Logarithmic kodaira dimension and zeros of holomorphic Log-One-Forms

We prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair of log-general type must be non-empty. This result is a natural generalization of the work of M. Popa and C. Schnell in 2015, which states that, the zero-locus of any global holomorphic one-form on a smooth projective variety of general type must be non-empty. We first give a simplified proof of the result of Popa and Schnell. Instead of using generic vanishing of mixed Hodge modules on abelian varieties, we obtain a new proof using only Kodaira-Saito Vanishing. To prove our result about log-one forms, we apply Saito's mixed Hodge modules theory. We prove some logarithmic comparison theorems and we use a filtered log-$\mathscr{D}$-module to represent a mixed Hodge module, instead of using filtered $\mathscr{D}$-modules. The structure of the proof still follows the general outline of the work of Popa and Schnell. There are two important applications of our main result. One of them is that we get an affirmative answer (in a much more general setting) to a question posed by F. Catanese and M. Schneider. Another application is that we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-terminal (klt) and of log-general type.

LOGARITHMIC KODAIRA DIMENSION AND ZEROS OF HOLOMORPHIC LOG-ONE-FORMS by Chuanhao Wei A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah August 2018 Copyright c Chuanhao Wei 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Chuanhao Wei has been approved by the following supervisory committee members: Christopher D. Hacon , Chair(s) April 24, 2018 Date Approved Aaron Bertram , Member April 24, 2018 Date Approved Tommaso de-Fernex , Member April 24, 2018 Date Approved Yuan-Pin Lee , Member Date Approved Ilya Zharov , Member Date Approved by Davar Khoshnevisan , Chair/Dean of the Department/College/School of Mathematics and by David B. Kieda , Dean of The Graduate School. ABSTRACT We prove that the zero-locus of any global holomorphic log-one-form on a projective logsmooth pair of log-general type must be non-empty. This result is a natural generalization of the work of M. Popa and C. Schnell in 2015, which states that, the zero-locus of any global holomorphic one-form on a smooth projective variety of general type must be non-empty. We first give a simplified proof of the result of Popa and Schnell. Instead of using generic vanishing of mixed Hodge modules on abelian varieties, we obtain a new proof using only Kodaira-Saito Vanishing. To prove our result about log-one forms, we apply Saito's mixed Hodge modules theory. We prove some logarithmic comparison theorems and we use a filtered log-D-module to represent a mixed Hodge module, instead of using filtered D-modules. The structure of the proof still follows the general outline of the work of Popa and Schnell. There are two important applications of our main result. One of them is that we get an affirmative answer (in a much more general setting) to a question posed by F. Catanese and M. Schneider. Another application is that we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-terminal (klt) and of log-general type. To my Grandfather, Chuanwen Tong. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTERS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. APPLICATIONS OF THE MAIN THEOREM . . . . . . . . . . . . . . . . . . . . . 5 2.1 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. TWO SPECIAL CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 The log-canonically-polarized case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Fibration over quasi-abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4. PROOF OF THE MAIN THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 A simplified proof of the non-log version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Proof of the Main Theorem (Theorem 1.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5. LOGARITHMIC COMPARISONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1 5.2 5.3 5.4 5.5 6. Kashiwara-Malgrange filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pushforward functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related vanishings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 32 37 45 48 PROOF OF THE MAIN CLAIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1 Connection to mixed Hodge modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Proof of the Main Claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 APPENDICES A. QUASI-ABELIAN VARIETIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 B. FILTERED LOG-D-MODULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 ACKNOWLEDGEMENTS Firstly, I would like to express my sincere gratitude to my advisor, Professor Chistopher Hacon, for his continuous support of my study and related research, for his motivation when I am not making progress in my research, and for his immense knowledge and insightful suggestions helping me finish the dissertation. Besides my advisor, I would like to thank the rest of my dissertation committee: Prof. Aaron Bertram, Prof. Tommaso de Fernex, and Prof. Yuan-Pin Lee, not only for their insightful comments, but also for the courses they taught during my five-year study in the University of Utah. I would like to express my special appreciation to Professor Christian Schnell for his hospitality during my visit in the Stony Brook University in 2016. I made essential progress in my research thanks to his advice. For the work in this dissertation, I have also benefited from discussion with Dr. Honglu Fan, Prof. Kalle Karu, Dr. Linquan Ma, Prof. Mihnea Popa, Dr. Lei Wu, and Ziwen Zhu. My sincere thanks also goes to my friends Yang Da, Dapeng Mu, and Junliang Shen for their encouragement and company to make my overseas graduate school study an enjoyable and memorable experience. Last but not least, I want to express my deep gratitude to my parents, Jiadi Tong, and Gengrui Wei, for all their continuous love. Without their support and understanding, it would be impossible for me to finish my doctoral degree. CHAPTER 1 INTRODUCTION In differential geometry, people use smooth differential one-forms to study the geometry of an analytic manifold. Those differential one-forms locally look like f1 dx1 + ... + fn dxn , where xi are local real coordinates and fi are local smooth functions. In the world of complex algebraic geometry, however, to study the geometry of a complex manifold, people usually use holomorphic one-forms to replace the smooth differential oneforms, and they locally look like f1 dz1 + ... + fn dzn , where zi are local complex coordinates and fi are local holomorphic functions. In [17], Popa and Schnell use the generic vanishing theorem on Saito's mixed Hodge modules [16] to prove the following Theorem 1.1. For any projective smooth variety of general type, there exists no nonvanishing global holomorphic one-form on it. For any projective complex smooth variety X, there is an universal projective morphism aX : X → AX , which is called the albanese map of X, where AX is an abelian variety called the albanese variety of X, such that all global holomorphic one-forms over X are the natural pullback of some global holomorphic one-forms over AX . Hence, to study the global holomorphic one-forms, it is natural to study those projective morphisms from a projective complex smooth variety to an abelian variety, and we get a direct corollary from the previous theorem. Corollary 1.2. There exists no smooth fibration of a smooth projective variety of general type over an abelian variety. When the given variety is not projective, or has singularities, people usually consider a pair (X, D), instead of X itself. Further, after proper compactification and resolution of singularities, we assume that X is smooth and projective, and D is a divisor over X 2 with simply normal crossing (SNC) support. We call such a pair (X, D) a projective logsmooth pair if we further assume that D is reduced. In this case, instead of considering the holomorphic one-forms over X, it is more natural to consider the logarithmic holomorphic one-forms over (X, D). In local coordinates z1 , ..., zd , we may assume D = {z1 · ... · zk = 0}. dzk 1 Under this setting, a log-holomorphic one-form is locally described by f1 dz z1 + ... + fk zk + fk+1 dzk+1 + ... + fn dzn . Definition 1.1. Given a projective log-smooth pair (X, D), we define the log-kodaira dimension of (X, D), κ(X, D) to be the Iitaka dimension of the line bundle ωX (D) on X. We say that (X, D) is of log-general type if dim X = κ(X, D). The main goal of this dissertation is to prove the following theorem, which is a natural generalization of the work of Popa and Schnell. Theorem 1.3. The zero-locus of any global holomorphic log-one-form on a projective logsmooth pair (X, D) of log-general type must be non-empty. Actually, we can prove a more general result: Theorem 1.4. Let (X, D) be a projective log-smooth pair, and let W ⊂ H 0 X, Ω1X (log D) be a linear subspace that consists of global holomorphic log-one-forms with empty zero-locus. Then the dimension of W can be at most dim X − κ (X, D). Similarly to the holomorphic one-form case where it is natural to study a projective morphism over an abelian variety, in this case, we need to introduce another class of smooth varieties called quasi-abelian varieties. They are defined as an extension of a d-dimensional abelian variety Ad by an algebraic torus Grm : 1 → Grm → T r,d → Ad → 1. In particular, T r,d is a principal Grm -bundle over Ad . We study a quasi-projective morphism over T r,d . 3 Theorem 1.4 is a corollary to the following main theorem of the dissertation. Given two log smooth pairs X, DX and Y, DY , and a morphism f : X → Y , we say that f is a morphism of log-pairs and denoted by f : X, DX → Y, DY , if f −1 DY := Supp f ∗ DY ⊂ DX . Given a holomorphic log-one-form θ on a log smooth pair (X, D), we use Z (θ) to denote the zero-locus of θ as a global section of the locally free sheaf Ω1X (log D). Theorem 1.5 (Main Theorem). Let (X, D) be a projective log-smooth pair, and P r,d , L be the canonical Pr -bundle compactification of a quasi-abelian variety T r,d , with boundary divisor L. Denote by p : P r,d → Ad , the natural projection. Given a morphism of log-pairs f : (X, D) → P r,d , L , if there is a positive integer k and an ample line bundle A on Ad , such that H 0 X, (ωX (D))⊗k ⊗ f ∗ p∗ A−1 ⊗ OP r,d (−L) 6= 0, then Z (f ∗ ω) 6= ∅, for any ω ∈ H 0 Ω1P r,d (log L) . Further, for generic such ω, we have Z (f ∗ ω) ∩ X \ f −1 (L) 6= ∅. Remark 1.1. Actually, we only need to show the case for r = 0 or 1. We refer to the proof of Theorem 2.1 below for the details. Remark 1.2. We will see that the two statements in the theorem are actually equivalent to each other. In particular, it is a problem that only depends on the morphism over T r,d . See Step 0 of the proof of this theorem in Section 4.2 for details. There are two important applications of Theorem 1.4. One of them is that we get an affirmative answer (in a much more general setting) to a question posed by F. Catanese and M. Schneider. Corollary 1.6. Let (X, D) be a projective log-smooth pair of log general type. Assume that we have a surjective projective morphism f : X \ D → T r,d , where T r,d is a quasi-abelian variety. Then, f is not smooth. We have a different proof of this corollary based on the structure of Higgs bundles without applying Theorem 1.3, which will be shown in Chapter 2. 4 Another application is that we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata log-terminal (klt) and of log-general type. Corollary 1.7. Given a log-smooth surjective morphism of log-pairs f : (X, D) → P r,d , L , with (X, Dred ) being log-smooth and the generic fiber (Xη , Dη ) being klt and of log-general type, then f is birationally isotrivial. E. Viehweg and K. Zuo have shown the previous two questions in the case that the base is one-dimensional and the general fibers are projective and smooth, [22]. In Chapter 2, we prove Theorem 1.4 assuming Theorem 1.5. Then we show the geometric applications mentioned above. In Chapter 3, to motivate the reader more about the technical details in the proof of Theorem 1.5, we first give a very short proof of Theorem 1.3 in the case that (X, D) is canonically polarized, i.e., ωX (D) is ample. Then we show Corollary 1.6 based on the structure of Higgs bundles without using Theorem 1.3. In Chapter 4, we first give a simplified proof of the main theorem in [17] without applying the generic vanishing of mixed Hodge modules on abelian varieties, [16]. Following the same ideas, assuming Claim 2 (Main Claim), we prove Theorem 1.5. In Chapter 5, we first show some logarithmic comparison theorems in mixed Hodge modules. Then we show a vanishing result that will be used to prove the Main Claim, which will be shown at the end of this chapter. In Appendix A, we first recall the definition of quasi-abelian varieties and some of their properties that are used in the dissertation. Then, in Appendix B, we recall some basic notions about filtered log-D-modules and the strictness in their derived category that is used in the logarithmic comparison theorems. Most results of this dissertation come from the author's recent preprints [23], [24], [25]. All the varieties that appear in the dissertation are assumed to be reduced but possibly reducible separated schemes of finite type over the field of complex number C. We use e strict right D-modules (Appendix A) to represent mixed Hodge modules, forgetting the weight filtration. All mixed Hodge modules in this paper are assumed to be algebraic. In particular, they are assumed to be extendable and polarizable, [21]. CHAPTER 2 APPLICATIONS OF THE MAIN THEOREM We show some geometric applications of the Main Theorem (Theorem 1.5) in this chapter. 2.1 Proof of Theorem 1.4 We start by showing that Theorem 1.5 implies the following. Theorem 2.1. Given a morphism of log smooth pairs f : (X, D) → P r,d , L , where P r,d , L is the canonical Pr -bundle compactification of a quasi-abelian variety T r,d , with boundary divisor L, then there exists a linear subspace W ⊂ H 0 P r,d , Ω1P r,d (log L) of co-dimension dim X − κ (X, D), such that Z (f ∗ ω) 6= ∅, for any ω ∈ W . Proof. Denote U = X \ D. The statement is vacuous when κ (X, D) = −∞, so we can assume that κ (X, D) ≥ 0. Let µ : (X 0 , D0 ) → (X, D) be a birational modification, such that g : X 0 → Z is a smooth model for the Iitaka fibration associated to the log-canonical bundle ωX (D). It is not hard to see that, if (Xz0 , Dz0 ) is a very general fiber over Z, then it is a log-smooth pair of dimension δ (U ) = dim U − κ (X, D) with log-Kodaira dimension 0. Hence, due to [11, Theorem 28], the quasi-albanese map of Xz0 \Dz0 is an open algebraic fiber space, and by [11, Theorem 27], it is not hard to see that its image in TUr,d is a quasi-abelian variety. Further, by Proposition 2.2 below, we know that there are at most countably many quasi-abelian sub-algebraic groups in TUr,d . Hence, the image in TUr,d of every fiber of g is a dense subset of a translate of a single quasi-abelian sub-algebraic group. Letting T be the quotient of TUr,d by that sub quasi-abelian sub-algebraic group, we have dim T ≥ m − δ (U ), where m = r + d = dim TUr,d . Hence, we get an induced rational map Z 99K T . Let (P, L) be the projective-space-bundle compactification of T . Though P r,d 99K P is just a rational map in general, after a toroidal log-resolution P̂ r,d , L̂ → P r,d , L , the induced map of 6 log-pairs P̂ r,d , L̂ → (P, L) is a morphism. After making a toroidal log resolution τ : X̂, D̂ → (X, D) , we can get a morphism of log-pairs X̂, D̂ → P̂ r,d , L̂ , by Lemma 2.3, or [1, Section 1, Remark 1.4] in general. Composing it with the previous one, we get a morphism of log-pairs fˆ : X̂, D̂ → (P, L) . Consider the following commutative diagram: (X 0 , D0 ) X̂, D̂ fˆ Z (P, L) . By replacing (X 0 , D0 ) → Z by another birationally equivalent model if necessary, we can assume that the horizontal maps above are actually morphisms. Since (X 0 , D0 ) → Z is an Iitaka fibration associated to the log-canonical bundle, it is not hard to see that, for any line bundle  on P , we have ⊗k ∗ ˆ ⊗ f  6= 0, H X̂, ωX̂ D̂ 0 for some integer k. Since τ : X̂, D̂ → (X, D) is a toroidal log-resolution, we have that the natural morphism τ ∗ Ω1X (log D) → Ω1X̂ log D̂ is surjective. Hence, θ ∈ H 0 X, Ω1X (log D) has zero-locus if and only if τ ∗ θ ∈ H 0 X̂, Ω1X̂ log D̂ has zero-locus. Hence, now we only need to show that, for any ω ∈ H 0 P, Ω1P (log L) , fˆ∗ ω ∈ H 0 X̂, Ω1X̂ log D̂ will have zero-locus. Now we can conclude the proof by Theorem 1.5. For the rest of the proof, we show that we actually just need the case that r = 0 or 1 in Theorem 1.5. Denote (P, L) by P r,d , L again to specify the dimension of the base quasi-abelian variety and projective space fiber, and T r,d := P r,d \ L. If r ≥ 2, we will argue that the statement can be reduced to the r = 1 case. Consider the subgroup Grm of T r,d , which is the fiber over 0, the unit of Ad . Fix {x1 , ..., xr }, a global algebraic coordinate system on Grm , and x1 = ... = xr = 1 gives the unit e of Grm . Given a vector a := [a1 , ..., ar ] ∈ Zr , 7 assuming a 6= 0, let's consider the divisor Ga of Grm defined by xa11 · ... · xar r = 1. It is evident that it is actually a subgroup of Grm , hence of T r,d . Let's denote the algebraic quotient by qa : T r,d → T r,d /Ga . It is evident that T r,d /Ga ' Ta1,d , for some quasi-abelian variety Ta1,d with indicated dimensions. Denote by Pa1,d , L , the canonical P1 -bundle compactification of Ta1,d . As in the argument above, replacing X̂, D̂ and P r,d , L by a toroidal log-resolution, we obtain morphisms of log-pairs: qa : P r,d , L → Pa1,d , L ga : X̂, D̂ → Pa1,d , L Denote Wa = qa∗ H 0 Pa1,d , Ω1P 1,d (log L) ⊂ H 0 P r,d , Ω1P r,d (log L) . a Note that ∪a Wa is a dense subset of H 0 P r,d , Ω1P r,d (log L) . This is because {dx1 /x1 , ..., dxr /xr } gives a basis of H 0 Ω1P r,d (log L) /p∗ H 0 Ad , Ω1Ad , and it is straightforward to check that Wa is the co-set C · (a1 dx1 /x1 + ... + ar dxr /xr ) + p∗ H 0 Ad , Ω1Ad . Since fˆ∗ ω has no zero-locus is an open condition on ω ∈ H 0 P r,d , Ω1P r,d (log L) , now it 1,d suffices to show the statement for each ga : X̂, D̂ → Pa , L , which is implied by Theorem 1.5. Remark 2.1. For a fixed quasi-abelian variety T r,d , fixing any smooth projective r-dimensional toric variety P r , we have the canonical P r -bundle compactification of T r,d , with a natural simply normal crossing boundary divisor L, and we also denote it by P r,d , L . All the P r,d , L that appear in this paper can be viewed in this setting. Proposition 2.2. There are at most countably many quasi-abelian sub-algebraic groups in a quasi-abelian variety. Proof. It is evident that we only need to show the case for any abelian variety and for any algebraic torus. For the abelian variety case, it is obvious by noting that there are at most 8 countably many sub-lattices in a given lattice of finite dimension. For the algebraic torus case, it is not hard to see that we only need to show that there are at most countably many algebraic group auto-morphism G1m → G1m . Fix a global coordinator t, the automorphism will be given by the map of function t 7→ P [t, t−1 ], where P [t, t−1 ] is a two-variable polynomial. Since it needs to be a group morphism, we have h i k P tk , t−k = P t, t−1 , for any k ∈ Z. Hence, it is not hard to conclude that P [t, t−1 ] = tm for some m ∈ Z, which has countably many choices. Recall that given a smooth quasi-projective variety U and a log-smooth compactification (X, D) of U , we have that T1 (U ) := H 0 X, Ω1X (log D) , which does not depend on the compactification, [6, 2.4]. Further, we canonically have a quasi-albanese map aU : U → TU , such that a∗U (T1 (TU )) = T1 (U ). TU is the quasi-albanese variety of U , which is a quasiabelian variety and the quasi-albanese map aU is algebraic. We refer to [6], [10] for the details of the quasi-albanese map. See also Appendix A. To prove Theorem 1.4 by applying the argument in the proof of the previous theorem, we need to construct such algebraic morphism of log-smooth pairs f : (X, D) → P r,d , L . Ideally, we want to directly use the quasi-albanese map aU : U → T r,d , and then compactify it and perform a log-resolution to get f . However, taking log-resolution may introduce new zero-loci for holomorphic log-one-forms. To keep track the zero-loci, we are only allowed to perform toroidal log-resolutions. That is the reason that we consider the following. Lemma 2.3. Fix a log smooth pair (X, D), and denote X \D = U . Assume that we have an algebraic morphism f : U → T r,d . Then there exists a toroidal log-resolution τ : X̂, D̂ → (X, D) such that f can be extended to get a morphism of log-pairs: fˆ : X̂, D̂ → P r,d , L . Proof. We first note that although we may not be able to extended f onto X, the morphism p ◦ f : U → Ad is well defined on X. This is because if not, by a sequence of blowing-ups of smooth centers, we can get a morphism. However, each rational curve will map to a point on an abelian variety. Hence, these blow-ups are not needed. Then we argue that it suffices to show the case that r = 1. Actually, it is not hard to see that we only need to show the case that P r,d , L is the (P1 )r -bundle compactification 9 of T r,d , Remark 2.1. Note that in this case, we can decompose P r,d = P11,d ×Ad P21,d ×Ad ... ×Ad Pr1,d , where Pi1,d is the P1 -bundle compatification of a quasi-abelian variety Ti1,d , with the indi cated dimension. Hence, to get a morphism of log-pairs fˆ : X̂, D̂ → P r,d , L , we only need to get a morphism of log-pairs X̂, D̂ → Pi1,d , L for each i. From now on, we only consider the case that r = 1. Denote the two components of L by L1 and L2 . It is evident that L separates into two components by the construction of P 1,d . We also have that L1 and L2 are linearly equivalent ([9, Exercise II.7.9]). Since f is well-defined on X except a co-dimension 2 locus. Hence, the zero order of f ∗ L1 and f ∗ L2 are well defined along each irreducible component Di of D. We denote by D0 and D00 the two sub-divisors of D that consist of those irreducible components with positive zero order of f ∗ L1 and f ∗ L2 , respectively. Since L1 ∩ L2 = ∅, D0 and D00 have no common components. We say that a co-dimension-2 stratum S of D is bad if S is the generic locus of D10 ∩ D100 , with D10 and D100 being irreducible components of D0 and D00 , respectively. For the rest of the proof, we will show that (1)If we have no bad stratum, then f is well defined on X. (2)We can find a sequence of toroidal resolutions on (X, D) to eliminate bad strata. For (1), if f is not well defined on X, by a sequence of blowing-ups of smooth locus on D, we can get a morphism of log-pairs fˆ : (X̂, D̂) → (P 1,d , L). Note that since we start with no bad stratum, it is evident that for each step of blow-up, we will not introduce new bad stratum. Let's consider the last step of the blow-ups and we denote it by π : (X̂, D̂) → (X, D), with E ⊂ D̂ being the exceptional divisor. We only need to show that each P1 on E maps to a point on P 1,d , which implies that this blow-up is not needed. Hence, we can conclude (1) by induction. Otherwise, since each P1 on E will map to a point on Ad , we can find a P1 on E maps to a P1 fiber over a point a ∈ Ad surjectively. In particular, there is one point e1 ∈ E that maps to L1 and another point e2 ∈ E maps to L2 . Consider the two sub-divisors of D̂: D̂0 := fˆ−1 L1 and D̂00 := fˆ−1 L2 . Obviously they have no common components, and E does not belong to either one. However, we can find two irreducible components D̂1 and D̂2 such that e1 ∈ D̂1 ⊂ D̂0 , and e2 ∈ D̂2 ⊂ D̂00 . We denote by D1 and D2 the two irreducible components of D with their strict transform on X̂ being D̂1 and 10 D̂2 , respectively. Let S be the stratum given by the generic locus of D1 ∩ D2 which is not empty by the construction. However, it is not hard to see that S is a bad stratum. Hence, we get a contradiction. For (2), since L1 and L2 are linearly equivalent, we can find a rational function y on P 1,d such that its zero-locus and pole locus are L1 and L2 , respectively. Since f is well defined on U and its image is in T 1,d , the zero-locus and the pole locus of the rational function f ∗ y on X has to be contained in D. Hence, around a stratum S, the generic locus of ∩1≤i≤n Di , up to a multiplication of a unit, we locally have f ∗ y = xa11 · ... · xann , where, for i = 1, ..., n, xi are local functions that define Di , the irreducible components of D, respectively, and ai ∈ Z. Note that ai does not depend on the choice of the stratum. Pick a bad stratum S of co-dimension-2. Assume S is the generic locus of D1 ∩ D2 . S being bad is equivalent to that a1 · a2 < 0. We can find two co-prime integers b1 , b2 , such that a1 · b2 + a2 · b1 = 0. By a blow-up of the ideal locally being xb11 , xb22 , which is toroidal, we get π : (X 0 , D0 + E) → (X, D), where D0 is the strict transform of D and E is the exceptional divisor. Now by induction, we only need to show that those newly introduced co-dimension-2 strata of D0 + E are not bad. It is evident by noticing that the zero order of (π ◦ f )∗ y along E is 0. Note that (X 0 , D0 + E) we construct above is not log-smooth in general, but after we eliminate all bad strata, we can perform a toroidal log-resolution to get a log-smooth pair we are after. Now, we are ready to show the proof of Theorem 1.4 assuming Theorem 1.5. Proof of Theorem 1.4. Following the argument of the proof of Theorem 2.1, we are reduced to consider the rational map ga : X 99K Pa1,d , which is well defined on U , ga U : U → Ta1,d . Now applying the previous lemma, we can find a toroidal log-resolution X̂, D̂ → (X, D) such that we honestly get a morphism of log pairs: ĝa : X̂, D̂ → Pa1,d , L . Hence, we can conclude the proof by the r = 0 or 1 case of Theorem 1.5 as in the proof of Theorem 2.1. 11 2.2 Applications We state couple corollaries that follow by Theorem 1.4. Corollary 2.4. Given a smooth projective variety X, a global holomorphic one-form θ on X with no zero-locus, and a smooth divisor D, if ωX (D) is big, then θ|D must have non-empty zero-locus as a global holomorphic one-form on D. Proof. Let x1 , ..., xn be a local holomorphic coordinate system of X and x1 defines D. Then {dx1 /x1 , dx2 , ..., dxn } gives a local basis of Ω1X (log D). Since we know that it has no zero-locus as holomorphic one-form, according to Theorem 1.3, θ has a zero-locus at some point p ∈ D as a global section of Ω1X (log D). Hence, it locally looks like θ = x1 g1 dx1 /x1 + g2 dx2 ... + gn dxn , where g2 (p) = ... = gn (p) = 0. Hence, θ|D ∈ H 0 (D, ΩD ) has zero-locus at p. Definition 2.1. Given a projective morphism of log pairs f : X, DX → Y, DY , assume X Y that X, Dred and Y, Dred are log-smooth. We say that f is log-smooth if the cokernel sheaf of the natural map f ∗ Ω1Y log DY → Ω1X log DX , is locally-free. Corollary 2.5. If f : (X, D) → P r,d , L is a log-smooth morphism, from a log-smooth pair (X, D) onto the canonical Pr -bundle compactification of a quasi-abelian variety T r,d , of dimension m = r + d, then m ≤ dim X − κ (X, D). Proof. It follows immediately by Theorem 1.4. See also Remark 2.1. We apply Theorem 1.5 to give another proof of the following. Corollary 2.6 (=Corollary 1.6). Let (X, D) be a projective log-smooth pair of log general type. Assume that we have a surjective projective morphism f : X \ D → T r,d , where T r,d is a quasi-abelian variety. Then, f is not smooth. Proof. We can find a log-resolution X̂, D̂ → (X, D) such that it induces a morphism of log-pairs 12 fˆ : X̂, D̂ → P r,d , L , −1 ∗ ˆ ˆ with f L = D̂. Now by Theorem 2.1, we have Z f ω 6= ∅, for any ω ∈ H 0 Ω1P r,d (log L) . However, by Lemma 2.7, (see also Step 0 of the proof of Theorem 1.5 in Chapter 4,) for general ω ∈ H 0 Ω1P r,d (log L) , Z fˆ∗ ω ∩ (X̂ \ D̂) 6= ∅. We use the following lemma in the proof above. Lemma 2.7. Fix a morphism f : X → P r,d , where X is smooth and quasi-projective and P r,d is the Pr -bundle compactification of a quasi-abelian variety T r,d , with L being the boundary divisor on P r,d . Let D = (f ∗ L)red and assume that (X, D) is log-smooth. Then for general θ ∈ H 0 Ω1P r,d (log L) , f ∗ θ ∈ H 0 Ω1X (log D) has a simple log pole on every point of D. In particular, it does not vanish at any point of D. Proof. Since we can cover P r,d by finite trivial Pr -bundles, we only need to prove the lemma over each of them. However, to show the trivial bundle case, it is not hard to see that we only need to show the case that P r,d = Pr , and L = L0 + ... + Lr is r + 1 hyperplanes of general position. After changing coordinates, we can assume that Li is defined by xi = 0 and Ui := Pr \ Li for i = 0, 1, ..., r. After giving a finite affine covering {Xi,j } of Xi = f −1 (Ui ), we only need to show the claim is true for each f |Xi,j : Xi,j → Ui . Without loss of generality, we pick U0 which is defined by x0 6= 0, hence H 0 (Pr , Ω1Pr (log L))|U0 is a C-vector space with a base {dxi /xi }, for 1 ≤ i ≤ r. Pick one of those X0,j , denoting it by V , and denote f |X0,j by g : V → U0 . Abusing the notation a little bit, we let Li be the P hyperplane defined by xi = 0 on U0 and denote g ∗ (Li ) = j mi,j Di,j with mi,j > 0. Set xi,j as the function on V that defines Di,j . Then, by a local computation, we have g ∗ xi = ui Y m xi,ji,j , (i,j) where ui is a nowhere vanishing function on V . Hence, g ∗ (dxi /xi ) over any x ∈ g −1 (Li ) is of the form 13 X mi,j (i,j)|x∈Di,j dxi,j + holomorphic part. xi,j Hence, it is clear that, for a general linear combination of dxi /xi , its pullback will have a pole on every point of g −1 (L1 + ... + Lr ) and the coefficients of the poles do not have zero locus on that divisor. Another application of Theorem 1.4 is that we can give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with general fibers being of log-general type, which is the special case of the following theorem by taking T r,d = G1m or Ad . We refer to [8, §16] for details of this topic. Note that a projective morphism of log-smooth pairs f : (U, DU ) → G1m being log-smooth is equivalent to the existence of (X, DX ), a projective log-smooth compactification of (U, DU ) such that g : (X, DX ) → (P1 , L), the induced morphism of log-smooth pairs by f , is log-smooth. It can be very complicated in the higher dimensional cases. A similar question has been asked and we refer to [2] for the details on this topic. Definition 2.2. Given a flat dominant projective morphism of log-smooth pairs f : (X, DX ) → (Y, DY ), with connected fibers and the generic fiber (Xη , DηX ) being Kawamata-logterminal (klt) and of log-general type, we say that it is birationally isotrivial if the two log-fibers (Xa , DaX ) and (Xb , DbX ) have the same log-canonical model ([3]), for any two general closed points a and b on Y . Corollary 2.8 (=Corollary 1.7). Given a log-smooth surjective morphism of log-pairs f : (X, D) → P r,d , L , with (X, Dred ) being log-smooth and the generic fiber (Xη , Dη ) being klt and of log-general type, then f is birationally isotrivial. Proof. We only need to show Var(f ) = 0, in the sense of [13, Definition 9.3]. By Corollary 2.5, we have κ (X, D) ≤ dim Xη . By Theorem 2.9, we have κ (X, D) ≥ κ (Xη , Dη ) + Var(f ). Since dim Xη = κ (Xη , Dη ) by the assumption of the theorem, we get that Var(f ) = 0. 14 Remark 2.2. If we only assume that the generic fiber (Xη , Dη ) is log-canonical and of loggeneral type, since (Xη , Dη ) being of log-general type is an open condition on the coefficients of the boundary divisor, we can consider f 0 : (X, (1 − ) D) → P r,d , L , for 0 < 1. Then the generic fiber of f 0 is klt and of log-general type. Following the same arguement above, we can show that f 0 is birationally isotrivial. In the proof of the previous theorem, we used the following theorem, which can be easily deduced from [13, Theorem 9.5, Theorem 9.6]. Theorem 2.9. Let f : (X, D) → (Y, E) be a surjective morphism of projective log canonical pairs with both (X, Dred ) and (Y, E) are log-smooth. Assume that bDc contains f −1 Ered and the generic fiber (Xη , Dη ) is of log-general type. Then κ (X, D) ≥ κ (Xη , Dη ) + κ (Y, E) . Further, if (Xη , Dη ) is klt, then κ (X, D) ≥ κ (Xη , Dη ) + max{κ (Y, E) , Var(f )}. CHAPTER 3 TWO SPECIAL CASES To give readers some intuition about the idea behind the proof of Theorem 1.5, we show the proof of two special cases in this chapter. 3.1 The log-canonically-polarized case In this section, we give a short proof of the following theorem, which is a special case of Theorem 1.3. The proof is based on the argument that appears in [7]. Theorem 3.1. Let (X, D) be a log smooth pair with log canonical bundle ωX (D), which contains an ample line bundle. Then, for any global log-one-form θ ∈ H 0 (Ω1X (log D)), the zero locus of θ must be non-empty. Proof. Assume θ ∈ H 0 (Ω1X (log D)) is a global log-one-form that vanishes nowhere. We have the following exact Koszul complex defined by θ: ∧θ ∧θ ∧θ 0 → OX −→ Ω1X (log D) −→ ... −→ ΩnX (log D) → 0. Let E 0 = 0, E n = ΩnX (log D), and E i = ker(∧θ) : ΩiX (log D) → Ωi+1 X (log D), for i = 1, ..., n − 1. Since ωX (D) contains an ample line bundle, we can write ωX (D) = OX (L + E), for some ample divisor L and some effective (possibly trivial) divisor E on X. Twisting the Koszul complex above by OX (L−D), we have the following exact sequence: ∧θ ∧θ ∧θ 0 → OX (L − D) −→ Ω1X (log D)(L − D) −→ ... −→ ΩnX (log D)(L − D) → 0. 16 This exact sequence can be decomposed into the following short exact sequences: 0 → E i (L − D) → ΩiX (log D)(L − D) → E i+1 (L − D) → 0. By Akizuki-Kodaira-Nakano vanishing [5, 6.4 Corollary], we have H n−i+1 (ΩiX (log D)(L − D)) = 0. Hence, the natural differential map associated to the short exact sequences H n−i (E i+1 (L − D)) → H n−i+1 (E i (L − D)) is surjective. Note that H 1 (E n (L − D)) = H 1 (ΩnX (log D)(L − D)) = 0. Since H 1 (E n (L − D)) → H n (E 1 (L − D)) is surjective, we also have that H n (E 1 (L − D)) = 0. However, since E 1 = OX , and OX (L + E) = ωX (D), we have E 1 (L − D) = ωX (−E). Hence, by Serre duality, we have H 0 (OX (E)) ' H n (ωX (−E))∨ = 0, which contradicts the fact that E is effective. 3.2 Fibration over quasi-abelian variety In this section, we apply the Higgs bundle construction from [18], and use the first reduction step in [17], which also appeared in [22], to prove Corollary 1.6 Proof of Corollary 1.6. Assume that the morphism f : X \ D → T r,d is smooth. After blowing up X along some locus contained in D and taking a log-resolution if necessary, we can extend f to X → P r,d ([9, II, Example 7.17.3]), which we also denote by f : X → P r,d . Since the logarithmic Kodaira dimension does not depend on the smooth compactification, we still have that (X, D) is of log-general type. Denote the boundary divisor on P r,d by L, which has r + 1 irreducible T r,d -invariant components and consists of r + 1 general hyperplanes when it is restricted to each Pr fiber. 17 Denote p : P r,d → A the natural projection. Pick any ample effective divisor E on A and let G = OP r,d (L) ⊗ p∗ OA (E) be a line bundle on P r,d . Since ωX (D) is big, there exists a positive integer k such that H 0 (X, (ωX (D))k ⊗ f ∗ G −1 ) 6= 0. Let [2k] : P r,d → P r,d be the finite morphism, which is defined in Appendix A. Let φ : X 0 → X be the corresponding base change followed by normalization and a log-resolution, and let (X 0 , D0 ) be the induced log-smooth pair. We may assume that X 0 \ D0 = X \ D ×T r,d T r,d . Let f 0 : X 0 → P r,d be the induced morphism. X0 φ f0 P r,d X f [2k] P r,d Since [2k] is étale over T r,d , we have that f 0 is smooth over T r,d . Since D0 ⊂ f 0∗ p∗ L, ωP r,d ' OP r,d (−L) and consider [12, II, Proposition 5.20], then φ∗ ωX (D) ⊂ ωX 0 (D0 ) ⊂ ωX 0 /P r,d . Further, [2k]∗ L = 2kL and [2k]∗ E − 2kE is linear equivalent to an effective divisor, [14, II, 6]. Hence, k 0∗ −2k φ∗ (ωX (D)k ⊗ f ∗ G −1 ) ⊂ ωX ). 0 /P r,d ⊗ f (G Let B = ωX 0 /P r,d ⊗ f 0∗ (G −2 ). Then, by the above inclusion, we see that H 0 (X 0 , B k ) 6= 0. Now, let's apply [18, Theorem 9.4 (a)] to our situation. Note that even through f 0 does not have connected fibers, [18, Theorem 9.4] still applies with minor changes. (See the remark below.) In our case, we can let Df 0 = L, which, by assumption, contains the image of all singular fibers. Hence, F0 ⊃ G 2 (−Df 0 ) ⊃ G Since G is big, by [18, Theorem 9.4 (c) and Theorem 19.1], we conclude that ωP r,d (Df 0 ) ' OP r,d is big, which is absurd. Remark 3.1. In [18, Theorem 9.4], if we drop the assumption that f has connected fibers, we only lose the property that f∗ OY = OX . The proof still works except that we must replace the result (a) by: One has L(−Df ) ⊗ f∗ OY ⊂ F0 ⊂ L ⊗ f∗ OY . As long as f is 18 surjective, we still have f∗ OY ⊃ OX . Hence, L(−Df ) ⊂ F0 , which is what is needed in the proof. CHAPTER 4 PROOF OF THE MAIN THEOREM In this chapter, we first give a simplified proof of the main theorem in [17]. Then we give a proof of the Main Theorem (Theorem 1.5) assuming a claim. 4.1 A simplified proof of the non-log version In this section, we show a simplified proof of the the following theorem, which is the main result of [17], without using the generic vanishing of mixed Hodge modules on abelian varieties, which they introduced in [16]. The method is still mainly based on [16]. Theorem 4.1. Given a morphism f : X → A, if there is a positive integer k and an ample line bundle A on A, such that ⊗k ⊗ f ∗ A−1 6= 0, H 0 X, ωX then Z (f ∗ ω) 6= ∅, for any ω ∈ H 0 A, Ω1A . Proof. Step 1. We have an étale morphism [k] : A → A, which is defined by multiplying by k. Apply the finite base change and let f 0 : X 0 → A be the fiber product. Since [k] is étale, we have that X 0 is smooth and it suffices to prove the theorem for f 0 : X 0 → A. X0 φ f0 A X f [k] A We have an inclusion A⊗k → [k]∗ A [14, II, 6]. Hence, ⊗k φ ∗ ωX ⊗ f ∗ A−1 ⊗k =φ∗ ωX ⊗ f 0∗ [k]∗ A−1 ⊗k ⊂ ωX 0 ⊗ f 0∗ A−1 which, by assumption, has a global section. 20 Hence, we reduce to the case that there exists a positive integer k such that ⊗k H 0 X, ωX ⊗ f ∗ A−1 6= 0. Step 2. Let B = ωX ⊗ f ∗ A−1 , a line bundle over X. Do the cyclic cover induced by a section s of B ⊗k and resolve it. We get ψ : Y → X. Hence, we have the following commutative diagram Y ψ X g f A. By the construction, there is a tautological section of ψ ∗ B on Y . Hence, we have a natural injection, ψ ∗ B −1 → OY , which induces the following injection ψ ∗ B −1 ⊗ ΩkX → ΩkY . (4.1) Actually, both of these injections are isomorpisms over the complement of the zero-locus of s. Step 3. Let d = dim A. We have that Ω1A is a trivial d-dimensional vector bundle over A. Denote V = H 0 A, Ω1A , and denote the space of cotangent of A by TA∗ . Hence, we have TA∗ = A × V. Consider the following commutative diagram, which contains all the morphisms we will need in the proof: pY Y ψ g X ψ̂ pX X ×V ĝ fˆ f A Y ×V pA A×V pV V, where all the morphisms are natural projections or naturally induced by f and ψ. 21 Let n = dim X = dim Y . Denote CY = [p∗Y OY → p∗Y Ω1Y → ... → p∗Y ΩnY ], placed in cohomological degrees −n, −n + 1, ..., 0, which is the Koszul complex given by the pullback of the tautological section of p∗A Ω1A . Note that pY is an affine morphism, and pY ∗ (CY ) is the following graded complex CY,• = [OY ⊗ S•−d → Ω1Y ⊗ S•−d+1 → ... → ΩnY ⊗ S•−d+n ], where S• := SymV ∗ . The differential in the complex is induced by the evaluation morphism V ⊗ OY → Ω1Y . We define CX , CX,• in a similar way. Denote L = g ∗ A and L̂ = p∗Y (L) . Claim 1. pV ∗ R0 ĝ∗ L̂−1 ⊗ CY is torsion free. Step 4. We continue the proof assuming the claim. We set F as the image of the map R0 ĝ∗ L̂−1 ⊗ ψ̂ ∗ p∗X B −1 ⊗ CX → R0 ĝ∗ L̂−1 ⊗ CY induced by the injection (4.1). Pick any global one-form θ ∈ H 0 A, Ω1A = V. The restriction of p∗X B −1 ⊗ CX on the fiber of pV ◦ fˆ : X × V → V over θ is just B −1 → B −1 ⊗ Ω1X → ... → B −1 ⊗ ΩnX , the Koszul complex defined by f ∗ (θ) and twisted by B −1 . Since the zero-locus of f ∗ (θ) is empty if and only if the above complex is exact, to prove the theorem, it suffices to prove pV (suppF) = V . 22 Note that pA and pY are affine, so we have that pA∗ and pY ∗ are exact, and pA∗ ◦ R0 ĝ∗ = R0 g∗ ◦ pY ∗ . Hence, pA∗ F is a graded pA∗ OA×V -module F• given by the image of pA∗ R0 ĝ∗ L̂−1 ⊗ ψ̂ ∗ p∗X B −1 ⊗ CX → pA∗ R0 ĝ∗ L̂−1 ⊗ CY , which is the same as the image of R0 g∗ L−1 ⊗ ψ ∗ B −1 ⊗ CX,• → R0 g∗ L−1 ⊗ CY,• . If pV (SuppF) 6= V , the subsheaf pV ∗ F ⊂ pV ∗ R0 ĝ∗ L̂−1 ⊗ CY would then be torsion and hence zero. Therefore, since V is a vector space, H 0 (A, pA∗ F) = H 0 (A, F• ) = 0. Recall that B = ωX ⊗ f ∗ A−1 and L = g ∗ A. We have F−n+r = g∗ L−1 ⊗ ψ ∗ B −1 ⊗ ωX = g ∗ OY This forces H 0 (g∗ OY ) = 0, which is absurd. Step 5. Finally, we show the proof of Claim 1. Denote E = RpV ∗ R0 ĝ∗ L̂ ⊗ CY . We first show that for l > 0, Hl E = 0. (4.2) In particular, E is a sheaf. Since both pA and pY are affine, we have H l A × V, R0 ĝ∗ L̂ ⊗ CY l 0 'H A, pA∗ R ĝ∗ L̂ ⊗ CY 'H l A, R0 g∗ pY ∗ L̂ ⊗ CY =H l A, R0 g∗ CY,• ⊗ A =0 The last vanishing is due to Laumon's formula ([17, Lemma 15.1], or Proposition 6.1) and [17, Lemma 2.5] or Proposition 6.2. More precisely, we have eY , R0 g∗ CY,• ' GrF• R0 g+ ω 23 and H l GrF• R0 g+ ω eY ⊗ A = 0, for l > 0. Hence, due to the degeneration of the Leray spectral sequence induced by pV ∗ , we have H 0 V, Hl E =H 0 V, Rl pV ∗ R0 ĝ∗ L̂ ⊗ CY 'H l A × V, R0 ĝ∗ L̂ ⊗ CY =0, and so that (4.2) follows. pV ∗ R0 ĝ∗ L̂−1 ⊗ CY being torsion free is due to the following relation: pV ∗ R0 ĝ∗ L̂−1 ⊗ CY = (−1V )∗ R0 Hom (E, OV ) , (4.3) where (−1V ) is the involution on V by multiplying −1. To show (4.3), by Grothendieck duality, we have that RHom (E, OV ) =DV RpV ∗ R0 ĝ∗ L̂ ⊗ CY [−d] 'RpV ∗ DA×V R0 ĝ∗ L̂ ⊗ CY [−d]. =RpV ∗ RHom R0 ĝ∗ L̂ ⊗ CY , OA×V [d] . By [16, Proposition 2.11] (or Proposition 5.11 by taking DY = 0 and r = 0), we have R0 ĝ∗ (CY ) ' G H0 g+ ω eX , the corresponding coherent sheaf on TA∗ = A×V of the mixed Hodge module H0 g+ ω eX on A. By the formula of taking duality in mixed Hodge modules [16, Theorem 2.3] (or Corollary 5.15 by taking H = 0), and considering that the dual Hodge module of R0 g+ ω eX is itself up to a Tate twist, we obtain RHom (E, OV ) = (−1V )∗ RpV ∗ R0 ĝ∗ L̂−1 ⊗ CY , which implies (4.3). 24 4.2 Proof of the Main Theorem (Theorem 1.5) The proof is inspired by M. Popa and C. Schnell's work [17]. The idea of the proof is similar to the proof of Theorem 4.1. Proof. Step 0. Let's consider the set S = {ω ∈ H 0 Ω1P r,d (log L) | Z (f ∗ ω) 6= ∅}. It is closed as a subset of H 0 Ω1P r,d (log L) . Considering Lemma 2.7, it is easy to see that the two statements in the theorem are equivalent to each other. Hence, we can focus on the part over T r,d . In other words, we only need to prove the theorem for any log-smooth model (X 0 , D0 ) over (X, D), that is étale over T r,d . In the rest of the proof, we use P and T to replace P r,d and T r,d , respectively, to simplify the notations. Actually, we can also restrict ourselves to the case that r = 0 or 1, which suffices to show the statement in general. See the second half of the proof of Theorem 2.1 in Chapter 2 for details. Step 1. We have a finite morphism [n] : P → P , which is defined in the following way. In the interior part T , it is defined by multiplying by n (if we use addition for the group structure on T ). It can be canonically extended onto the boundary L. Note that it is only ramified over the boundary L of degree n. Take n = k, and apply the finite base change, and let (Xn , Dn ) be the nomalization of the fiber product. fn : Xn → P is the induced morphism. Let (X 0 , D0 ) be a log-resolution of (Xn , Dn ), which can be achieved by only blowing up loci contained in fn−1 (L). Since the induced morphism φ : X 0 → X is étale over T , according to Step 0, it suffices to prove the theorem for f 0 : X 0 → P . φ X0 π Xn X ×P P X fn f f0 P [k] P Note that we have φ∗ ωX (D) ⊂ ωX 0 (D0 ), and we have an inclusion p∗ A⊗k → [k]∗ p∗ A [14, II, 6]. Hence, 25 φ∗ (ωX (D))⊗k ⊗ f ∗ p∗ A−1 ⊗ OP r,d (−L) = (φ∗ ωX (D))⊗k ⊗ f 0∗ [k]∗ OP (−L) ⊗ f 0∗ [k]∗ p∗ A−1 ⊗k ⊂ ωX 0 D0 ⊗ f 0∗ OP (−kL) ⊗ f 0∗ p∗ A−k ⊗k = ωX 0 D0 ⊗ f 0∗ OP (−L) ⊗ f 0∗ p∗ A−1 , which, by assumption, has a global section. Denote E 0 = D0 −f 0−1 L. Note that OX 0 (D0 − E 0 ) ⊂ f 0∗ OP (L), we have that ⊗k 6= 0. H 0 X 0 , ωX 0 E 0 ⊗ f 0∗ p∗ A−1 Hence, we reduce to the case that there exists a positive integer k such that ⊗k 6= 0, H 0 X, ωX (E) ⊗ f ∗ p∗ A−1 where E = D − f −1 L. Step 2. Let B = ωX (E) ⊗ f ∗ p∗ A−1 , a line bundle over X. Do the cyclic cover induced by a section s of B ⊗k and resolve it. We get ψ : Y → X. Denote DY = ψ −1 D, which can also be assumed to have normal crossings after further blowing up if necessary. We denote by g : Y, DY → (P, L) the induced morphism. Hence, we have the following commutative diagram Y, DY ψ (X, D) g f (P, L) . By the construction, there is a tautological section of ψ ∗ B on Y . Hence, we have a natural injection, ψ ∗ B −1 → OY , which induces the following injection ψ ∗ B −1 ⊗ ΩkX (log D) → ΩkY log DY . Actually, both of them are isomorpisms over the complement of the zero-locus of s. (4.4) 26 Step 3. Let m = r + d = dim P. We have that Ω1P (log L) is a trivial m-dimensional vector bundle over P . Denote V = H 0 P, Ω1P (log L) , and denote the log-cotangent space ∗ of (P, L) by T(P,L) . Hence, we have ∗ = P × V. T(P,L) Consider the following commutative diagram, which contains all the morphisms we will need in the proof: pY Y ψ g X ψ̂ pX pP p Ad X ×V ĝ fˆ f P Y ×V P ×V pV V p̂ pA Ad × V where all the morphisms are natural projections or naturally induced by f and ψ. Let n = dim X = dim Y . Denote CY,DY = [p∗Y OY → p∗Y Ω1Y log DY → ... → p∗Y ΩnY log DY ], placed in cohomological degrees −n, −n + 1, ..., 0, which is the Koszul complex given by the tautological section of p∗P Ω1P (log L). Note that pY is an affine morphism, and pY ∗ CY,DY is the following graded complex CY,DY ,• = [OY ⊗ S•−m → Ω1Y log DY ⊗ S•−m+1 → ... → ΩnY log DY ⊗ S•−m+n ], where S• := SymV ∗ . The differential in the complex is induced by the evaluation morphism V ⊗ OY → Ω1Y log DY . We define CX,D , CX,D,• in a similar way. Denote L = g ∗ p∗ A ⊗ OY g −1 L , and L̂ = p∗Y (L) . Claim 2 (Main Claim). If r = 0 or 1, pV ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY is torsion free. 27 Step 4. We continue the proof assuming the claim. We set F as the image of the map −1 −1 ∗ ∗ −1 0 0 R ĝ∗ L̂ ⊗ ψ̂ pX B ⊗ CX,D → R ĝ∗ L̂ ⊗ CY,DY induced by the injection (4.4). Pick any global log-one-form θ ∈ H 0 P, Ω1P (log L) = V. The restriction of p∗X B −1 ⊗ CX,D on the fiber of pV ◦ fˆ : X × V → V over θ is just B −1 → B −1 ⊗ Ω1X (log D) → ... → B −1 ⊗ ΩnX (log D) , the Koszul complex defined by f ∗ (θ) and twisted by B −1 . Since the zero-locus of f ∗ (θ) is empty if and only if the above complex is exact, to prove the theorem, it suffices to prove pV (SuppF) = V . Note that pP and pY are affine, so we have that pP ∗ and pY ∗ are exact, and pP ∗ ◦ R0 ĝ∗ = R0 g∗ ◦ pY ∗ . Hence, pP ∗ F is a graded pP ∗ OP ×V -module F• given by the image of pP ∗ R0 ĝ∗ L̂−1 ⊗ ψ̂ ∗ p∗X B −1 ⊗ CX,D → pP ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY , which is the same as the image of R0 g∗ L−1 ⊗ ψ ∗ B −1 ⊗ CX,D,• → R0 g∗ L−1 ⊗ CY,DY ,• . If pV (SuppF) 6= V , the subsheaf pV ∗ F ⊂ pV ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY would then be torsion and hence zero. Therefore, since V is a vector space, H 0 (P, pP ∗ F) = H 0 (P, F• ) = 0. Recall that B = ωX (E) ⊗ f ∗ p∗ A−1 and L = g ∗ p∗ A ⊗ OY g −1 L . We have F−n+r =g∗ L−1 ⊗ ψ ∗ B −1 ⊗ ωX (D) =g∗ OY ψ ∗ (D − E) − g −1 L =g∗ OY ψ ∗ f −1 L − g −1 L This forces H 0 P, g∗ OY ψ ∗ f −1 L − g −1 L = 0, which is absurd, since ψ ∗ f −1 L − g −1 L is effective over Y . CHAPTER 5 LOGARITHMIC COMPARISONS The goal of this chapter is to show some logarithmic comparison theorems for mixed Hodge modules. We follow the notations that are introduced in Appendix B. 5.1 Kashiwara-Malgrange filtration In this section, we recall some basic properties of R-indexed Kashiwara-Malgrange e filtration on a coherent strict D-module with respect to a smooth divisor from [20, 3.1]. Then we define multi-indexed rational Kashiwara-Malgrange filtration with respect to a e simply normal crossing divisor when the D-module is of normal crossing type, and recall some properties that we will need in later sections. We say that an R-indexed increasing filtration V is indexed by A + Z, where A is a finite subset of [−1, 0), if graV := Va /Va<0 = 0 if and only if a ∈ / A + Z. All R-indexed filtrations in this paper are indexed by A + Z for some finite set A ⊂ [−1, 0). Fix a smooth variety X and a smooth divisor H on X. Let t be a local function that defines H, and ∂t be a local vector field satisfying [∂t , t] = 1. eX -module. We say that a rationally indexed Definition 5.1. Let M be a coherent strict D increasing filtration V•H on M is a Kashiwara-Malgrange filtration with respect to H, if e(X,H) -module. (1)∪a∈R VaH M = M, and each filtered piece Va M is a coherent D eX ⊂ V H M for all a ∈ R, i ∈ Z, and V H M t = V H M if a < 0. (2) VaH M ViH D a a−1 a+i H (3)The action t∂et − a over grVa M is nilpotent for any a ∈ R. HD eX , ∂et ∈ V H D eX , hence condition (3) implies Note that t ∈ V−1 1 H H H H (4)t : grVa M → grVa−1 M and ∂et : grVa−1 M → grVa Mz are bijective for a 6= 0. H M = (V M) V H D eX for a ≥ 0, i ≥ 0. (5)Va+i a i Further, all the conditions above are independent from the choice of t and ∂et . We know that given a Mixed Hodge Modue M, in particular being strictly R-specializable 29 along H, there exists a rational Kashiwara-Malgrange filtration with respct to H and it is H unique. ([20, 3.1.2. Lemme], [19, 7.3.c]). Further, we know that every graV M is a strict H eX -module. [19, Proposition 7.3.26] gr0V D eX -module M equipped with a Kashiwara-Malgrange Recall that given a coherent D filtration V•H , we have a free resolution: eX -module that is equipped with the Kashiwara-Malgrange Proposition 5.1. Fix M a strict D filtration V•H with respct to a smooth divisor H on X. Then locally over X we can find a resolution: M• , V•H → M, V•H , eX · z p , V H [a] , with −1 ≤ a ≤ where each Mi , V•H is a direct sum of finite copies of D 0, p ∈ Z. Further, the resolution is strict with respct to V•H , which means it is still a resolution after taking any filtered piece of V•H . Further, if we have that the multiplication by t induces an isomorphism t : grV0 M → grV−1 M, in particular, when M = M[∗H], the above resolution can be achieved with −1 < a ≤ 0. If we have that applying ∂ex induces an isomorphism ∂ex : grV−1 M → grV0 M · z, in particular, when M = M[!H], the above resolution can be achieved with −1 ≤ a < 0. Proof. Without the extra assumption, it is just [20, 3.3.9 Lemme], see also the proof of [20, 3.3.17 Proposition]. With the extra assumption, it is evident that we can also achieve the corresponding index interval in [20, 3.3.9 Lemme]. See [19, Proposition 9.3.4 and Proposition 9.4.2], for the corresponding isomorphism in the localization and dual localization case. Remark 5.1. Actually, the existence of the free resolution as above only depends on the strictness assumption (4) in Definition 5.1. See also [20, 3.3.6-3.3.9]. 30 Now we start to consider the case that D = D1 + ... + Dn is a normal crossing divisor on X with irreducible components Di . For any a = [a1 , ..., an ] ∈ Rn , we denote eX = ∩i V Di D eX . VaD D ai eX is just D e(X,D) . For a = 0 := [0, ..., 0], V0D D eX -module that possesses the Kashiwara-Malgrange filtration Given a strict coherent D V•Di with respct to all components Di of D, we define a multi-indexed Kashiwara-Malgrange filtration with respct to D by VaD M = ∩VaDi i M, for any a = [a1 , ..., an ] ∈ Rn . It is not hard to see that V•D M is a multi-indexed graded eX , in the sense that module over V•D D eX ⊂ VD M, VaD M · VbD D a+b for any a, b ∈ Rn . Given a subset S ⊂ {1, ..., n}, we denote DS = P i∈S Di . For any such DS , we denote VaDS M := ∩i∈S VaDi i M. Further, we say that b < a if bi < ai for all i. We denote D V<a M := ∪b<a VbD M. In general, the n filtrations V•Di do not behave well between each other, in the sense that when we look V•D1 as a filtration on V0D2 M, it does not have good strictness conditions as in eX -module of normal crossing type with Definition 5.1. However, if M is a strict coherent D respct to D, then locally we have the following strictness relations ([21, 3.11 Proposition], [19, Lemma 11.2.11]) ∼ D xi : VaD M − → Va−1 i M (ai < 0) Di Di ∼ ∂ei : graVi −1 VaD−Di M − → graVi VaD−Di M · z (ai > 0), where 1i :=[0, ..., 0, 1, 0, ..., 0], with the 1 at the i-th position. (5.1) (5.2) 31 Further, if we have the isomorphisms ∼ Di xi : V0Di M − → V−1 M, for every i, (5.3) in particular when M = M[∗D], the identity (5.1) still holds when ai = 0 If we have Di Di V−1 M∂ei + V<0 M · z = V0Di M · z, (5.4) in particular when M = M[!D], the identity (5.2) still holds when ai = 0. Note that, to make the multi-indexed Kashiwara-Malgrange filtration satisfy (5.1) and (5.2), instead of requiring that M is of normal crossing type with respct to D, we only need to assume that M is of normal crossing type with respct to D0 , where D0 is a normal crossing divisor that contains D. One advantage of such an assumption is that it behaves well when we do induction on the number of components on the boundaries. eX · z p , VD [a] , to denote DX equipped with shifted filtraWe introduce the notation D • tions: eX · z p , VD [a] := VD D eX · z p . VbD D • b−a As Proposition 5.1, we have a similar free resolution: eX -module of normal crossing type with respct Proposition 5.2. Fix M, a strict coherent D to D0 , a normal crossing divisor that contains D as above. Then locally over X we can find a resolution: M• , V•D → M, V•D , eX · z p , VD [a] , with −1 ≤ a ≤ where each Mi , V•D is a direct sum of finite copies of D 0, p ∈ Z. Further, the resolution is strict with respct to V•D , which means it is still a resolution after taking any filtered piece of V•D . Further, if we have the isomorphisms (5.3), the above resolution can be achieved with −1 < a ≤ 0. If we have the isomorphisms (5.4), the above resolution can be achieved with −1 ≤ a < 0. D Proof. Let Λ = {(λ1 , ..., λn ) ∈ Rn | −1 ≤ λi ≤ 0, grλVi i M = 6 0, for all i = 1, ..., n}, which is a finite set. 32 As in the proof of [20, 3.3.9 Lemme], we only need to show that locally, the natural morphism: eX · z p , VD [a] → M u : ⊕p∈Z,a∈Λ Fp Va M ⊗ D • is strictly surjective, i.e., it is still surjective after taking the Va filtered piece, for any a ∈ Rn . It is obvious for −1 ≤ a ≤ 0. Due to the strictness condition (5.1) and (5.2), it is also true for general a. 5.2 Comparison theorem Fix a smooth variety X and a normal crossing divisor D = D1 + ... + Dn on X as in the previous sections. We start with the following vanishing. e(X,D) -module M. Let DS = D −D1 and x1 be a local function Lemma 5.3. Given a strict D that defines D1 . Assume that GrF M is torsion free with respct to x1 , then we have i L e H M ⊗De D(X,DS ) = 0, (X,D) for i 6= 0. Proof. Take the canonical left resolution of M as in (B.1): e(X,D) M 'Sp(X,D) M ⊗ D tens e ={0 → M ⊗ D(X,D) ⊗ ∧n Te(X,D) → ... tens e(X,D) e(X,D) → M⊗D ⊗ Te(X,D) → M ⊗ D tens e(X,D) -module structure on M ⊗ D e(X,D) Recall that the D tens tens → 0}. ⊗ ∧i Te(X,D) is the trivial one. e(X,D) and D e(X,D ) are locally free over O eX , we have Since both D S −• e e e M ⊗ D(X,D) ⊗ ∧ T(X,D) ⊗L e(X,D) D(X,DS ) D tens e(X,D ) ' M⊗D ⊗ ∧−• Te(X,D) S tens e(X,D ) 'Sp(X,D) M ⊗ D . S tens e(X,D ) Now we only need to show that GrF Sp(X,D) M ⊗ D S tens However, it is just K x1 ∂e1 , ..., xn ∂en , ∂en+1 , ..., ∂edX ; N , has no higher cohomology. 33 e(X,D ) the Koszul complex of N := GrF M ⊗ D S tens , with x1 ∂e1 , ..., xn ∂en , ∂en+1 , ..., ∂edX e(X,D) . More precisely, for any element m ⊗ P ∈ N , we have the as the elements in GrF1 D relations (m ⊗ P ) x1 ∂e1 = mx1 ∂e1 ⊗ P − mx1 ⊗ ∂e1 P, (m ⊗ P ) xi ∂ei = mxi ∂ei ⊗ P − m ⊗ xi ∂ei P, for i = 2, ..., n (m ⊗ P ) ∂ej = m∂ej ⊗ P − m ⊗ ∂ej P, for j = n + 1, ..., dX . Note that these elements act on N homogeneously. We know that e(X,D ) , ∂e1 , x2 ∂e2 , ..., xn ∂en , ∂en+1 , ..., ∂edX ∈ GrF1 D S (5.5) is a regular sequence on N , which is due to e(X,D ) M ' Sp(X,DS ) M ⊗ D S tens , e(X,D ) . Now we need to show that or by an easy argument on the natural grading on GrF D S e(X,D) x1 ∂e1 , ..., xn ∂en , ∂en+1 , ..., ∂edX ∈ GrF1 D is also a regular sequence on N . Since we have that e(X,D) , x2 ∂e2 , ..., xn ∂en , ∂en+1 , ..., ∂edX ∈ GrF1 D is a regular sequence on N , it suffices to show that x1 ∂e1 is torsion free on N := N /N x2 ∂e2 , ..., xn ∂en , ∂en+1 , ..., ∂edX . Note that due to the relations we have above, it is not hard to see that N ' GrF M ⊗ OX [∂e1 ]. However, by the assumption that x1 is torsion-free on GrF M, we can conclude the proof. 34 For the rest of this section, we will consider M as a mixed Hodge module on X. Given a divisor D with normal crossing support on X, we denote M[∗D] = i∗ i−1 M, and M[!D] = i! i−1 M being the localization and dual localization of M on X \ D [21, 2.11 Proposition], [19, Chapter 9]. Note that both M[∗D] and M[!D] are mixed Hodge modules, in particular, eX -modules. strict coherent D Theorem 5.4 (Comparison Theorem with normal crossing boundary). Notations as above. Given two subset S ⊂ I ⊂ {1, ..., n} and a mixed Hodge module M of normal crossing type with respct to D0 , a normal crossing divisor that contains D, then we have the following e(X,D ) : two quasi-isomorphisms in DGcoh D X V0DI (M[∗D]) ⊗L e D e(X,D ) ' VDS (M[∗D]) . D 0 S (5.6) DI V<0 (M[!D]) ⊗L e D e(X,D ) ' VDS (M[!D]) . D <0 S (5.7) (X,DI ) (X,DI ) Proof. We only show (5.6) here, (5.7) follows similarly. Since M[∗D] = M[∗D][∗DI ], we can assume DI = D without loss of generality. Further, by induction, we only need to show the case that DS = D − D1 , V0D (M[∗D]) ⊗De (X,D) e(X,D ) ' VDS (M[∗D]) . D 0 S (5.8) Due to the strictness condition on the multi-indexed Kashiwara-Malgrange filtration (5.1), xi is torsion-free on GrF V0D M[∗D] for any local function xi that defines Di , hence so is any V•D filtered piece of M[∗D]. Now applying the previous lemma, we get V0D (M[∗D]) ⊗L e D (X,D) e(X,D ) ' VD (M[∗D]) ⊗ e D 0 S D (X,D) e(X,D ) . D S Further, by (5.2), we have e(X,D ) ⊂ M[∗D]. V0DS (M[∗D]) ' V0D (M[∗D]) · D S To show (5.8), now we only need to show that V0D M[∗D] ⊗De (X,D) e(X,D ) = VD M[∗D] · D e(X,D ) . D 0 S S (5.9) 35 Note that we have a natural injection V0D (M[∗D]) → V0D (M[∗D]) ⊗De (X,D) e(X,D ) . D S Similarly to the proof of [19, 9.3.4 (7)], we give an integer-indexed increasing filtration on N := V0D (M[∗D]) ⊗De (X,D) e(X,D ) by: D S ( V D1 M[∗D] ∩ V0D (M[∗D]) , for k ≤ 0, UkD1 N = Pk k D ej j=0 V0 (M[∗D]) ⊗ ∂1 , for k ≥ 1. (5.10) where k ∈ Z and ∂1 is a differential operator satisfying [x1 , ∂1 ] = 1, [xi , ∂1 ] = 0, for i 6= 1, where x1 is a local holomorphic function that defines D1 . It is obvious that ∪k UkD1 (N ) = N . We give a similar filtration on V0DS M[∗D] by: UkD1 ( V D1 M[∗D] ∩ V0D (M[∗D]) , for k ≤ 0, V0DS M[∗D] = Pk k D ej j=0 V0 (M[∗D]) ∂1 , for k ≥ 1. We have that ∪k UkD1 V0DS M[∗D] = V0DS M[∗D] due to (5.2). Consider the following commutative diagram, GrU 0 D1 ∂e1k (N ) GrU k D1 (N ) ∼ GrU 0 D1 where GrU k := Uk /Uk−1 . V0DS M[∗D] ∂e1k GrkU D1 V0DS M[∗D] , Since the upper horizontal map is surjective and the lower horizontal map is isomorphic by the construction of the filtrations, we get that all the arrows appearing in the commutative diagram are isomorphisms, which concludes (5.9), hence (5.8). Applying the Spencer functor on both sides, we can recover the classical logarithmic comparison theorem: eX -module that represents a mixed Hodge module on Proposition 5.5. Let M be a right D a smooth variety X. Let D be a normal crossing divisor on X. Assuming that M is of 36 normal crossing type with respct to a normal crossing divisor D0 , which contains D, then we have Sp(X,D) V0D (M[∗D]) ' SpX (M[∗D]) D Sp(X,D) V<0 (M[!D]) ' SpX (M[!D]) e X , the derived category of graded C e X -modules. in DG C Proof. By the definition of the Spencer functor, we have Sp(X,D) V0D (M[∗D]) 'V0D (M[∗D]) ⊗L e D OeX 'V0D (M[∗D]) ⊗L e D eX ⊗L OeX D e D (X,D) (X,D) 'M[∗D] ⊗L eX D X OeX 'SpX (M[∗D]) . The third identity is due to the comparison theorem, by taking DI = D, and DS = 0. The dual localization case follows similarly. When the boundary divisor D := H is smooth, the strictness conditions (5.1) and (5.2) are satisfied by Definition 5.1. Hence, we have Theorem 5.6 (Comparison Theorem with smooth boundary). Given a mixed Hodge module eX : M, we have the following two quasi-isomorphisms in DG D V0H (M[∗H]) ⊗L e D eX ' M[∗H], D H V<0 (M[!H]) ⊗L e D eX ' M[!H]. D (X,H) (X,H) Proof. It follows by a similar argument as we show (5.8) in the Comparison Theorem with normal crossing boundary. Remark 5.2. Actually, both Theorem 5.4 and Theorem 5.6 can also be proved by using the free resolution in Proposition 5.1 and Proposition 5.2, respectively. More precisely, for the 37 M[∗H] case, we can get a resolution with each term being a direct sum of finite copies of eX · z p , V H [a] , with −1 < a ≤ 0, p ∈ Z. Note that in this case, we have D eX · z p , V H [a] = D e(X,D) . V0H D e(X,D) -module resolution. A similar Hence, by taking the V0H piece, we also get a free D argument will be used in proving Proposition 5.14. 5.3 Pushforward functor eX -module of normal crossing type on a log-smooth pair Fix M, a strict coherent D (X, D0 ). Let D be a divisor with its support contained in D0 , and D = r · Di := r1 D1 + ... + rn Dn be the decomposition of reduced and irreducible components. Let i : (X, D) → (Y, H) be the graph embedding given by the local function y = xr11 ...xrnn , where each xi is a local function that defines the divisor Di on X. We first build a relation between the Kashiwara-Malgrange filtration V•D on M and V•H on i+ M. eY that are mutually commutative and satisfying Let ∂e1 , ..., ∂en , ∂ey ∈ D [∂i , xi ] =[∂y , y] = 1, [∂i , xj ] =[∂i , y] = [∂y , xi ] = 0.for i 6= j Let u = xr11 ...xrnn − y, and consider x1 , ..., xn , u being a local coordinate system on Y . We eY that satisfies can get ∂ex1 , ..., ∂exn , ∂eu ∈ D (m ⊗ δ) ∂xi =m∂xi ⊗ δ, (m ⊗ δ) ∂u =m ⊗ δ∂u , and we can further require that [∂xi , xi ] =[∂u , u] = 1, [∂xi , xj ] =[∂xi , u] = [∂u , xi ] = 0, for i 6= j. 38 eY = D eX [u]h∂eu i-module. By changing of We can write i# M = ⊕k∈N M ⊗ δ ∂euk as a D coordinates, we have the following relations: (m ⊗ δ) ∂eyk = m ⊗ δ ∂euk , (m ⊗ δ) ∂ei = m∂exi ⊗ δ − (mri xr11 ...xrnn /xi ) ⊗ δ ∂eu , (m ⊗ δ) y = (mxr11 ...xrnn ) ⊗ δ, (m ⊗ δ) OeX = mOeX ⊗ δ, and the usual commutation rules. Lemma 5.7. Notations as above, we can get the Kashiwara-Malgrange filtration V•H on i+ M from V•D on M, by D e(Y,H) , for a < 0. VaH i+ M = Va·r M⊗δ ·D (5.11) For a ≥ 0, we can get it inductively, by H H VaH i+ M · z = V<a (i+ M) · z + Va−1 (i+ M) ∂ey . If we further have that Di xi : V0Di M → V−1 M is an isomorphism, for every component Di , in particular, when M = M[∗D], we have that (5.11) still holds, when a = 0. Proof. The proof is similar to the proof in [19, Theorem 11.3.1]. However, since our settings are a little bit more general, we spell out the details here. We need to check that the filtration we defined above satisfies the Condition (1), (2), (3) in Definition 5.1. We only check the case with the extra condition here. The other case follows similarly. D M over D e For Condition (1), if a ≤ 0 and if m1 , ..., ml generate Va·r (X,DX ) , then m1 ⊗ e δ, ..., ml ⊗ δ generate VaH i+ M over D (Y,DY ) , due to the fact that we have the relation mxi ∂exi ⊗ δ = (m ⊗ δ) xi ∂ei + ri y ∂ey . e Hence, we can conclude that, for every a, VaH i+ M is coherent over D (Y,DY ) . To show that ∪a VaH i+ M = i+ M, 39 we only need to show that M ⊗ δ ⊂ ∪a VaH i+ M. Note that eX = M, V0D M · D which is due to the assumption that M, hence M [∗D] is of normal crossing type, hence the strictness condition (5.2). Then, we can use the relation m∂exi ⊗ δ = (m ⊗ δ) ∂ei + (mri xr11 ...xrnn /xi ) ⊗ δ ∂eu , to get that eY ⊃ M ⊗ δ. ∪a VaH i+ M ⊃ V0D M ⊗ δ · D For Condition (2), we have that, for any a, H VaH i+ M y ⊂ Va−1 i+ M, which can be checked by using the relation (m ⊗ δ) y = (mxr11 ...xrnn ) ⊗ δ. Further, the equality for a < 0 can also be deduced from the corresponding properties on M. We are left to show H Va−1 i+ M ∂ey ⊂ VaH i+ M. When a > 0, it follows by definition. When a ≤ 0, for any m ⊗ δ ∈ VaH i+ M, considering the relation [19, 11.2.10] and our extra assumption (5.12), we obtain D VaD M xi = Va−1 i M, H i M, we can write for a ≤ 0. Hence, for any m0 ⊗ δ ∈ Va−1 + m0 ⊗ δ = (mri xr11 ...xrnn /xi ) ⊗ δ, D for some m ∈ Va·r−1 i M. Now using the relation (m ⊗ δ) ∂ei = m∂ex ⊗ δ − (mri xr1 ...xrn /xi ) ⊗ δ ∂eu , i 1 n we obtain m0 ⊗ δ ∂y = m∂exi ⊗ δ − (m ⊗ δ) ∂ei ∈ VaH i+ M, which concludes the proof of the Condition (2). 40 For Condition (3), it is straightforward to check that we have 1 e 1 e (m ⊗ δ) (y ∂y − az) = m xi ∂xi − ri az ⊗ δ − xi m ⊗ δ ∂ei . ri ri Using this relation and assuming a ≤ 0, we can get that, if νr a Di i ex − ri az i ⊂ V<r M, VrD M x ∂ i i ia ia we have that D Va·r M⊗δ νr a i D e(Y,H) , y ∂ey − az ⊂ Va·r−¯ M ⊗ δ ·D i where ¯i := [0, ..., 0, , 0, ..., 0], where 0 < 1 is located at the i-th position. Denote P µa = i νri a , do the computation above inductively, and we obtain D Va·r M⊗δ µ a D e(Y,H) . y ∂ey − az ⊂ V<a·r M⊗δ ·D H Hence, y ∂ey − az is nilpotent on grVa i+ M, for a ≤ 0. When a > 0, we can get the same conclusion by induction and using the relation ∂ey y ∂ey − az = y ∂ey − (a − 1) z ∂ey . e(X,D) -module N . Assume GrF N is torsionLemma 5.8. Notations as above. Fix a strict D free with respct to every xi , then we have i# N ' H0 i# N . Proof. Let Di0 be the divisors on Y that are defined by the the local functions xi . Denote D0 := D10 + ... + Dn0 . We can decompose i : (X, D) → (Y, H) into j : (X, D) → (Y, D0 + H) and id : (Y, D0 + H) → (Y, H). By (B.2), we have L −1 e e j# N ' j∗ N ⊗De OX ⊗f −1 Oe f D(Y,D0 +H) . (X,D) Y However, it is straightforward to check that the natural map dj : j ∗ ΩY log (D0 + H) → ΩX (log D) is surjective. Hence, its dual ∂j : T(X,D) → j ∗ T(Y,H) 41 has a locally free cokernal and we denote it by N . Then we have e(Y,D0 +H) ' D e(X,D) ⊗ Sym N e ·z . OeX ⊗f −1 Oe f −1 D Y e(Y,D0 +H) is locally free over D e(X,D) , hence we In particular, we have that OeX ⊗f −1 Oe f −1 D Y obtain j# N ' H0 j# N . For id : (Y, D0 + H) → Y part, by (B.2), we have id# j# N ' j# N ⊗L e D (Y,D0 +H ) e(Y,H) . D However, by the assumption that GrF N is torsion-free with respct to xi , hence so is GrF j# N . By applying Lemma 5.3 indectively, we obtain id# j# N ' H0 id# j# N . Lemma 5.9. Notations as in Lemma 5.7. We have D i# Va·r M ' VaH i+ M for a < 0. Further, if the morphism Di xi : V0Di M → V−1 M (5.12) is an isomorphism for every component Di , e.g., M = M[∗D], the second identity still holds when a = 0 D M is x torsion free for a < 0 (resp. a ≤ 0 with the extra assumption), Proof. Since GrF Va·r i due to the strictness condition (5.1), applying the previous lemma, we get D D i# Va·r M ' H0 i# Va·r M, for a < 0 (resp. a ≤ 0). 42 We have D D H0 i# Va·r M ' i∗ Va·r M ⊗De (X,D) e(Y,H) OeX ⊗i−1 Oe i−1 D Y , and eY i+ M ' i∗ M ⊗De OeX ⊗i−1 De D . Y X There is a natural morphism D e(Y,H) eY i∗ Va·r M ⊗De OeX ⊗i−1 De D → i∗ M ⊗De OeX ⊗i−1 De D , Y (X,D) X Y D M, and its image is which is an injection due to the torsion free condition on GrF Va·r exactly D e(Y,H) . Va·r M⊗δ ·D We can conclude the proof by using Lemma 5.7. Fix a projective morphism of two smooth varieties f : X → Y . Given DX , DY two reduced divisors with normal crossing support on X, Y respectively, we say that f : X, DX → Y, DY is a projective morphism of log-smooth pairs if we further have that f −1 DY ⊂ DX . Proposition 5.10. Notations as above, assume that f −1 DY = DX . Fix a mixed Hodge module M on X. Then we have i h Hi f+ M ∗DX = Hi f+ M [∗DY ], h i Hi f+ M !DX = Hi f+ M [!DY ]. Proof. By [21, 2.11. Proposition], we only need to check the identities above at the level of perverse sheaves, which is obvious by considering the following commutative diagram: X \ DX iX f0 Y \ DY X f iY Y. Noticing that iX and iY are affine and f and f 0 are projective, we have Rf∗ RiX∗ K =RiY ∗ Rf∗0 K, Rf∗ RiX! K =RiY ! Rf∗0 K, for any perverse sheaf K on X \ DX . 43 Proposition 5.11. Fix a projective morphism of log-smooth pairs f : X, DX → Y, DY and a mixed Hodge module M on X. Assume DY is smooth. If DX is not smooth, we further assume that M is of normal crossing type with respct to a divisor D0 with normal crossing support that contains DX . Then we have h i X Y Hi f# V0D M ∗DX = V0D h i DX DY Hi f# V<0 M !DX = V<0 Hi f+ M [∗DY ]. Hi f+ M [!DY ]. In particular, the direct image funtor f# is strict on h i h i X DX V0D M ∗DX and V<0 M !DX . Proof. We only show the proof of the first identity here. The second one follows similarly. Consider the morphism of log pairs induced by the identity map id : X, DX → X, f −1 DY , X and its induced direct image of V0D M[∗DX ]. We can forget the logarithmic structure on those components of DX − f −1 DY . More precisely, we have X X id# V0D M[∗DX ] 'V0D M[∗DX ] ⊗L e D (X,DX ) 'V0f −1 D Y e D (X,f −1 DY ) M[∗DX ]. The second identity is due to the comparison theorem. Hence, without loss of generality, we can further assume that DX = f −1 DY . Consider the following commutative diagram i X, DX S, H S g f Y, D Y j T, H T , where i : X, DX → S, H S is the graph embedding given by a local function f ∗ y, where y is a local function on Y that defines DY , and j : Y, DY → T, H T , 44 is the graph embedding given by y. g : S, H S → T, H T is the naturally induced morphism, and it is straightforward to check that we have g ∗ H T = H S . By the previous lemma, we have i# V0D X h i S M ∗DX ' V0H i+ M[∗H S ] . Now by [19, 7.8.5] or [20, 3.3.17], we have h i h i S T Hi g# V0H i+ M ∗H S ' V0H (Hi (g ◦ i)+ M) ∗H T . Hence, we obtain (g ◦ i)# V0D X h i h i T M ∗DX ' V0H (Hi (g ◦ i)+ M) ∗H T . (5.13) Note that since both DY and H T are smooth and j ∗ H T = DY , it is evident that the natural map dj : j ∗ ΩT logDT → ΩY logDY e e is surjective. Hence, as in Lemma 5.8, we have that D (Y,DY )→(T,H T ) is a flat left D(Y,DY ) module. Hence, i h i h X X j# Hi f# V0D M ∗DX ' H0 j# Hi f# V0D M ∗DX . By the degeneration of Leray spectral sequence and (5.13), we obtain h i h i X T j# Hi f# V0D M ∗DX ' V0H Hi (j ◦ f )+ M ∗H T . Applying the previous lemma again, we have h i Y h i X j# Hi f# V0D M ∗DX ' j# V0D Hi f+ M ∗DY . By the construction of j, it is not hard to see that j# is a fully faithful functor (see also the remark below), hence we can conclude that h i h i X Y Hi f# V0D M ∗DX ' V0D Hi f+ M ∗DY . e In particular, it is a strict D (Y,DY ) -module. 45 Remark 5.3. Note that to deduce j# being fully faithfully, we need to use the assumption that DY is smooth, hence by changing of coordinates, we have e e e e D (Y,DY )→(T,H T ) ' D(Y,DY ) ⊗ OY [∂u ]. e Now the fully faithfulness is evident by a local computation. If DY is not smooth, D (Y,DY )→(T,H T ) e is not flat as a left D (Y,DY ) -module in general. See also the computation in Lemma 5.8. Corollary 5.12. Notations as in the previous theorem. Instead of assuming that DY is smooth, we assume that Hi f+ (M[∗DX ]) (resp. Hi f+ (M[!DX ])) is an admissible variation of mixed Hodge structure restricted on Y \ DY . Then we have i ∨∨ h X Y = V0D Hi f+ M [∗DY ]. Hi f# V0D M ∗DX i ∨∨ h DX X DY i Y i = V<0 H f+ M [!D ]. Resp. H f# V<0 M !D Proof. Due to the extra assumption, we have that V0D Y Hi f+ M [∗DY ] is locally free, hence reflexive. Now we only need to show the identity out of a closed codimension 2 subset on Y . This is implied by the previous theorem. Consider the associated graded complex and, by Proposition B.4, we obtain Corollary 5.13. Notations as in Theorem 5.11 and Proposition B.4, we have i h Y F DX X ' GrF V0D Hi f# Gr V M ∗D e 0 i h DY F DX X Hi f# Gr V M !D ' GrF V<0 e <0 5.4 f [∗DY ], Hi f+ M f [!DY ]. Hi f+ M Dual functor Proposition 5.14. Let M be a mixed Hodge module on X, H be a smooth divisor on X. eX be the dual mixed Hodge module of M. We have Let M0 = RHomDe M, ω e [dX ] ⊗ D X D(X,H) V0H M[∗H] H ' V<0 M0 [!H], H D(X,H) V<0 M[!H] ' V0H M0 [∗H]. H M[!H] are strict. In particular, we have that both D(X,H) V0H M[∗H] and D(X,H) V<0 Proof. We only prove the first equation here, the second one follows similarly. 46 Apply the resolution in Proposition 5.1 on M[∗H], locally we have M• , V•H → M[∗H], V•H , eX · z p , V H [a] , with −1 < a ≤ where each Mi , V•H is a direct sum of finite copies of D 0, p ∈ Z. Now we only need to show that RHomDe (X,H) H =V<0 RHomDe X e(X,H) V0H M• , V•H , ωX [dX ] ⊗ D eX . M• , V•H , ωX [dX ] ⊗ D (5.14) However, for −1 < a ≤ 0, we have eX · z p , V H [a] = D e(X,H) · z p . V0H D Hence, RHomDe (X,H) eX · z p , V H [a] , ωX [dX ] ⊗ D e(X,H) = ωX [dX ] ⊗ D e(X,H) · z −p . V0H D On the other hand, to get the Kashiwara-Malgrange filtration V•H on M0 , we set RHomDe X eX · z −p , V H [−1 − a] , eX · z p , V H [a] , ωX [dX ] ⊗ D eX = ωX [dX ] ⊗ D D which gives a filtration on the complex RHomDe X eX . M• , V•H , ωX [dX ] ⊗ D This filtration is actually strict and induces the Kashiwara-Malgrange filtration V•H on M0 by [20, 5.1.13. Lemme.]. Note that Saito only showed the pure Hodge module case there, but the proof still works in the mixed Hodge module case by induction on the weights. Hence, for −1 < a ≤ 0, we have H RHomDe V<0 X eX · z p , V H [a] , ωX [dX ] ⊗ D eX = ωX [dX ] ⊗ D e(X,H) · z −p . D Now (5.14) is clear. Passing to the associated graded pieces, by Proposition B.5, we obtain 47 Corollary 5.15. Notations as in the previous theorem, we have H GrF V0H M0 [∗H] 'RHomA(X,H) GrF V<0 M[!H], ωX [dX ] ⊗OX A(X,H) , H GrF V<0 M0 [!H] 'RHomA(X,H) GrF V0H M[∗H], ωX [dX ] ⊗OX A(X,H) . ∗ ∗ If we consider both sides as graded OT(X,D) -complexes on T(X,D) , we get RHomOT ∗ G V0H M0 [∗H] ' (−1)∗T ∗ (X,H) (X,H) H RHomOT ∗ G V<0 M0 [!H] ' (−1)∗T ∗ (X,H) (X,H) H ∗ G V<0 M[!H] , p∗X ωX [dX ] ⊗ OT(X,H) , ∗ G V0H M[∗H] , p∗X ωX [dX ] ⊗ OT(X,H) . For the normal crossing case, by using the free resolution in Proposition 5.2 as in Theorem 5.14, we have Proposition 5.16. Let M be a mixed Hodge module on X, D be a normal crossing divisor on X. Assume that M is of normal crossing type with respct to a normal crossing divisor D0 ⊃ D. Let M0 be the dual mixed Hodge module of M: M0 := RHomDe X eX . M, ω e [dX ] ⊗ D Similarly, we have D M0 [!D], D(X,D) V0D M[∗D] ' V<0 D D(X,D) V<0 M[!D] ' V0D M0 [∗D]. D M[!D] are strict. In particular, we have that both D(X,D) V0D M[∗D] and D(X,D) V<0 Corollary 5.17. Notations as in the previous theorem, we have D GrF V0D M0 [∗D] 'RHomA(X,D) GrF V<0 M[!D], ωX [dX ] ⊗OX A(X,D) , D GrF V<0 M0 [!D] 'RHomA(X,D) GrF V0D M[∗D], ωX [dX ] ⊗OX A(X,D) . ∗ ∗ If we consider both sides as graded OT(X,D) -complexes on T(X,D) , we get G V0D M0 [∗D] ' (−1)∗T ∗ RHomOT ∗ (X,D) (X,D) ∗ D 0 G V<0 M [!D] ' (−1)T ∗ RHomOT ∗ (X,D) (X,D) D ∗ G V<0 M[!D] , p∗X ωX [dX ] ⊗ OT(X,D) , ∗ . G V0D M[∗D] , p∗X ωX [dX ] ⊗ OT(X,D) 48 5.5 Related vanishings Now we are ready to show some vanishing results that will be used to prove the main claim. The main vanishing result is the following theorem. Note that by taking D = 0, it is Kodaira-Saito vanishing . Theorem 5.18. Fix a mixed Hodge module M, a possibly un-reduced effective divisor D and a semi-ample line bundle L on a smooth projective variety X. Assume further that OX (D) ⊗ L is an ample line bundle, then we have the following vanishing: Hi GrFp SpX M[∗D] ⊗ L =0, for i > 0, Hi GrFp SpX M[!D] ⊗ L−1 =0, for i < 0. Proof. The proof is similar to the proof of Kodaira-Saito vanishing in [21, 2.33. Proposition]. See also [15]. Since both taking (dual) localization and GrFp SpX are exact, by a standard reduction, we only need to show the case that M is a pure Hodge module with strict support Z ⊂ X. If Z ⊂ D, the vanishings are trivial. Further, by Grothendieck-Serre duality and its compatibility with the dual functor in mixed Hodge modules, it suffices to show the second vanishing. Since Lm is globally generated for some integer m, by Bertini's theorem, we can find a global section s ∈ H 0 (X, Lm ), which defines a smooth hypersurface Y that is noncharacteristic for both M and M[!D]. We have a finite covering: π : X := SpecX ⊕0≤i<m L−i → X, ramified along Y . Denote U = X\Y and j : V = X\(D+Y ) → X to be the open embedding. Note that V is affine by the ampleness assumption. Because π is non-characteristic by the construction with respct to M, the Hodge module π∗π∗M is well defined. Further, we have a natural injection M → π∗π∗M. Denote the Hodge module M = Coker (M → π∗ π ∗ M) . Taking the dual localization at D, we have the following short exact sequence: 0 → M[!D] → π∗ π ∗ M[!D] → M[!D] → 0. 49 By the construction, M is the unique extension of M U onto X with strict support, since the eigenspace with eigenvalue 0 of the monodromy operator on ψY SpX M, the nearby cycle of SpX M with respct to Y , is empty. In particular, we have M[!D] = M[!(D + Y )]. Hence, by Artin's vanishing, we have Hi SpX M[!D] = Hic SpV M V = 0, for i < 0. Further, by the strictness of a+ M[!D], where a : X → pt, Hi GrFp SpX M[!D] is a sub quotient of Hi SpX M[!D] . Hence, Hi GrFp SpX M[!D] = 0, for i < 0. (5.15) On the other hand, we have the short exact sequence 0 → M[!D] → (M[!D]) [∗Y ] → i∗ H1 i! (M[!D]) → 0, with H1 i! M[!D] being a mixed Hodge module by the non-charactericity of i, where i : Y → X is the close embedding. We have i∗ H1 i! (M [!D]) = i∗ H1 i! M [!D] , which can be checked at the level of perverse sheaves by [21, 2.11. Proposition]. Note that i∗ H1 i! M has support on Z ∩ Y . By induction on the dimension, now we only need to show Hi GrFp SpX ((M[!D]) [∗Y ]) ⊗ L−1 = 0, for i < 0. (5.16) f in [21, (2.33.2)] by M[!D], (see also [15, (8.8)],) we have However, by replacing M GrFp M[!D] ' GrFp ((M[!D]) [∗Y ]) ⊗ L, where L := Coker OX → π∗ OX ' ⊕0<i<m L−i . Now (5.16) follows by (5.15) and (5.17), which concludes the proof. (5.17) 50 Corollary 5.19. Fix P r,d , L , p : P r,d → Ad as in Theorem 1.5. Fix a mixed Hodge module M on P r,d and an ample line bundle A over Ad . Assume that r = 0 or 1, or M is of normal crossing type with respct to a divisor L0 that contains L. Then we have H i P r,d , GrFp M∗L ⊗ p∗ A = 0, for i > 0, H i P r,d , GrFp M!L ⊗ p∗ A−1 = 0, for i < 0, for any integer k. Proof. Note that if r = 0, it is just [16, Lemma 2.5]. We use P and A to replace P r,d and Ad to simplify the notations. We only show the vanishing of the i > 0 case here, the other case follows similarly. Let m = r + d = dim P . We have that GrFp Sp(P,L) M∗L = [GrFp M∗L ⊗ ∧m T(P,L) → GrFp+1 M∗L ⊗ ∧m−1 T(P,L) → ... → GrFp+m M∗L ], placed in cohomological degree −m, ..., 0. Note that OP (L) ⊗ p∗ A is ample and p∗ A is semi-ample. According to the previous vanishing theorem and Proposition 5.5, for i > 0, we have that Hi GrFp Sp(P,L) M∗L ⊗ p∗ A = 0. (5.18) We have that T(P,L) ' OP⊕m . Since Fp M∗L = 0, for p 0, take p+m to be the smallest inte ger such that GrFp+m M∗L is not trivial. Then by (5.18), we get that H i P, GrFp+m M∗L ⊗ p∗ A = 0, for i > 0. By induction, we can conclude the vanishing of GrFp M∗L for any p. Corollary 5.20. Assume that we have a morphism of log smooth pairs f : (X, D) → P r,d , L , with r = 0 or 1, with p : P r,d → Ad , the natural projection. Then we have −1 H i P r,d , Hk f# L ⊗ p∗ A = 0, e ωX f for any ample line bundle A over Ad , all k ∈ Z and i > 0. Proof. By Proposition 5.11 and Proposition B.4, we have k H f# e ωX f −1 k F k −1 F H f+ (e ωX [!D]) L ' Gr H f# ω eX f L ' Gr . Now it follows by the previous corollary. ∗L CHAPTER 6 PROOF OF THE MAIN CLAIM We show the proof of the Main Claim (Claim 2) in this chapter, which concludes the proof of the Main Theorem (Theorem 1.5). 6.1 Connection to mixed Hodge modules Proposition 6.1. Let g : Y, DY → P r,d , L be a morphism between two log-smooth pairs, with r = 0 or 1. Denote E Y = DY − g −1 L. We have Y , g# ' Rg∗ CY,DY ,• ⊗ OY E Y − DY e ωY E and dually Y Y , g# ' Rg∗ CY,DY ,• ⊗ OY −E Y e ωY D − E where CY,DY ,• is defined in the proof of Theorem 1.5 (Section 4.2). In particular, we have F Gr , H g+ ω eY [∗D ] ' Ri g∗ CY,DY ,• ⊗ OY E Y − DY i Y !L and F Gr i Y H g+ ω eY [!D ] ∗L ' Ri g∗ CY,DY ,• ⊗ OY −E Y . Proof. This follows exactly as [16, 2.11] by considering Proposition 5.11 and Proposition B.4. More precisely, we have Y g# e ωY E Y L ∗ 'Rg∗ ωY E ⊗A g A(P r,d ,L) (Y,DY ) h i Y −• L ∗ 'Rg∗ ωY E ⊗OY ∧ T(Y,DY ) ⊗OY A(Y,DY ) ⊗A g A(P r,d ,L) (Y,DY ) h i n+• L ∗ Y Y Y g A(P r,d ,L) 'Rg∗ OY E − D ⊗OY ΩY log D ⊗OY A(Y,DY ) ⊗A (Y,DY ) 'Rg∗ CY,DY ,• ⊗ OY E Y − DY , 52 where the second identity is due to the canonical resolution B.1 and the third identity is due to [9, Exercise II 5.16 (b)]. The second identity of the proposition can be shown similarly. Proposition 6.2. Notations as in the previous proposition, let p : P r,d → Ad be the natural projection, and A be an ample line bundle over Ad . Then we have that for any i and l > 0, H l X, Ri g∗ CY,DY ,• ⊗ OY −E Y ⊗ p∗ A = 0. Proof. It follows by the previous proposition and Proposition 5.20. 6.2 Proof of the Main Claim Proof of Claim 2 (Main Claim). Denote E = RpV ∗ R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY . We first show that for l > 0, Hl E = 0. (6.1) In particular, E is a sheaf. Since both pP and pY are affine and by Proposition 6.2, we have H l P × V, R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY 'H l P, p1∗ R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY 'H l P, R0 g∗ pY ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY 'H l P, R0 g∗ CY,DY ,• ⊗ OY −E Y ⊗ p∗ A =0, where E Y = DY − f −1 L. Hence, due to the degeneration of the Leray spectral sequence induced by pV ∗ , we have H 0 V, Hl E =H 0 V, Rl pV ∗ R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY 'H l P × V, R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY =0, and so that (6.1) follows. 53 To prove that pV ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY is torsion free, now it suffices to show that pV ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY = R0 Hom (E, OV ) . (6.2) By Grothendieck Duality, we have that RHom (E, OV ) 'DV RpV ∗ R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY [−m] [−m]. 'RpV ∗ DP ×V R0 ĝ∗ L̂ ⊗ p∗Y OY −DY ⊗ CY,DY Note that by definition, L̂ ⊗ p∗Y OY −DY = p∗Y OY −E Y ⊗ g ∗ p∗ A . ∗ By Proposition 6.1 and T(P,L) = P × V , we have 0 R ĝ∗ CY,DY ⊗ 'G H g+ ω eY [∗D ] , E −D !L 0 Y ∗ Y 'G H g+ ω eY [!D ] . ⊗ pY OY −E p∗Y OY R0 ĝ∗ CY,DY Y Y 0 Y ∗L Since up to a Tate twist, we have DP H0 g+ ω eY [!DY ] = H0 g+ ω eY [∗DY ], by Proposition 5.14 and Corollary 5.15, we have , p∗P ωP [m] RHom R0 ĝ∗ CY,DY ⊗ p∗Y OY −E Y 0 Y 'G D(P,L) H g+ ω eY [!D ] ∗L 'G H0 g+ ω eY [∗DY ] !L 'R0 ĝ∗ CY,DY ⊗ p∗Y OY E Y − DY . Comparing it with (6.3), and by definition L̂−1 = p∗Y OY E Y − DY ⊗ g ∗ p∗ A−1 , we obtain RHom (E, OV ) ' RpV ∗ R0 ĝ∗ L̂−1 ⊗ CY,DY , which implies (6.2). (6.3) APPENDIX A QUASI-ABELIAN VARIETIES In this appendix, we recall the definition of quasi-abelian varieties in the sense of Iitaka, [10], and recall a couple of propositions that will be later used. We refer to Fujino's survey article [6] for details. Definition A.1. T r,d is an quasi-abelian variety (in the sense of Iitaka), if it is an extension of a d-dimensional abelian variety Ad by algebraic tori Grm , i.e., it is a connected commutative algebraic group that has the following Chevalley decomposition 1 → Grm → T r,d → Ad → 1. In particular, T r,d is a principal Grm -bundle over Ad . Remark A.1. The commutativity of T r,d is not necessary for the definition, since it can be deduced from the remaining conditions [6]. We consider the following group homomorphism: ρ : Grm → P GL(r, C), given by  1  λ1  ρ(λ1 , ..., λr ) =   0 0 ..      . λr Let P r,d = T r,d ×ρ Pr = T r,d × Pr /Grm , which is a Pr -bundle over Ad . We can view P r,d as a compactification of T r,d that naturally carries the T r,d action on it and denote the boundary divisor L, which has simple normal crossings and is T r,d -invariant for each stratum. Abusing the notations a little bit, over T r,d , we have the morphism [k] : T r,d → T r,d given by multiplication by k, if we use addition for the group operation on T r,d . We can 55 also define [k] : Pr → Pr , given by the Fermat's morphism [z0 , ..., zr ] → [z0k , ..., zrk ]. It is not hard to check that, combining these two morphisms, we get a finite morphism [k] : P r,d → P r,d , which is étale over T r,d , and ramified along the boundary divisor L by degree k. Proposition A.1. The logarithmic cotangent bundle Ω1P r,d (log L) over P r,d is a globally free vector bundle. In particular, we have ωP r,d (L) ' OP r,d . Proof. Since T r,d is a symmetric space, the global T r,d -invariant 1-forms generate the cotangent space Ω1T r,d of T r,d . This free vector space can be extended onto P r,d viewed as a sub-bundle of Ω1P r,d (∗L). Now we need to show that this sub-vector bundle, denoted as T , is Ω1P r,d (log L). To see that T ⊂ Ω1P r,d (log L), pick any local section of Ω1P r,d (∗L), which is T r,d -invariant, we only need to show that it has at most a simple log pole along L. Restricted onto each Pr -fiber of P r,d , its non-holomorphic part is a linear combination of P r,d , its non-holomorphic part is a linear combination of dfi fi , define L. For Ω1P r,d (log L) ⊂ T , we only need to show that dxi xi . Hence, locally on where fi are local functions that dfi fi is a local section of T . Since we know that L is T r,d -invariant for each irreducible component, dfi fi is T r,d -invariant. For any smooth quasi-projective variety V , we use the notation T1 (V ) to denote the space of global holomorphic one-forms on V that can be extended to a log-one-form onto a smooth compactification of V . This space does not depend on the smooth compactification, [10]. Further, we can canonically define a quasi-albanese morphism aV : V → ÃV , where ÃV is a quasi-abelian variety and we call it the quasi-albanese variety of V . In particular, if V is projective, aV is just the usual albanese morphism. Proposition A.2. We have that a∗V : T1 (ÃV ) → T1 (V ) is an isomorphism. We refer to [6, 3.12] for the proof. APPENDIX B FILTERED LOG-D-MODULES In this appendix, we define the basic notions that are used in Chapter 5. B.1 Definition of filtered log-D-modules In this section, we first define the log-D-module corresponding to a log-smooth pair. e Then we recall the notions appeared in [19, Appendix A], but in the log D-modules setting. We will also define some basic functors: the Spencer functor, the pushforward functor, and e modules. If we set the boundary divisor the dual functor, on the derived category of log-D e modules. D = 0, we will get the same functors on the derived category of D Fix a log-smooth pair (X, D), i.e., a smooth variety X with a normal crossing divisor D on X, with D = D1 + ... + Dn being its decomposition of irreducible components. Denote dim X = dX . Working locally in the analytic topology on X, we can assume that X is just a polydisc ∆dX in CdX . Let x1 , ..., xdX be a local analytic coordinate system on X and D is the simply normal crossing (SNC) divisor that is defined by y := x1 ·...·xn = 0. Let ∂1 , ..., ∂dX be the dual basis of dx1 , ..., dxdX . We define T(X,D) , the log-tangent bundle sheaf on (X, D), to be the locally free sheaf that is locally generated over OX by {x1 ∂1 , ..., xn ∂n , ∂n+1 , ..., ∂dX }. T(X,D) is naturally a sub-sheaf of TX , the sheaf of the tangent bundle over X. Similarly, we define D(X,D) , the sheaf of logarithmic differential operators on (X, D), to be the subOX -algebra of DX that is locally generated by {x1 ∂1 , ..., xn ∂n , ∂n+1 , ..., ∂dX }. If D = 0, D(X,D) = DX . There is a natural filtration F on DX given by the degree of the differential operators. We denote eX = RF DX := ⊕p Fp DX , D the Rees algebra induced by (DX , F ) . For any differential operator ∂t ∈ F1 DX , we denote eX . ∂e := ∂t · z ∈ RF D 57 eX , the category of graded D eX modules whose morphisms are graded Denote by M G D morphisms of degree zero, and we call it the category of associated Rees modules of the eX is an abelian category. See category of filtered DX -modules M F (DX ). Note that M G D eX , [19, Definition A.2.3] for details. Hence, we can define the derived category of M G D eX . We also use D∗ G D eX , ∗ ∈ {−, +, b} to denote the and we denote it by DG D bounded above, bounded below, bounded condition, respectively, on the derived category. eX is a coherent D eX -module if M is locally finitely Recall that, we say M ∈ M G D presented, i.e., if for any x ∈ X, there exists an open neighborhood Ux of x and an exact sequence: eq D X ep →D X →M . eX , the full subcategory M G D eX with the objects that are We denote by M Gcoh D eX is also an abelian category. We refer to [19, coherent. Note that we know M Gcoh D Ux Ux Ux A.9, A.10] for a discussion on this topic. e Fix M ∈ M G DX , we can write Mp , the p-th graded piece as Mp = F p M · z p , eX -module, then Fp M is a coherent where Fp M is an OX -module. If M is a coherent D OX -module. Similarly, we have a natural filtration F on D(X,D) , and denote e(X,D) = RF D(X,D) := ⊕p Fp D(X,D) . D e(X,D) , D∗ G D e(X,D) to denote the corresponding categories We similarly use M G(coh) D as above. Let OeX be the sheaf given by the graded ring OX [z] = RF OX , where the filtration F on e X = CX [z] as a graded sub-ring of OeX . Further, given OX is the trivial one. We denote C a coherent OX -module L, we denote Le = L[z], the induced graded OeX -module. From now on, we will omit the word "graded" as long as it is clear from the context. B.2 Basic functors e(X,D) As in [19, A.5.] and [4, 3.1], we define the logarithmic Spencer complex on a D module M, by Sp(X,D) (M) := {0 → M ⊗ ∧n Te(X,D) → ... → M ⊗ Te(X,D) → M → 0}, 58 e X -linear differential map locally given by with the C m ⊗ ξ1 ∧ ... ∧ ξk 7→ k X (−1)i−1 mξi ⊗ ξ1 ∧ ... ∧ ξˆi ∧ ... ∧ ξk + i=1 X (−1)i+j m ⊗ [ξi , ξj ] ∧ ξ1 ∧ ... ∧ ξˆi ∧ ... ∧ ξˆj ∧ ... ∧ ξk , 1≤i<j≤k where ξ1 , ..., ξdX is a local basis of T(X,D) · z. We have the following ([4, Theorem 3.1.2.]) e(X,D) is a resolution of OeX as left D e(X,D) -modules. Theorem B.1. Sp(X,D) D e(X,D) , the complex of its Proof. We only need to show it with respct to GrF Sp(X,D) D associated graded pieces. However, it is straightforward to check that it is just e(X,D) , K ξ1 , ..., ξdX ; GrF D e(X,D) , with ξ1 , ..., ξd ∈ GrF D e the Koszul complex of GrF D 1 (X,D) . It is obvious that ξ1 , ..., ξdX X e(X,D) over GrF OeX . Hence, is a regular sequence and generates GrF D e(X,D) ' GrF OeX . GrF Sp(X,D) D Hence, we can view the logarithmic Spencer complex in the following way. We have the right exact functor: ⊗De (X,D) e(X,D) → M G C eX , OeX : M G D by OeX . M 7→ M ⊗De (X,D) We can define its left derived functor Sp(X,D) = ⊗L e D (X,D) e(X,D) → D∗ G C eX , OeX : D∗ G D by Sp(X,D) M• = M• ⊗L e D (X,D) OeX . e(X,D) is locally free over D e(X,D) . Hence, due to Theorem Note that each term of Sp(X,D) D B.1, we have Sp(X,D) (M) ' M ⊗Le D(X,D) OeX 59 To relieve the burden of notation, in the rest of the paper, ⊗ means ⊗Oe or ⊗O depending on the context if not otherwise specified. eX case [20, 2.4.1. 2.4.2] [19, Exercise A.3.9], there are two right D e(X,D) As in the D e(X,D) that are defined by : module structures on M ⊗ D ( (m ⊗ P ) ·triv f = m ⊗ (P f ) (right)triv : (m ⊗ P ) ·triv ξ = m ⊗ (P ξ) ( (m ⊗ P ) ·tens f = mf ⊗ P (right)tens : (m ⊗ P ) ·tens ξ = mξ ⊗ P − m ⊗ (ξP ) . e(X,D) -module structure, and the second one the right We call the first one the trivial right D e(X,D) -module structure induced by tensor product. D e(X,D) → M ⊗ D e(X,D) that Proposition B.2. There is a unique involution τ : M ⊗ D exchanges both structures and is the identity on M ⊗ 1. It is given by (m ⊗ P ) 7→ mP ⊗ 1 − m ⊗ P. Since it can be checked directly, we omit the proof here. See [20, 2.4.2] for details. Consider the identity functor: e(X,D) → D∗ G D e(X,D) . ⊗OeX : D∗ G D e(X,D) , we get that, for any M ∈ M G D e(X,D) , Replacing OeX by Sp(X,D) D e(X,D) M 'Sp(X,D) M ⊗ D triv e(X,D) '{0 → M ⊗ D ⊗ ∧n Te(X,D) → ... triv e(X,D) e(X,D) → M⊗D ⊗ Te(X,D) → M ⊗ D triv triv → 0}, e(X,D) -modules where D e(X,D) acts on which is a resolution of M by right D e(X,D) M⊗D triv ⊗ ∧i Te(X,D) e(X,D) -module structure: using the induced tensor D (m ⊗ P ⊗ t) f =mf ⊗ P ⊗ t, (m ⊗ P ⊗ t) Q =mQ ⊗ P ⊗ t − m ⊗ QP ⊗ t, for any function f and differential operator P . 60 Applying the involution in Proposition B.2 onto the previous Spencer complex, we get that e(X,D) M 'Sp(X,D) M ⊗ D tens e '{0 → M ⊗ D(X,D) ⊗ ∧n Te(X,D) → ... tens e(X,D) e(X,D) ⊗ Te(X,D) → M ⊗ D → M⊗D tens (B.1) tens → 0}. e(X,D) -modules where D e(X,D) acts on It is also a resolution of M by right D e(X,D) M⊗D tens ⊗ ∧i Te(X,D) e(X,D) -module structure: using the trivial D (m ⊗ P ⊗ t) f = m ⊗ P f ⊗ t, (m ⊗ P ⊗ t) Q = m ⊗ P Q ⊗ t. Now we start to define the pushforward functor. Given a morphism of log-smooth e pairs f : X, DX → Y, DY , we can similarly define M G f −1 D (Y,DY ) , the category of e f −1 D (Y,DY ) -modules, which is an abelian category. Hence, we can define the corresponding e derived category D∗ G f −1 D Y (Y,D ) . We denote −1 e e e D D(Y,DY ) , (X,DX )→(Y,DY ) = OX ⊗f −1 Oe f Y −1 D e e which is a left D (X,DX ) , right f (Y,DY ) -module. We define the relative logarithmic Spencer functor: ∗ −1 e e D(Y,DY ) , Sp(X,DX )→(Y,DY ) : D∗ G D (X,DX ) → D G f by Sp(X,DX )→(Y,DY ) (M• ) = M• ⊗L e D (X,DX ) e D (X,DX )→(Y,DY ) . The topological direct image gives a functor e e f∗ : M G f −1 D → M G D Y Y (Y,D ) (Y,D ) . e Since M G f −1 D Y (Y,D ) is an abelian category with enough injectives, we can define the right derived functor of f∗ : + e e Rf∗ : D+ G f −1 D (Y,DY ) → D G D(Y,DY ) 61 We define the pushforward functor + e e D → D G f# : D+ G D Y X (Y,D ) (X,D ) being the composition of the following two functors: Rf Sp(X,DX )→(Y,DY ) ∗ + −1 e + e e − − − − − − − − − − − − → D G f D − − → D G D D+ G D (Y,DY ) (Y,DY ) (X,DX ) Putting them together, we have f# M = Rf∗ M• ⊗L e D • (X,DX ) e OeX ⊗f −1 Oe f −1 D (Y,DY ) Y . (B.2) e(X,D) , then each cohomology of f# M is an object in If M is an object in M G(coh) D e(X,D) , [19, Theorem A.10.26]. M G(coh) D Note that if both DX and DY are trivial, we have f# = f+ , where f+ is the pushforward e functor on the derived category of D-modules. (In [19], they use the notation D f∗ instead of f+ .) Proposition B.3. Let f : X, DX → Y, DY and g : Y, DY → Z, DZ be two morphisms of log-smooth pairs, and assuming that f is proper, we have a functorial canonical isomorphism of functors (g ◦ f )# ' g# ◦ f# . e e Proof. Since D (X,DX ) is locally free over OX , we have L e e D f −1 D (X,DX )→(Y,DY ) ⊗f −1 D (Y,DY )→(Z,DZ ) e Y (Y,D ) L −1 e eY ⊗ −1 e g −1 D e = OeX ⊗f −1 Oe f −1 D ⊗ f O Y Z (Y,D ) (Z,D ) e g OZ f −1 D Y (Y,DY ) −1 e −1 e L −1 e L f D = OeX ⊗L ⊗ f O ⊗ (g ◦ f ) D Y Z Y (Y,D ) (Z,D ) e f −1 Oe f −1 D (g◦f )−1 Oe Y 'OeX ⊗L (g◦f )−1 OeZ (Y,DY ) (g ◦ f ) −1 Z e D (Z,DZ ) e =D (X,DX )→(Z,DZ ) . −1 e e The equalities above are as left D D(Z,DZ ) bi-modules. Now by the (X,DX ) , right (g ◦ f ) definition (B.2) and the projection formula, we can conclude the proof. We refer to [19, Theorem A.8.11 Remark A.8.12] for a more detailed explanation on the composition of direct images. 62 Recall that we define the dual functor ([20, 2.4.3]) eX , eX → D+ G D DX : D − G D by DX M• = RHomDe X eX M• , ω eX [dX ] ⊗ D tens , where ω eX [dX ] means shifting the sheaf ω eX to the cohomology degree −dX . Similarly, we define the dual functor + e e D(X,DX ) : D− G D → D G D X X (X,D ) (X,D ) , by D(X,DX ) M• = RHomDe (X,DX ) e M• , ω eX [dX ] ⊗ D (X,DX ) tens , (B.3) e e We use the induced tensor right D eX ⊗ D (X,DX ) -module structure on ω (X,DX ) when we take the RHomDe (X,DX ) e (M• , −) functor, and we use the trivial right D (X,DX ) -module • e e structure on ω eX ⊗ D (X,DX ) to give the right D(X,DX ) -module structure on D(X,DX ) M . B.3 Strictness Condition In this section, we recall the definition of the strictness condition. We also recall the associated graded functor in this section. At the end of the section, we explicitly write down the formulae for the induced direct image functor and dual functor on the associated graded module, under certain strictness condition. They can be very useful in some geometric applications. Fix a filtered ring (A, F ). We always assume the filtration F is exhaustive, which means ∪i Fi A = A, and F 0 = 0. Denote Ae := RF A, the associated Rees algebra. We define the e by [19, Definition A.2.7.]: strictness condition on M G(A) Definition B.1. e is said to be strict if it has no C[z]-torsion, i.e., comes from a (1) An object of M G(A) filtered A-module. e is said to be strict if its kernel and cokernel are strict. (Note (2) A morphism in M G(A) that the composition of two strict morphisms need not be strict). e is said to be strict if each of its cohomology modules is (3) A complex M• of M G(A) 63 e (Hence we can use the same definition for the strictness of a strict object of M G(A). e M• ∈ D∗ G(A). We have a right exact functor F e f e Gr : M G(A) → M G Gr A , F e C. e It naturally induces a derived functor: defined by GrF (M) = M ⊗Ce C/z F F ∗ ∗ f e f e Gr : D G(A) → D G Gr A , given by e C. e f F M• = M• ⊗L C/z Gr e C e is strict, which means M is the associated In particular, if we assume that M ∈ M G(A) Rees module of a filtered A-module, we have F f M. GrF M ' Gr f F functor on M is the same as taking the Hence, it is not hard to see that taking the Gr associated graded module. e we have a spectral sequence Now given M• ∈ DG(A), F F f Hq M• ⇒ Hp+q Gr f M• . E2p,q = Hp Gr If M• is strict, we have the spectral sequence degenerates at the E2 page, which implies f F functor under the strictness condition: the commutativity of taking cohomology and Gr f F Hi M• = Hi Gr f F M• . Gr e(X,D) , we have In the case that Ae = D e(X,D) = SymT(X,D) . fF D A(X,D) := Gr We denote ∗ T(X,D) = Spec A(X,D) , the space of log-cotangent bundle over (X, D). Hence, we have a canonical functor e(X,D) → D∗ G A(X,D) , f F : D∗ G D Gr ∗ We denote by G (M• ) , the object in the derived category of OT(X,D) -modules that is induced f F M• . by Gr 64 Proposition B.4 (Laumon's formula). Let f : X, DX → Y, DY be a morphism of • e log-smooth pairs. Let M ∈ M G D Y (Y,D ) . Assume that both M and f# M are strict. Then we have i F i F H f# e Gr M := R f∗ Gr M ⊗L A(X,DX ) ∗ f A(Y,DY ) ' GrF Hi f# M. Proof. Denote N • = M• ⊗L e D (X,DX ) e OeX ⊗f −1 Oe f −1 D (Y,DY ) . Y By the associativity of tensor product and M being strict, we have f F N • = GrF M ⊗L Gr A (X,DX ) f ∗ A(Y,DY ) . Further, by the projection formula, we have f F Rf∗ N . f F N ' Hi Gr Ri f∗ Gr Now we only need to show that f F Rf∗ N ' Gr f F Ri f∗ N . Hi Gr f F functor under the It follows evidently by the commutativity of taking cohomology and Gr strictness assumption on f# M• ' Rf∗ N . e(X,D) and assume D(X,D) M ∈ M G D e(X,D) , Proposition B.5. Fix a M ∈ M G D which means it has only one nontrivial cohomology at cohomology degree 0. We further assume that both M and D(X,D) M are strict. Then we have GrF D(X,D) M ' RHomA(X,D) GrF M, ωX [dX ] ⊗OX A(X,D) . Note that the sections of A(X,D),k := Symk T(X,D) act on the right-hand with an extra factor of (−1)k . It is due to that we are using the induced tenser log-D module structure on the ∗ right-hand side. If we consider both sides as OT(X,D) -complexes, we get ∗ G D(X,D) M ' (−1)∗T ∗ RHomOT ∗ G(M), p∗X ωX [dX ] ⊗ OT(X,D) . (X,D) (X,D) Proof. Since M is strict, we have e e C e RHomDe M, ω eX [dX ] ⊗ D(X,D) ⊗L C/z e C (X,D) tens e C e , ωX [dX ] ⊗O A(X,D) , 'RHomA(X,D) M ⊗Ce C/z X e(X,D) -modules. Now the which can be checked locally by taking a resolution of M by free D statement is clear by the strictness assumption on D(X,D) M. REFERENCES [1] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math., 139 (2000), pp. 241-273. [2] D. Abramovich, K. 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