOU Definition 2.3. We define a Q-line bundle ΛB (i.e. Hodge line bundle) on B as follows: 1 K + D = π∗Λ : X =B r B The CM-line bundle satisfies the base change property, see [CP21, Lemma 3.5, Proposition 3.8]. When B is a projective normal variety, by Riemann-Roch formula, cf [CP21, Appendix], we have d+1 (1) λd+1 = π∗(L ); d d+1 1 d (2) λd = 2 π∗(L ) + 2 π∗(−KX =BL ); ~ d (3) λd = π∗(L D): Remark 2.4. Notation as above, if B is a projective normal base, we have the following simple computations: d−1 d d(KXt Lt ) d+1 (1) M1 = (d + 1)π∗(L KX =B) − d π∗L : Lt d−1 d d(DtLt ) d+1 (2) M2 = −(d + 1)π∗(L D) − d π∗L : Lt c d−1 d c d(KXt + r Dt)Lt d+1 (3) λCM;c := (d + 1)π∗(L (KX =B + r D)) − d π∗L : Lt c d+1 (4) If c 2 [0; 1) and L = −(KX =B + r D), then λCM;c = −π∗L : c d+1 (5) If c > 1 and L = KX =B + r D, then λCM;c = π∗L : d 1 (6) If c = 1, then λCM;1 = (d + 1)π∗(L (KX =B + r D)): Lemma 2.5.
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