SUBADDITIVITY OF KODAIRA DIMENSIONS FOR FIBRATIONS OF THREE-FOLDS IN POSITIVE 6 CHARACTERISTICS 1 0 2 LEIZHANG p e S 6 Abstract. In this paper, wewillstudy subadditivity of Kodairadimensions 2 inpositivecharacteristics. Weprovethatforaseparablefibrationf :X →Y fromasmoothprojectivethree-foldtoasmoothprojectivesurfaceoracurve, ] G overanalgebraicallyclosedfieldk withchark>5,ifY isofgeneraltypeand A S0(Xη¯,lKXη¯)6=0forsomepositiveintegerlwhereXη¯denotesthegeometric genericfiber,then h. κ(X)≥κ(Y)+κ(Xη¯,KXη¯) t under certain technical assumptions. We also get some general results under a nefnessandrelativesemi-amplenessconditions. m At the end of this paper we show a numerical criterion for a fibration to [ be birationally isotrivial, and give a new proof to C3,1 under the situation thatgeneralfibersaresurfacesofgeneraltype,whichhasbeenprovenbyEjiri 3 ([14]). v Keywords: Kodaira dimension; positive characteristic; weak positivity; mini- 7 malmodel. 0 MSC:14E05;14E30. 9 6 1. Introduction 0 . 1 Let X be a projective variety over a field k, D a Q-Cartier divisor on X. The 0 D-dimension κ(X,D) is defined as 6 1 −∞, if for every integer m>0,|mD|=∅; κ(X,D)= v: (cid:26) max{dimkΦ|mD|(X)|m∈Z and m>0}, otherwise. Xi IfX hasasmoothprojectivebirationalmodelX˜,theKodairadimensionκ(X)ofX r isdefinedasκ(X˜,KX˜)whereKX˜ denotesthecanonicaldivisor. Kodairadimension a is one of the most important birational invariant in the classification theory. Let f :X →Y be a morphism between two schemes. For y ∈Y, let X denote y the fiber of f overy; and for a divisor D (resp. a sheafF) on X,let D (resp. F ) y y denotetherestrictionofD (resp. F)onthefiberX . Throughoutthispaper,since y Y frequentlyappearsasanintegralscheme,weusethe specialnotationη andη¯for the generic and geometric generic point of Y respectively. We say f is a fibration if f is a projective morphism such that f O =O . ∗ X Y Forafibrationbetweentwoprojectivevarietiesoverthefieldofcomplexnumbers C, Iitaka conjectured that subadditivity of Kodaira dimensions holds: Conjecture 1.1 (Iitaka conjecture). Let f : X → Y be a fibration between two smooth projective varieties over C, with dimX =n and dimY =m. Then C :κ(X)≥κ(Y)+κ(X ). n,m η¯ 1 2 LEIZHANG This conjecture has been studied by Kawamata ([23], [24], [25]), Koll´ar ([27]), Viehweg ([39], [40], [40]), Birkar ([5]), Chen and Hacon ([9]), etc. We refer readers to [12] for a collection of results over C. In positive characteristics, analogously it is conjectured that Conjecture 1.2 (Weak Subadditivity). Let f : X → Y be a fibration between smooth projective varieties over an algebraically closed field k of positive character- istic, with dimX =n and dimY =m. Assume that the geometric generic fibre X η¯ is integral and has a smooth projective birational model X˜ . Then η¯ WC :κ(X)≥κ(Y)+κ(X˜ ). n,m η¯ Remark 1.3. The condition that X is integral is equivalent to that X is reduced, η¯ η¯ and also is equivalent to that f is separable ([[30], Sec. 3.2.2]). If dimY =1 then the fibration f is separable by [[3], Lemma 7.2], thus X is integral. η¯ The reason why we assume the existence of smooth birational models is to guar- anteethatWC makes sense, becausethegeometricgenericfibreX is notneces- n,m η¯ sarily smooth (which is true over C). In positive characteristics, smooth resolution of singularities has been proved in dimension ≤3 ([10] and [11]). Notice that if both X and Y are smooth, then the dualizing sheaf of X is η¯ invertible, which corresponds to a Cartier divisor K . It is reasonable to ask Xη¯ whether the following is true. Conjecture 1.4. Let f :X →Y be a fibration between smooth projective varieties over an algebraically closed field k of positive characteristic, with dimX = n and dimY =m. Then C :κ(X)≥κ(Y)+κ(X ,K ). n,m η¯ Xη¯ It is known that C implies WC by [[12], Corollary 2.5], and we call the n,m n,m inequalityWC weaksubadditivity. UptoapowerofFrobeniusbasechangesand n,m asmoothresolution,toproveweaksubadditivityWC isequivalenttoproveC n,m n,m for another fibration with smooth geometric generic fiber ([[7], proof of Corollary 1.3]). It is much easier to treat a fibration with smooth geometric generic fiber, becausethenonecantakeadvantageofmodulitheoryandpositivityresultsproved recentlybyPatakfalvi[32]andEjiri[14]. Uptonow,thefollowingresultshavebeen proved: (i) WC and C by Chen and Zhang ([12]); n,n−1 2,1 (ii) WC by Birkar, Chen and Zhang over F¯ ,p>5 ([7]); 3,1 p (iii) WC under the situation that X˜ is of general type and chark > 5 by 3,1 η¯ Ejiri ([14], a new proof is given in Appendix 7); (iv) C underthe situationthatf isseparable,dim S0(X ,K )>0and n,m k(η¯) η¯ Xη¯ K is big by Patakfalvi ([33]). Y ThispaperaimstostudysubadditivityofKodairadimensionsforfibrationswith singular geometric generic fibers. Our main result is the following theorem, which generalizes the result (iv) above due to Patakfalvi, and can be applied to study fibrations of 3-folds. Theorem 1.5. Let f :X →Y be a separable fibration between two normal projec- tive varieties over an algebraically closed field k with chark =p>0. Assume either that Y is smooth or that f is flat. Let D be a Cartier divisor on X. SUBADDITIVITY OF KODAIRA DIMENSIONS 3 If there exist an effective Q-Weil divisor ∆ on X and a big Q-Cartier divisor A on Y such that (1) K +∆ is Q-Cartier and p∤ind(K +∆) ; X X η (2) D−K −∆−f∗A is nef and f-semi-ample; X/Y (3) dim S0 (X ,D )>0, k(η¯) ∆η¯ η¯ η¯ then κ(X,D)≥dimY +κ(X ,D ). η¯ η¯ In particular, if D is nef and f-big, and conditions (1) and (2’) D−K −∆−f∗A is nef X/Y hold, then D is big. Remark 1.6. Setting ∆ = 0,D = K and A = K , applying the theorem above X Y we get the result (iv) mentioned above ([[33], Theorem 1.1]). Remark 1.7. For a separable fibration f :X →Y, there always exists a birational modification Y′ →Y such that the main component X′ of X× Y′ is flat over Y′ Y ([[1], Lemma 3.4]). Since Kodaira dimension is invariant under birational modifi- cation, we can reduceto a flat fibration. The advantage of flat fibrations lies in that the relative canonical sheaves behave well under base changes (cf. Proposition 2.1). Combining recent results of minimal model theory in dimension 3 (cf. [18], [6]), we can prove Corollary 1.8. Let f : X → Y be a separable fibration from a smooth projective 3-fold to a smooth projective curve or a surface over an algebraically closed field k with chark =p>5. Assume that (1) K is big; Y (2) S0(X ,lK )6=0 for some positive integer l; and η¯ Xη¯ (3) if dimY = 2, assume moreover that K is pseudo-effective, and that there X exists a birational map σ : X 99K X¯ to a minimal model of X such that, the restriction σ| is an isomorphism to its image. Xη Then κ(X)≥κ(Y)+κ(X ,K ). η¯ Xη¯ Remark1.9. IfY isnon-uniruled,thenassumption(3)aboveissatisfied(Theorem 2.14). Varieties ofmaximal Albanesedimension arenon-uniruled. Theresultabove may be used to study abundance of a 3-fold with non-trivial Albanese map. Corollary 1.10. Let f : X →Y be a separable fibration from a smooth projective 3-fold to a smooth projective curve or a surface, over an algebraically closed field k with chark =p>5. Assume that K is big and Y is non-uniruled. Then Y (1) C is true if K is f-big; and 3,n X/Y (2) WC is true. 3,1 Idea of the proof: By a standard approach proposed by Viehweg in [42], granted the bigness of K , to prove subadditivity of Kodaira dimensions, we only Y need to prove the weak positivity of f ωl . Unfortunately, in positive character- ∗ X/Y istics, if fibers have bad singularities, the sheaf f ωl is not necessarily weakly ∗ X/Y positive (see Raynaud’s example 1.14 below). To overcome this difficulty, stimu- lated by [34] and [33], we prove a positivity result (Theorem 1.11 below) without singularityconditions,butatthecostofassumingotherconditionslikenefnessand relative semi-ampleness. These conditions are closely related to minimal model 4 LEIZHANG theory. For a fibration of a 3-fold, by passing to a minimal model, we can prove that the sheaf Fg∗f (ωl ⊗ωl−1) contains a non-zero weakly positive sub-sheaf Y ∗ X/Y Y under certain situations (say, when ω is f-big), which plays a similar role as X/Y the sheaf f ωl does in proving subadditivity of Kodaira dimensions. ∗ X/Y The positivity result mentioned above is stated as follows. Theorem 1.11. Let f : X → Y be a separable surjective projective morphism between two normal projective varieties over an algebraically closed field k with chark =p>0. Assume that Y is Gorenstein. Let ∆ be an effective Q-Weil divisor on X such that K +∆ is Q-Cartier and p∤ind(K +∆) . If D is a Cartier X/Y X/Y η divisor on X suchthat D−K −∆is nefandf-semi-ample, then for sufficiently X/Y divisible g, thesheaf Fg∗f O (D)containsaweaklypositive sub-sheafSgf O (D) Y ∗ X ∆ ∗ X of rank dim S0 (X ,D ). k(η¯) ∆η¯ η¯ η¯ Moreover if Y is smooth, then Sgf O (D) is FWP. ∆ ∗ X Remark1.12. (1)PleaserefertoSec. 2.2and2.3forthedefinitionsofSgf O (D), ∆ ∗ X S0 (X ,D ) and FWP. The property FWP is mildly stronger than weak positivity, ∆η¯ η¯ η¯ andiseasiertoleadtosubadditivityofKodairadimensioninpositivecharacteristics (cf. Theorem 4.1). (2) In [[34], Theorem D and Theorem E], the authors got similar results under the assumptions that f is flat, relatively G and S , p ∤ ind(K +∆) and D− 1 2 X/Y K −∆ is nef and f-ample. And in [[33], Sec. 6], Patakfalvi proved the weak X/Y positivity of Sgf ω under some mild assumptions. ∗ X/Y (3) Patakfalvi [32] andEjiri [14] proved that, under assumptions that (X ,∆ )is η¯ η¯ sharplyF-pure,K +∆ isampleandp∤ind(K +∆),thesheaff O (m(K + Xη¯ η¯ X/Y ∗ X X/Y ∆)) is weakly positive for sufficiently divisible m. (4) The main idea of the proof is to consider the tracemaps of relative Frobenius iterations, similarly as in [34] and [33]; and the proof is simplified by use of the criterion of weak positivity suggested by Ejiri ([[14], Sec. 4]). Applying the theorem above to log minimal models, immediately we get Corollary 1.13. Let f : X → Y be a separable surjective projective morphism between two normal projective varieties over an algebraically closed field k with chark = p > 0. Let ∆ be an effective Q-Weil divisor on X such that K +∆ is X Q-Cartierandp∤ind(K +∆) . AssumethatK +∆isnefandf-semi-ampleand X η X Y is Gorenstein. Then for a positive integer l such that l(K +∆) is Cartier and X sufficiently divisible g, the sheaf Fg∗(O (l(K +∆))⊗ωl−1) contains a weakly Y X X/Y Y positive sub-sheaf of rank dim S0 (X ,l(K +∆) ). k(η¯) ∆η¯ η¯ X/Y η¯ Let’s recallRaynaud’sexample,whichgivesa minimalsurfaceS of generaltype over a curve C, with Fe∗f ωl being negative while f ωl ⊗ωl−1 being nef for C ∗ S/C ∗ S/C C l≥2. Example 1.14 ([35], [43] Theorem 3.6). Let C be a Tango curve with g(C) ≥ 2 over an algebraically closed field k with chark = p ≥ 3. Then there exists a line bundle L on C such that K ∼pL. We have a non-trivial extension C 0→O →E →L→0 C such that SympE ⊗L−p has a non-zero section. SUBADDITIVITY OF KODAIRA DIMENSIONS 5 Let X = P (E∗) = Proj ⊕ SymlE, g : X → C the natural projection, E C OC l the natural section such that E ∼ O (1) and C′ a smooth curve on X such that X C′ ∼pE−pf∗L. Then E and C′ are disjoint to each other. Let p+1 M ∼ E−pf∗L′ 2 where L′ is a line bundle on C such that 2L′ ∼L. Denoteby π :S →X the smooth double cover induced by the relation 2M ∼ E+C′, and by f : S → C the natural fibration. Then we have that π ωl ∼=O (l(K +M))⊕O (lK +(l−1)M), ∗ S/C X X/C X X/C thus by K ∼−2E+g∗detE ∼−2E+g∗L, X/C f ωl ∼=g O (l(K +M))⊕O (lK +(l−1)M)) ∗ S/C ∗ X X/C X X/C l(p−3) ∼=g (O ( E+g∗(2−p)lL′)) ∗ X 2 lp−p−3l−1 ⊕g (O ( E+g∗((2−p)l+p)L′)) ∗ X 2 ∼=(Syml(p2−3)E ⊗(2−p)lL′)⊕(Symlp−p−23l−1E ⊗((2−p)l+p)L′). We can see that for any positive integers e and l, the sheaf Fe∗f ωl is negative, C ∗ S/C while the sheaf f∗ωSl/C ⊗ωCl−1 ∼=Syml(p2−3)E ⊗(lp+2l−2p)L′⊕Symlp−p−23l−1E ⊗(lp+2l−p)L′ is nef for l ≥2. Conventions: Foramorphismbetweenschemes,weoftenusethesamenotationfortherestric- tion map to a sub-scheme. For a notherianscheme X,we denote by CDiv(X) the additive groupof Cartier divisors. An element D in CDiv(X)⊗Q is called a Q-Cartier divisor, the index of D is the smallest positive integer n such that nD is Cartier, which is denoted by ind(D). LetX be anoetherianG andS schemeoverafieldk offinite type andofpure 1 2 dimension. An almost Cartier divisor (AC divisor for short) on X is a reflexive coherentO -submoduleofthesheafoftotalquotientringK(X)suchthatinvertible X incodimensionone. DenotebyWSh(X)thesetofACdivisors,whichisanadditive group. For D ∈ WSh(X), we denote by O (D) the coherent sheaf defining D; if X O ⊂ O (D), we say D is effective. For two AC divisors D and D on X such X X 1 2 that E =D −D is an effective AC divisor, then O ⊂O (E) induces a natural 2 1 X X inclusionO (D )⊂O (D ). AnelementofWSh(X)⊗QiscalledaQ-ACdivisor. X 1 X 2 NaivelywecandefineeffectivenessinWSh(X)⊗Q. Foraflatmorphismg :W →X, the pull-back g∗ : WSh(X)⊗Q → WSh(W)⊗Q makes sense. For more details, please refer to for example [31]. Let X be a normal variety. Denote by WDiv(X) the additive group of Weil divisors. A Weil divisor D on X defines a reflexive, invertible in codimension one, coherent sub-sheaf O (D) of the constant sheaf K(X), via X O (D) :={f ∈K(X)|((f)+D)| ≥0 for some open set U containing x}. X x U 6 LEIZHANG So we can regard a Weil divisor as an AC divisor. For D ∈ WDiv(X)⊗R, [D] denotes the integral part of D. We use ∼ (resp. ∼ ) for linear (resp. Q-linear) equivalence between AC (resp. Q Q-AC) divisors. We use ≃ for quasi-isomorphism between objects in a derived category,and use ∼= for the isomorphism between sheaves or schemes. Acknowledgments. The author expresses his gratitudetoDr. ShoEjiri for many useful discussions, and to Prof. Zsolt Patakfalvi and Chenyang Xu for some useful communications about numerical criterion of isotriviality. The author is supported by grant NSFC (No. 11401358 and No. 11531009). 2. Preliminaries 2.1. Canonical sheaf and relative canonical sheaf. Let f : X → Y be a projective morphism between noetherian schemes of pure dimension. Let r = dimX −dimY. By Grothendieck duality theory (cf. [19]), there exists a func- tor f! :D+(Y)→D+(X) such that for F ∈D−(X) and G∈D+(Y), Rf RHom (F,f!G)≃RHom (Rf F,G). ∗ X Y ∗ The relative dualizing sheaf is defined as ωo =H0(f!O [−r]). X/Y Y Canonical divisor and relative canonical divisor are defined as follows. (1) If Y is the spectrum of a field and X is G and S , then there exists a 1 2 Gorenstein open set X ⊂ X such that codim (X \ X ) ≥ 2. So f!O | ∼= 0 X 0 Y X0 ωo [r]| , and ωo | is an invertible sheaf on X . The canonicalsheafof X is X/Y X0 X/Y X0 0 defined as ω = i ωo | where i : X ֒→ X denotes the open immersion. The X ∗ X/Y X0 0 canonical divisor K is defined as an AC divisor such that O (K )∼=ω . X X X X (2) If either Y is Gorenstein and X is G and S , or f is flat and relatively G 1 2 1 andS ,similarlythereexistsanopenimmersioni:X ֒→X suchthatcodim (X\ 2 0 X X )≥ 2, f!O | ∼= ωo [r]| , and ωo | is an invertible sheaf. The relative 0 Y X0 X/Y X0 X/Y X0 canonicalsheafω and relative canonicaldivisor K can be defined similarly X/Y X/Y as in (1). If moreover X and Y are projective schemes over a field and Y is Gorenstein, then K ∼K −f∗K . X/Y X Y (3) If f : X → Y is a finite morphism, then ω is defined via f ω ∼= X/Y ∗ X/Y Hom (f O ,O ) ≃ f f!O . And if moreover both X and Y are G and S OY ∗ X Y ∗ Y 1 2 projective schemes over a field, then we can define canonical divisor K as in X/Y (1), which satisfies that K ∼K −f∗K . X/Y X Y It is known that relative dualizing sheaf is compatible with a flat base change (cf. [[19], Chap. III Sec. 8]). For a non-flat base change, we have the following result which is similar to [[12], Theorem 2.4]. Proposition2.1. Letf :X →Y beaflatprojectivemorphismbetweentwonormal varieties. Let ∆ be an effective Q-Weil divisor on X such that K +∆ is Q- X/Y Cartier. Let π :Y′ →Y be a smooth modification, X¯′ =X× Y′ and σ :X′ →X¯ Y the normalization morphism, which fit into the following commutative diagram σ′ X′ ❙❙❙❙❙❙❙σ❙❙f❙′❙❙❙// X❙¯❙′❙❙=❙X(cid:15)(cid:15)fׯ′Y Y′ π′ ((//X(cid:15)(cid:15)f ))Y′ π //Y SUBADDITIVITY OF KODAIRA DIMENSIONS 7 where π′ and f¯′ denote the natural projections, and f′ =f¯′◦σ. Then there exist an effective σ′-exceptional Cartier divisor E′ and an effective divisor ∆′ on X′ such that KX′/Y′ +∆′ =σ′∗(KX/Y +∆)+E′. Proof. Denote by Y the smooth locus of Y, and let Y′ = π−1Y , X = X × Y , 0 0 0 0 Y 0 X¯0′ =X¯′×Y′ Y0′ and X0′ =X′×Y′ Y0′. By arguing in codimension one, we assume X is Gorenstein, hence f| is a flat Gorenstein morphism by [[19], p.298 (Ex. 0 X0 9.7)]. Then by remarks of [[19], p.388], we have KX¯′/Y′ =π′∗KX/Y|X¯′. 0 0 0 Since X′ → X¯′ is the normalization, by results of [[36], Sec. 2], there exists an 0 0 effective divisor C′ such that σ′∗KX/Y|X¯′ =σ∗KX¯′/Y′ =KX′/Y′ +C′. 0 0 0 0 0 Since K + ∆ is assumed to be Q-Cartier, its pull-back makes sense. By X/Y argument above, there exists an effective divisor ∆′ on X′ such that 0 0 KX0′/Y0′ +∆′0 =σ′∗(KX/Y +∆)|X¯0′. Let D′ be the closure of ∆′ in X′, which is a Q-Weil divisor. Let B′ = 0 σ′∗(KX/Y +∆)−(KX′/Y′+D′). If B′ =0, then we are done. Otherwise, since f′ is equi-dimensional, the supportof B′ is mapped via f′ to a codimensionone cycle contained in Y′ \Y′. Since Y′ is smooth, we can find an effective π-exceptional 0 Cartier divisor E on Y′ such that D′′ = f′∗E −B′ ≥ 0. Let E′ = f′∗E which is σ′-exceptional and ∆′ =D′+D′′. Then we are done. (cid:3) 2.2. Trace maps of Frobenius iterations. Throughoutthissubsection,letk be an algebraically closed field of characteristic p>0. Let f :X →Y be a morphism of schemes over k. We will use the following notation: (1) Fe : X → X for the eth absolute Frobenius iteration, and sometimes, to X avoid confusions, we use Xe for the source scheme in the morphism Fe :X →X; X (2)X forthe fiberproductX× Ye ofmorphismsf :X →Y andFe :Ye → Ye Y Y Y, f :X →Y and πe :X →X for the natural projections; e Ye Y Ye (3) Fe :X →X for the eth relative Frobenius iteration over Y. X/Y Ye We will discuss the trace maps of (relative) Frobenius iterations in different settings. Please referto [32], [33], [34]and [14]for more details andrelatedresults. 2.2.1. Trace maps of absolute Frobenius iterations. Notation 2.2. Let X be a G and S projective scheme over k of finite type 1 2 and of pure dimension. Denote by X a Gorenstein open subset of X such that 0 codim (X\X ) > 1. Let ∆ be an effective Q-AC divisor such that K +∆ is X 0 X Q-Cartier and p ∤ ind(K +∆). Then there exists a positive integer g such that X (1−peg)(K +∆) is Cartier for every positive integere, in particular (peg−1)∆| X X0 is an effective Cartier divisor. Let D be a Cartier divisor on X. Since X is G and S , the composite map of the natural inclusion 1 2 FegO ((1−peg)(K +∆))| ֒→Feg O ((1−peg)K ) X∗ X X X0 X0∗ X0 X0 8 LEIZHANG and the trace map TrFXeg0 : FXeg0∗OX0((1−peg)KX0) → OX0 extends to a map on X: Treg :FegO ((1−peg)(K +∆))→O . X,∆ X∗ X X X Twisting the trace map Treg above by O (D) induces a map X,∆ X Treg (D):FegO (1−peg)(K +∆)⊗O (D) X,∆ X∗ X X X ∼=FegO ((1−peg)(K +∆)+pegD)→O (D), X∗ X X X then taking global sections gives H0(Treg (D)):H0(X,FegO ((1−peg)(K +∆)+pegD))→H0(X,D). X,∆ X∗ X X Let Seg(X,D)=ImH0(Treg (D)) and S0(X,D)=∩ Seg(X,D). ∆ X,∆ ∆ e≥0 ∆ If ∆=0, we usually use the notation S0(X,D) instead of S0(X,D). 0 For e′ >e, the map Tre′g (D) factors as X,∆ Tre′g (D):FegF(e′−e)gO ((1−pe′g)(K +∆)+pe′gD) X,∆ X∗ X∗ X X −F−Xe−g∗−T−r−X(e−,′∆−−e−)g−(−(1−−−p−eg−)−(K−X−−+−∆−)+−−pe−g−D→) FegO ((1−peg)(K +∆)+pegD) X∗ X X Treg (D) −−−X−,∆−−→O (D). X So there is a natural inclusion Se′g(X,D) ⊂ Seg(X,D), thus for sufficiently large ∆ ∆ e, Seg(X,D)=S0(X,D). ∆ ∆ Proposition 2.3. Let the notation be as in Notation 2.2. Then (1) There exists an ideal σ(X,∆), namely, the non-F-pure ideal of (X,∆), such that for sufficiently divisible e, ImTreg =σ(X,∆)=Treg Feg(σ(X,∆)·O ((1−peg)(K +∆))). X,∆ X,∆ X∗ X X (2) If D is ample, then for sufficiently large l S0(X,lD)=H0(X,σ(X,∆)·O (lD)). ∆ X (3) If X is integral and D is big, then for sufficiently large l, S0(X,lD)6=0. ∆ (4) Assume moreover that X is integral. And let X′ be another integral G and 1 S projective scheme over k, granted a birational morphism σ : X′ → X. If there 2 exist an effective Cartier divisor E and an effective Q-AC divisor ∆′ on X′ such that KX′ +∆′ =σ∗(KX +∆)+E, then dimS∆0(X,D)≤dimS∆0′(X′,σ∗D+E). Inparticular, ifX′ isthenormalization ofX,thenthereexistsaneffectiveQ-AC divisor ∆′ such that KX′ +∆′ =σ∗(KX +∆), thus dimS∆0(X,D)≤dimS∆0′(X′,σ∗D). SUBADDITIVITY OF KODAIRA DIMENSIONS 9 Proof. For (1), please refer to [[16], Lemma 13.1]. For (2), fix a sufficiently divisible g such that for every positive integer e the trace map below is surjective Feg(σ(X,∆)·O ((1−peg)(K +∆)))→σ(X,∆), X∗ X X and denote by Beg the kernel. By Fujita vanishing, we can find an integer l such 0 that for every integer l>l , 0 H1(X,Bg⊗O (lD))=0, X thus for e=1 the trace map H0(Trg (lD)):H0(X,Fg (σ(X,∆)·O ((1−pg)(K +∆)))⊗O (lD)) X,∆ X∗ X X X →H0(X,σ(X,∆)⊗O (lD)) X is surjective. Then applying the arguments of [[32], Sec. 2.E], by induction we can prove the surjection of H0(Treg (lD)) for every e. X,∆ For (3), assume that X is integral and D is big. Then we can find an ample Q- CartierdivisorH suchthat D−H is effective. Takeasufficiently divisible positive integer l such that, both E =l(D−H) and lH are Cartier. By the natural map H0(X,Fg O ((1−pg)(K +∆))⊗O (lH))⊗H0(X,O (E)) X∗ X X X X →H0(X,Fg O ((1−pg)(K +∆))⊗O (lH +E)) X∗ X X X H0(Trg (D)) −−−−−−X−,∆−−−→H0(X,O (lH +E))=H0(X,O (D)), X X for a non-zero s ∈H0(X,O (E)), we get an injective map below E X S0(X,lH)−⊗−s−→E S0(X,lH+E)=S0(X,lD). ∆ ∆ ∆ Then we can conclude assertion (3) by (2). We are left to prove (4). Let K = K(X) and Kp1g = K(Xg). Denote by ζ = specK the generic point of X and X′. Regard HomK(Kp1g,K) and K as constant quasi-coherent sheaves of both X and X′. We have the following natural commutative diagram Trg X,∆ FXg∗OX((1−pg)(KX +∆))(cid:31)(cid:127) //FXg∗ωX1−pg ∼=HomO(cid:127)_X(FXg∗OX,OX) ++//OX(cid:127)_ ωζ1−pg ∼=Hom(cid:15)(cid:15) K(Kp1g,K) Trζg=ev(1)//K(cid:15)(cid:15)OO OO FXg′∗OX′((1−pg)(KX′+∆′)(cid:31)(cid:127) //FXg′∗(ωX1−′pg)∼=Hom(cid:31)? OX′(FXg′∗OX′,OX′) //33O(cid:31)?X′ TrXg′,∆′ 1 where ev(1) denotes the evaluation map at 1∈Kpg. Since(1−pg)(KX′+∆′)+pg(σ∗D+E)−σ∗((1−pg)(KX+∆)+pgD)=E isan effective Cartier divisor, with the two sheaves below naturally seen as sub-sheaves of HomK(Kp1g,K)∼=Kp1g on X′, we see that FXg′∗σ∗OX((1−pg)(KX+∆)+pgD)⊆FXg′∗OX′((1−pg)(KX′+∆′)+pg(σ∗D+E)). 10 LEIZHANG By the natural inclusions OX′(σ∗D)⊆OX′(σ∗D+E)⊆K, we have the following commutative diagram σ∗Trg (D) σ∗OX((1−pg)(K(cid:127)_X +∆)+pgD) X,∆ // OX′(σ(cid:127)_∗D) . OX′((1−pg)(KX′ +(cid:15)(cid:15)∆′)+pg(σ∗D+ET)r)Xg′,∆′(σ∗D+E//)OX′(σ∗(cid:15)(cid:15)D+E) Taking global sections of the traces maps above, we conclude the injection σ∗S∆0(X,D)֒→S∆0′(X′,σ∗D+E), thus dimS∆0(X,D)≤dimS∆0′(X′,σ∗D+E). In particular, if σ : X′ → X is the normalization map, then by [[36], Sec. 2] there exists an effective AC divisor C on X′ such that σ∗ωX =ωX′(C). The remaining assertion is an easy consequence. (cid:3) 2.2.2. Trace maps of relative Frobenius iterations I. Notation 2.4. Let f : X → Y be a surjective projective morphism between two schemes over k of finite type and of pure dimension. Assume that X is G and S 1 2 and that Y is integral and regular. Let ∆, g and D be assumed as in Notation 2.2. By assumption Feg is a flat morphism, so X also satisfies G and S , and Y Yeg 1 2 K =πeg∗K . By easy calculation we have that XYeg/Yeg Y X/Y K =(1−peg)K and Feg∗ πeg∗D =pegD. Xeg/XYeg Xeg/Yeg X/Y Y Similarly as in 2.2.1, we get the trace map Treg (D):Feg O ((1−peg)(K +∆)+pegD)→O (πeg∗D). X/Y,∆ X/Y∗ X X/Y XYeg Y Applying f to the above map, we get eg∗ f Treg (D):f O ((1−peg)(K +∆)+pegD) ∗ X/Y,∆ ∗ X X/Y ։Segf O (D)֒→f O (πeg∗D)∼=Feg∗f O (D). ∆ ∗ X eg∗ XYeg Y Y ∗ X where Segf O (D), introduced by [[34], Def. 6.4] with slightly different notation, ∆ ∗ X denotes the image of f Treg (D). If ∆ = 0, we use the notation Segf O (D) ∗ X/Y,∆ ∗ X