Subadditivity of Kodaira dimensions for fibrations of three-folds in positive characteristics

0.38 MB·

by  Lei Zhang

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