Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic II

0.2 MB·English

by  Qihong Xie

Strongly Liftable Schemes and the Kawamata-Viehweg
Vanishing in Positive Characteristic II ∗
Qihong Xie
1
Abstract
1
0
AsmoothschemeX overafieldkofpositivecharacteristicissaidtobestrongly
2
liftable, if X andallprime divisorsonX canbe lifted simultaneouslyoverW2(k).
n In this paper, first we prove that smooth toric varieties are strongly liftable. As a
a
corollary,weobtaintheKawamata-Viehwegvanishingtheoremforsmoothprojec-
J
tive toric varieties. Second, we prove the Kawamata-Viehweg vanishing theorem
0
for normal projective surfaces which are birational to a strongly liftable smooth
1
projectivesurface. Finally,wededucethecycliccovertrickoverW2(k),whichcan

] be used to construct a large class of liftable smooth projective varieties.
G
A
1 Introduction
.
h
t
a Throughoutthispaper,wealwaysworkoveran algebraically closed field k of character-
m
istic p > 0 unless otherwise stated. A smooth scheme X is said to be strongly liftable,
[
if X and all prime divisors on X can be lifted simultaneously over W2(k). This notion
2 was first introduced in [Xie10c] to study the Kawamata-Viehweg vanishing theorem in
v
positive characteristic, furthermore, some examples and properties of strongly liftable
4
2 schemes were also given in [Xie10c].
0 In this paper, we shall continue to study strongly liftable schemes. First of all, we
3
find an important class of strongly liftable schemes with simple structures.
.
8
0
Theorem 1.1. Smooth toric varieties are strongly liftable over W (k).
0 2
1
As a consequence, we obtain the Kawamata-Viehweg vanishingtheorem on smooth
:
v
projective toric varieties forampleQ-divisorswhich arenotnecessarily torusinvariant.
i
X
r Corollary 1.2. Let X be a smooth projective toric variety of dimension d, H an ample
a
Q-divisor on X, and D a simple normal crossing divisor containing Supp(hHi). Then
Hi(X,Ωj (logD)(−pHq)) = 0 holds for any i+j < inf(d,p).
X
In particular, Hi(X,K +pHq) = 0 holds for any i> d−inf(d,p).
X
Second,wegeneralize[Xie10c,Theorem1.4]slightlytothecasewherenosingularity
assumption is made.
∗This paper was partially supported by the National Natural Science Foundation of China (Grant
No. 10901037) and the Ph.D. Programs Foundation of Ministry of Education of China (Grant No.
20090071120004).
1
Theorem 1.3. Let X be a normal projective surface and H a nef and big Q-divisor
on X. If X is birational to a strongly liftable smooth projective surface Z, then
H1(X,K +pHq) = 0 holds.
X
As a corollary, we obtain the Kawamata-Viehweg vanishing theorem for rational
surfaces, which is a generalization of [Xie10b, Theorem 1.4].
Corollary 1.4. Let X be a normal projective rational surface and H a nef and big
Q-divisor on X. Then H1(X,K +pHq) = 0 holds.
X
Finally, we deduce an explicit statement of the cyclic cover trick over W (k) (see
2
Theorem 4.1 for more details). By means of the cyclic cover trick, we can construct a
large class of liftable smooth projective varieties from certain strongly liftable varieties
with simple structures, e.g. toric varieties. Namely, we have the following corollary.
Corollary 1.5. Let X be a smooth projective toric variety, and L an invertible sheaf
on X. Let N be a positive integer prime to p, and D an effective divisor on X with
LN = O (D) and Sing(D ) = ∅. Let π : Y → X be the cyclic cover obtained by
X red
taking the N-th root out of D. Then Y is a liftable smooth projective scheme.
Especially, it follows from Corollary 1.5 that there do exist many liftable smooth
projective varieties of general type, whose existence is helpful for studying birational
geometry of algebraic varieties in positive characteristic. In general, the schemes ob-
tained by taking cyclic covers over strongly liftable schemes are no longer strongly
liftable (see Remark 4.6 for more details), which shows that the class of strongly
liftable schemes is really restrictive.
In §2, we will recall some definitions and preliminary results of liftings over W (k).
2
§3 is devoted to the proofs of the main theorems. The cyclic cover trick over W (k)
2
will be treated in §4. In §5, we will give some corrections to the mistakes in [Xie10c].
For the necessary notions and results in birational geometry, we refer the reader to
[KMM87] and [KM98].
Notation. We use ∼ to denote linear equivalence, ≡ to denote numerical equivalence,
and [B]= [b ]B (resp.pBq = pb qB , hBi= hb iB , {B}= {b }B )to denote
P i i P i i P i i P i i
the round-down (resp. round-up, fractional part, upper fractional part) of a Q-divisor
B = b B , where for a real number b, [b] := max{n ∈ Z|n ≤ b}, pbq := −[−b],
P i i
hbi := b−[b] and {b} := pbq−b. We use Sing(D ) to denote the singular locus of the
red
reduced part of a divisor D.
Acknowledgments. I would like to express my gratitude to Professor Luc Illusie for
pointing out some errors in [Xie10c] and an earlier version of this paper. I would also
like to thank Professors Osamu Fujino and Fumio Sakai for useful comments.
2 Preliminaries
Definition 2.1. LetW (k) bethe ringof Witt vectors of length two of k. ThenW (k)
2 2
is flat over Z/p2Z, and W (k)⊗ F = k. For the explicit construction and further
2 Z/p2Z p
properties of W (k), we refer the reader to [Se62, II.6]. Thefollowing definition [EV92,
2
Definition8.11]generalizesthedefinition[DI87,1.6]ofliftingsofk-schemesoverW (k).
2
Let X be a noetherian scheme over k, and D = D a reduced Cartier divisor
P i
on X. A lifting of (X,D) over W (k) consists of a scheme X and closed subschemes
2
e
2
D ⊂ X, all defined and flat over W (k) such that X = X × Speck and
ei e 2 e SpecW2(k)
D = D × Speck. We write D = D and say that (X,D) is a lifting of
i ei SpecW2(k) e P ei e e
(X,D) over W (k), if no confusion is likely.
2
Let L be an invertible sheaf on X. A lifting of (X,L) consists of a lifting X of X
e
over W (k) and an invertible sheaf L on X such that L| = L. For simplicity, we say
2 X
e e e
that L is a lifting of L on X, if no confusion is likely.
e e
Let X be a lifting of X over W2(k). Then OXe is flat over W2(k), hence flat over
Z/p2Z. Neote that there are an exact sequence of Z/p2Z-modules:
0 → p·Z/p2Z → Z/p2Z →r Z/pZ→ 0
and a Z/p2Z-module isomorphism p :Z/pZ → p·Z/p2Z. Tensoring the above by Oe,
X
we obtain an exact sequence of Oe-modules:
X
r
0→ p·OXe → OXe → OX → 0, (1)
and an Oe-module isomorphism
X
p : OX → p·OXe, (2)
where r is the reduction modulo p satisfying p(x) = px, r(x)= x for x ∈ OX, x ∈ OXe.
The following lemma has already been proved in [eEV9e2, Lemmas 8.13 ande8.14].
Lemma 2.2. Let (X,D) be a lifting of (X,D) as in Definition 2.1. If X is smooth
e e
over k and D ⊂X issimple normal crossing, then X is smooth over W (k) and D ⊂X
2
e e e
is relatively simple normal crossing over W (k).
2
Proof. Since the statement is local, we may assume that there is an ´etale morphism
ϕ : X → An = Speck[t ,··· ,t ], such that ϕ∗(t ) give a regular system of parameters
k 1 n i
(x ,··· ,x ) on X, and D = D ⊂ X is defined by the equation x ···x = 0 for
1 n P i 1 r
some r ≤ n. Take xi ∈ OXe with r(xi)= xi (1 ≤ i≤ n), and define
e e
∗
ϕ : W2(k)[t1,··· ,tn] → OXe
e
∗
as a W (k)-algebra homomorphism by ϕ (t ) = x (1 ≤ i ≤ n), which gives rise to a
2 i i
morphism ϕ : X → An . First of all,ewe shalleprove that ϕ is an ´etale morphism,
e e W2(k) e
which implies that X is smooth over W (k).
2
e
Without loss of generality, we may assume that OX is a free OAn-module with
k
generators g1,··· ,gm. Take gi ∈ OXe lifting gi (1 ≤ i ≤ m), then for any x ∈ OXe,
we can write x = r(xe) = Pmi=e1λigi for some λi ∈ OAnk. Take λei ∈ OAnW2(k) leifting λi
(1 ≤ i ≤ m), then by the exact sequence (1) we have x−Pmi=1λigi ∈ p·OXe. Thus we
can find µi ∈ OAn (1 ≤ i ≤ m), such that µi = r(µei) and e e
e W2(k) e
m m m
x−Xλigi = p(Xµigi)= Xpµigi,
e i=1e e i=1 i=1 e e
hence we have xe = Pmi=1(λei +pµei)gei. Assume Pmi=1λeigei = 0 for some λei ∈ OAnW2(k).
Then λ = r(λ ) satisfy m λ g = 0, which implies λ = 0 (1 ≤ i ≤ m). Thus we
i i Pi=1 i i i
e
3
can find µi ∈ OAn (1 ≤ i ≤ m), such that µi = r(µi) and λi = p(µi)= pµi, hence
e W2(k) e e e
m m m
p(Xµigi) = Xpµigi = Xλigi = 0.
i=1 i=1 e e i=1e e
Sincepisanisomorphism,wehave m µ g = 0,henceµ = 0andλ = 0(1 ≤ i ≤ m).
Pi=1 i i i i
e
TheSrienfocreeϕO:XeXis→aAfrneeisO´etAanWle2(,kw)-me hoadvueleΩa1nd =ϕe iϕs∗flΩa1t. = n O dx . By definition,
k X/k An/k Li=1 X i
k
we have ϕe∗Ω1AnW2(k)/W2(k) = Lni=1OXedxei. Combining the exact sequence (1) and the
isomorphism (2), we obtain an exact sequence of Oe-modules:
X
p r
0 → OX → OXe → OX → 0. (3)
Tensoring the exact sequence (3) by Ω1 , we obtain an exact sequence:
e
X/W2(k)
Ω1 →p Ω1 →r Ω1 → 0.
X/k Xe/W2(k) X/k
We shall prove that the natural homomorphism ϕ∗Ω1 → Ω1 is sur-
e AnW2(k)/W2(k) Xe/W2(k)
jective. For any ω ∈ Ω1 , we can write ω = r(ω) = n λ dx ∈ Ω1 for some
λi ∈ OX. Take λei ∈ OXeXe/Wl2if(tki)ng λi (1 ≤ i ≤ n), theen ω −PPi=1ni=1iλidixi ∈XI/mk(Ω1X/k →p
Ω1Xe/W2(k)). Thusewe can find µei ∈ OXe (1 ≤ i≤ n) suchethat µi =er(µeei) and
n n n
ω−Xλidxi = p(Xµidxi) = Xpµidxi,
e i=1e e i=1 i=1 e e
hence ω = n (λ +pµ )dx , so ϕ∗Ω1 → Ω1 is surjective. By the
e Pi=1 ei ei ei e AnW2(k)/W2(k) Xe/W2(k)
first exact sequence associated to ϕ :X → An [Ha77, Proposition II.8.11], we have
Ω1 = 0, which together witeh thee flatnWes2s(ko)f ϕ implies that ϕ is´etale (see [Ha77,
Xe/An
W2(k) e e
Exercise III.10.3]), so X is smooth over W (k).
2
e
By assumption, D is defined by the equation x = 0 (1 ≤ i ≤ r). Since D is a
i i i
e
lifting of Di over W2(k), the flatness of OXe and ODei over W2(k) implies that the ideal
sheaf IDei of Dei is flat over W2(k). Thus we have an exact sequence:
r
0→ p·IDei → IDei → IDi → 0,
and an isomorphism p : IDi → p· IDei, where IDi is the ideal sheaf of Di. For any
ge ∈ IDei, we can write g = r(ge) = λixi for some λi ∈ OX. Take λei ∈ OXe lifting λi
(µ1 =≤ri(≤µ r))a,ntdhen ge−λeixei ∈ p·IDei. Thus we can find µei ∈ OXe (1 ≤ i ≤ r) such that
i i
e g−λ x = p(µ x ) =pµ x ,
i i i i i i
e e e e e
hence eg = (λei +pµei)xei, so IDei is generated by xei, i.e. xei = 0 is a defining equation
for D (1 ≤ i ≤ r). Thus D = D ⊂ X is relatively simple normal crossing over
i P i
e e e e
W (k).
2
4
We can use Lemma 2.2 to deduce Bertini’s theorem for ample invertible sheaves on
smooth projective schemes over W (k).
2
Theorem 2.3. Let X be a smooth projective scheme over W (k), and L an ample
2
invertible sheaf on X.eThen there is a positive integer m such that Lm isevery ample
and associated to a geeneral section s ∈H0(X,Lm), the divisor of zeroes D = div (s) is
0
smooth over W (k). e e e e e
2
r
Proof. The natural surjection W (k) → k induces the closed immersion ι : Speck ֒→
2
SpecW (k). Let X = X × Speck, ι : X ֒→ X the closed immersion, and
∗ 2 e SpecW2(k) e
L = ι L =L| the induced invertible sheaf on X.
X
Sincee Leis ample, there is a positive integer m such that Lm is very ample, hence
induces a eclosed immersion ϕe : Xe → PNW2(k) with ϕe∗OPNW2(k)(1e) = Lem. Let ι : PNk ֒→
PN be the closed immersion induced by ι : Speck ֒→ SpecW (k). Then it is easy
W2(k) 2
to see that X = X ×PN PNk . Let ϕ : X → PNk be the induced closed immersion.
e W2(k)
X(cid:127)_(cid:31)(cid:127) ϕ //PNk(cid:127)_ // Spec(cid:127)_ k
ι ι ι
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
X(cid:31)(cid:127) ϕe //PNW2(k) //SpecW2(k)
e
Then we have ϕ∗OPN(1) = Lm, hence L is ample. Taking m sufficiently large, we may
k
assume that Lm is very ample and H1(X,Lm) = 0. Tensoring the exact sequence (3)
by Lm, we obtain an exact sequence:
e
0 → Lm →p Lm →r Lm → 0,
e
which implies that H0(X,Lm)→r H0(X,Lm) is surjective.
e e
Since X is a smooth projective scheme over k, it follows from Bertini’s theorem
[Ha77,TheoremII.8.18]thatassociatedtoageneralsections ∈ H0(X,Lm),thedivisor
of zeros D = div (s) is smooth over k. Take a section s ∈ H0(X,Lm) with r(s) = s,
0
then the divisor of zeros D = div (s) is a lifting of D oever W (ke). eBy Lemmae2.2, D
0 2
is smooth over W (k). e e e
2
Definition 2.4. Let X be a smooth scheme over k. X is said to be strongly liftable
over W (k), if there is a lifting X of X over W (k), such that for any prime divisor D
2 2
e
on X, (X,D) has a lifting (X,D) over W (k) as in Definition 2.1, where X is fixed for
2
e e e
all liftings D.
e
Let X be a smooth scheme over k, X a lifting of X over W (k), D a prime divisor
2
e
on X and L = O (D) the associated invertible sheaf on X. Then there is an exact
D X
sequence of abelian sheaves:
q ∗ r ∗
0 → O → O → O → 1, (4)
X Xe X
where q(x) = p(x)+1 for x ∈ OX, p : OX → p·OXe is the isomorphism (2) and r
is the reduction modulo p. The exact sequence (4) gives rise to an exact sequence of
cohomology groups:
H1(X,O∗ )→r H1(X,O∗ )→ H2(X,O ). (5)
e Xe X X
5
If r : H1(X,O∗ ) → H1(X,O∗ ) is surjective, then L has a lifting L , i.e. L is an
e Xe X D eD eD
invertible sheaf on X with L | = L . Tensoring the exact sequence (3) by L , we
D X D D
e e e
have an exact sequence of Oe-modules:
X
p r
0 → L → L → L → 0,
D D D
e
which gives rise to an exact sequence of cohomology groups:
H0(X,L ) →rD H0(X,L ) → H1(X,L ). (6)
D D D
e e
We recall here a sufficient condition for strong liftability [Xie10c, Proposition 3.4].
Proposition 2.5. Let X be a smooth scheme over k, and X a lifting of X over W (k).
2
e
Then X is strongly liftable if the following two conditions hold:
(i) r : H1(X,O∗ ) → H1(X,O∗ ) is surjective;
e Xe X
(ii) For any prime divisor D on X, there is a lifting L of L = O (D) such that
D D X
r :H0(X,L ) → H0(X,L ) is surjective. e
D D D
e e
Lemma 2.6. Let X be a strongly liftable smooth scheme, and X a lifting of X as in
Definition 2.4. Then the natural map r : H1(X,O∗ ) → H1(X,Oe∗ ) is surjective.
e Xe X
Proof. Let D = a D be an effective divisor on X. Then we can lift each distinct
P i i
prime divisor D to D on X, hence D has a lifting D = a D , which is an effective
i i P i i
e e e e
Cartier divisor on X. Let L be an invertible sheaf on X. Assume L = O (E −E ),
X 1 2
e
where E and E are effective divisors on X without common irreducible components,
1 2
then we can lift Ei to Ei on X. Define L = OXe(E1−E2). Then L is a lifting of L on
X. Thus r : H1(X,O∗e) → He1(X,O∗ ) ies surjectieve. e e
e e Xe X
Remark 2.7. Let X be a strongly liftable smooth scheme, X a lifting of X as in
e
Definition 2.4, L an invertible sheaf on X, and L a lifting of L on X. Then in general
the natural map r :H0(X,L)→ H0(X,L) is noet necessarily surjecteive. The following
e e
example is given by Illusie and Kato. Let C be a smooth projective curve of genus
g ≥ 1. Then C is strongly liftable by [Xie10c, Theorem 1.3(i)]. Let C be a lifting of
e
C. By the exact sequence (4), we have an exact sequence of cohomology groups:
0 → H1(C,O )→ H1(C,O∗)→r H1(C,O∗).
C e Ce C
Since dimH1(C,O ) = g ≥ 1, there exists a non-trivial invertible sheaf L on C such
C
that L = L| ∼= O . Then the section s ∈ H0(C,L) giving an isomorphisem Oe → L
C C C
cannot beelifted to a section of L. Thus r : H0(C,L)→ H0(C,L) is not surjective.
e e e
For convenience of the reader, we recall the Kawamata-Viehweg vanishing theorem
forsmoothprojectivevarietiesinpositivecharacteristic undertheliftingconditionover
W (k) of certain log pairs, which has first been proved by Hara [Ha98].
2
Theorem 2.8 (Kawamata-Viehweg vanishing in char. p > 0). Let X be a smooth
projective variety of dimension d, H an ample Q-divisor on X, and D a simple nor-
mal crossing divisor containing Supp(hHi). Assume that (X,D) admits a lifting over
W (k). Then
2
Hi(X,Ωj (logD)(−pHq)) = 0 holds for any i+j < inf(d,p).
X
In particular, Hi(X,K +pHq) = 0 holds for any i> d−inf(d,p).
X
6
IfX isasmoothprojectivevarietystronglyliftableoverW (k),thentheKawamata-
2
Viehweg vanishingtheorem holds on X, which is a direct consequence of Definition 2.4
and Theorem 2.8.
3 Proofs of the main theorems
In [Xie10c], it was proved that An, Pn, smooth projective curves, smooth rational
k k
surfaces, and certian smooth complete intersections in Pn are strongly liftable over
k
W (k). It follows easily from Proposition 2.5 that smooth affine schemes are strongly
2
liftable over W (k). In this section, we shall give further examples of strongly liftable
2
schemes and some applications to vanishing theorems.
Theorem 3.1. Any smooth toric variety is strongly liftable over W (k).
2
Proof. First of all, we recall some definitions and notations for toric varieties from
[Fu93]. Let N be a lattice of rank n, and M the dual lattice of N. Let ∆ be a fan
consisting of strongly convex rational polyhedral cones in N , and let A be a ring. In
R
general, we denote the toric variety associated to the fan ∆ over the ground ring A
by X(∆,A). More precisely, to each cone σ in ∆, there is an associated affine toric
∨
variety U = SpecA[σ ∩M], and these U can be glued together to form the
(σ,A) (σ,A)
toric variety X(∆,A) over SpecA. Note that almost all definitions, constructions and
results for toric varieties are independent of the ground ring A, although everything is
stated in [Fu93] over the complex number field C.
Let X = X(∆,k) be a smooth toric variety over k associated to a fan ∆. Let
∨ ∨
X = X(∆,W (k)). NotethatU = Speck[σ ∩M],U = SpecW (k)[σ ∩M]
e 2 (σ,k) (σ,W2(k)) 2
is flat over W (k) and U × Speck = U , hence X is a lifting of X
2 (σ,W2(k)) SpecW2(k) (σ,k) e
over W (k).
2
By [Fu93, Page 74, Corollary], we have H2(X,O ) = 0, which implies that any
X
invertible sheaf L on X has a lifting L on X by the exact sequence (5). In fact, we
e e
can prove the liftability of invertible sheaves in an explicit way. Let L be an invertible
sheaf on X. By [Fu93, Page 63, Proposition], we have an exact sequence:
0→ M → Div (X) → Pic(X) → 0.
T
Thereforethereexists atorusinvariant divisor D onX suchthat L= O (D). Assume
X
that {u(σ) ∈ M/M(σ)} ∈ projlimM/M(σ) determines the torus invariant divisor D.
Then the same data {u(σ) ∈ M/M(σ)} also determines a torus invariant divisor D
e
on X (we have only to change the base k into W (k)). Thus the invertible sheaf
2
e
L = OX(D) has a lifting L= OXe(D) on X.
e e e
Let E be a prime divisor on X, and L = O (E) the associated invertible sheaf on
X
X. Then we can take torus invariant divisors D and D as above such that L = O (D)
X
e
and the invertible sheaf L = OXe(D) lifts L. Let vi be the first lattice points in the
e e
edges of the cones of maximal dimension in ∆, D the corresponding orbit closures in
i
X, andD thecorrespondingorbitclosuresinX (1 ≤ i ≤ N). Thenthetorusinvariant
i
divisorseD = N a D and D = N a D edetermine a rational convex polyhedral
Pi=1 i i Pi=1 i i
e e
P in M defined by
D R
P = {u ∈M | hu,v i≥ −a , 1 ≤ i≤ N}.
D R i i
7
By [Fu93, Page 66, Lemma], we have
H0(X,D) = M k·χu, H0(X,D)= M W2(k)·χu.
e e
u∈PD∩M u∈PD∩M
Thus the map H0(X,D)→r H0(X,D) induced by the natural surjection W (k) →r k is
2
obviously surjectivee. Heence r : H0(X,L) → H0(X,L) is surjective. By Proposition
E
e e
2.5, X is strongly liftable over W (k).
2
As a consequence, we have the Kawamata-Viehweg vanishing theorem on smooth
projective toric varieties in positive characteristic for ample Q-divisors which are not
necessarily torus invariant, whereas Musta¸tˇa [Mu02] and Fujino [Fu07] have proved
certainmoregeneralversionsoftheKawamata-Viehwegvanishingtheoremoncomplete
toric varieties in arbitrary characteristic for nef and big torus invariant Q-divisors.
Corollary 3.2. Let X be a smooth projective toric variety of dimension d, H an ample
Q-divisor on X, and D a simple normal crossing divisor containing Supp(hHi). Then
Hi(X,Ωj (logD)(−pHq)) = 0 holds for any i+j < inf(d,p).
X
In particular, Hi(X,K +pHq) = 0 holds for any i> d−inf(d,p).
X
Proof. It follows from Theorems 2.8 and 3.1.
Corollary 3.3. Let X be a smooth toric variety of dimension d, π : X → S a pro-
jective toric morphism onto a toric variety S, and H a π-ample Q-divisor on X with
Supp(hHi) being simple normal crossing. Then Riπ∗OX(KX+pHq) = 0 holds for any
i > d−inf(d,p).
Proof. It follows from Corollary 3.2 and a similar proof to that of [KMM87, Theorem
1-2-3].
The following vanishing result [KK, Corollary 2.2.5] is useful, which holds in arbi-
trary characteristic.
Lemma 3.4. Let f : Y → X be a proper birational morphism between normal surfaces
with Y smooth and with exceptional locus E = ∪s E . Let L be an integral divisor
i=1 i
on Y, 0 ≤ b ,··· ,b < 1 rational numbers, and N an f-nef Q-divisor on Y. Assume
1 s
L ≡ KY +Psi=1biEi+N. Then R1f∗OY(L) = 0 holds.
The following lemma is a generalization of [EL93, (1.2)], which holds in arbitrary
characteristic.
Lemma 3.5. Let X be a normal projective surface, f :Y → X a resolution, and H a
Q-Cartier Q-divisor on X. If H1(Y,K +pf∗Hq) = 0, then H1(X,K +pHq) = 0
Y X
holds.
Proof. Let E = ∪s E be the exceptional locus of f. Write H = pHq− r b D ,
i=1 i Pj=1 j j
where D are distinct prime divisors and 0 < b < 1. Write f∗D = f−1D +
j j j ∗ j
s q E , where f−1D is the strict transform of D and q ≥ 0. Thus we have
Pi=1 ji i ∗ j j ij
r s r
f∗H = f∗pHq−Xbjf∗−1Dj −X(Xbjqji)Ei, (7)
j=1 i=1 j=1
s r
pf∗Hq = f∗pHq−X(cid:2)Xbjqji(cid:3)Ei. (8)
i=1 j=1
8
By (7) and (8), we have
s r r
KY +pf∗Hq ≡ KY +X(cid:10)Xbjqji(cid:11)Ei +(f∗H +Xbjf∗−1Dj),
i=1 j=1 j=1
which implies R1f∗OY(KY +pf∗Hq) = 0 by Lemma 3.4.
By [KM98, Propositions 5.75 and 5.77], the trace map TraceX/Y : f∗OY(KY) →
O (K ) is an injective homomorphism, furthermore, it is an isomorphism over the
X X
points where f−1 is an isomorphism. It follows from (8) that we have the following
injective homomorphism, which is an isomorphism over the points where f−1 is an
isomorphism:
f∗OY(KY +pf∗Hq) → OX(KX +pHq),
hence the quotient sheaf Q of OX(KX +pHq) by f∗OY(KY +pf∗Hq) is supported on
a finite set of points. By the Leray spectral sequence, we have
H1(X,f∗OY(KY +pf∗Hq)) = H1(Y,KY +pf∗Hq) =0,
which together with H1(X,Q) = 0 implies H1(X,K +pHq) = 0.
X
Thereisageneralization of[Xie10c,Theorem1.4], wherenosingularityassumption
is made.
Theorem 3.6. Let X be a normal projective surface and H a nef and big Q-divisor
on X. If X is birational to a strongly liftable smooth projective surface Z, then
H1(X,K +pHq) = 0 holds.
X
∗
Proof. Take a resolution f : Y → X such that Supp({f H}) is simple normalcrossing.
Thus(Y,{f∗H})isKawamata logterminal(KLT,forshort),andK +pf∗Hq−(K +
Y Y
{f∗H}) = f∗H is nef and big. By Kodaira’s lemma, there exists an effective Q-divisor
B on Y such that (Y,{f∗H}+B) is KLT and K +pf∗Hq−(K +{f∗H}+B) =
Y Y
f∗H−B is ample. By [Xie10c, Theorem 4.2], we have H1(Y,K +pf∗Hq) = 0, which
Y
implies H1(X,K +pHq) = 0 by Lemma 3.5.
X
There is also a generalization of [Xie10b, Theorem 1.4].
Corollary 3.7. Let X be a normal projective rational surface and H a nef and big
Q-divisor on X. Then H1(X,K +pHq) = 0 holds.
X
Proof. It follows from [Xie10c, Theorem 1.3] and Theorem 3.6.
By means of Lemma 3.5, we can give an alternative proof of [Sa84, Theorem 5.1].
Corollary 3.8. Let X be a normal projective surface over an algebraically closed field
k with char(k) = 0, and H a nef and big Q-divisor on X. Then H1(X,K +pHq) =0
X
holds.
∗
Proof. Take a resolution f : Y → X such that Supp(hf Hi) is simple normal crossing.
Sincef∗H is nef and big, by [KMM87, Theorem 1-2-3], we have H1(Y,K +pf∗Hq) =
Y
0, which implies H1(X,K +pHq) = 0 by Lemma 3.5.
X
9
4 Cyclic cover trick over W (k)
2
The cyclic cover trick is a powerful technique, which is used widely in algebraic ge-
ometry. The general theory of cyclic covers over a field of arbitrary characteristic has
already been given in [EV92, §3]. In this section, we shall deduce the cyclic cover
trick over W (k) and use it to study the behavior of cyclic covers over strongly liftable
2
schemes.
Theorem 4.1. Let X be a smooth scheme, and L an invertible sheaf on X. Let N be
a positive integer prime to p, and D an effective divisor on X with LN = O (D).
X
iD
(i) Let A = N−1L−i( ). Then A is an O -algebra. Let Y = SpecA. Then
Li=0 (cid:2)N (cid:3) X
Y is normal and the natural projection π :Y → X is a finite surjective morphism
of degree N, which is called the cyclic cover obtained by taking the N-th root out
of D. Furthermore, if Sing(D )= ∅, then Y is smooth.
red
(ii) If X has a lifting X over W (k), L has a lifting L on X, and D has a lifting D on
2
X with LN = OXe(eD), then Y is liftable over W2e(k). Ien fact, we have an indeuced
e e e
cyclic cover π : Y → X, such that π is a lifting of π over W (k). Furthermore,
2
if Sing(D )e=e∅, theen Y is smooteh over W (k), and π : Y → X has similar
red 2
properties to those of π :eY → X, i.e. the statements ofe[EVe92, Cleaim 3.13 and
Lemma 3.15] hold for π : Y → X, where the phrase “nonsingular” should be
replaced by “smooth oveer We(k)”.e
2
Proof. (i) The construction and the properties of the cyclic cover π :Y → X described
as above have already been given in [EV92, §3].
iD
(ii) Let Ae= LiN=−01Le−i((cid:2)Ne(cid:3)). By using the homomorphism se: OXe → OXe(De) =
LeN, we can prove that Ae is an OXe-algebra. Let Ye = SpecAe. Then Ye ×SpecW2(k)
Speck = Y. Since A is a locally free sheaf on X and X is smooth over W (k) by
2
e e e
Lemma 2.2, Y is flat over W (k). Thus Y is a lifting of Y over W (k). Let π :Y → X
2 2
be the naturael projection. Then we haveethe following cartesian square: e e e
(cid:31)(cid:127) ι //
Y Y
e
π πe
(cid:15)(cid:15) (cid:31)(cid:127) ι // (cid:15)(cid:15)
X X,
e
which implies that π is a lifting of π over W (k).
2
If Sing(D ) =e∅, then Y is smooth. By Lemma 2.2, Y is smooth over W (k). In
red 2
e
fact, thesmoothnessof Y canalsobeprovedbyalocal calculation. SinceSing(D )=
red
e
∅, by Lemma2.2, theirreduciblecomponents of D aredisjointandsmooth over W (k).
2
e
Since the statements of [EV92, Claim 3.13 and Lemma 3.15] (replacing “nonsingular”
by “smooth over W (k)”) are local problems, we may assume that X = SpecB, D =
2
α D , X = SpecB and D = α D . We can factorize π : Y → X into two parts: one
1 1 1 1
is an ´eteale cover πe : Z →e X ofedegree gcd(N,α ), aned thee otheer is a ramified cover
1 1
π : Y → Z of deegreeeN/gced(N,α ), hence the properties of π : Y → X follow from
2 1
aen alemost eidentical local calculation to that of [EV92, Claim 3e.13]e. e
10