Calabi-Yau threefolds in positive characteristic

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by  Yukihide Takayama

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Calabi-Yau threefolds in positive characteristic
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Yukihide Takayama

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G Expanded lecture notes of a course given as part of XVI International Workshop for Young
A Mathematicians“AlgebraicGeometry”heldin Krakow,September 18– 24,2016.
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h Abstract. Inthisnote,anoverview ofCalabi-Yauvarieties inpositive characteristic ispresented. Although
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a Calabi-Yau varieties in characteristic zero are unobstructed, there are examples of Calabi-Yau threefolds
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in positive characteristic which cannot be lifted to characteristic zero, although one-dimensional and two-
[ dimensional Calabi-Yau varieties, i.e.,elliptic curves and K3surfaces, are allliftable tocharacteristic zero.

1 Inthisrespect, Calabi-Yauthreefoldsinpositivecharacteristic areinteresting inviewofdeformation theory
v andtheyarestillverymysterious.
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Keywords: Calabi-Yau variety,positivecharacteristic, projectiveliftingproblem
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1 2010MathematicsSubject Classification:14J32,14J28,14G17,14M20
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Contents
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1 Introduction 3
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2 Unique features ofCY 3-foldsinpositiv characteristic 3
2.1 HodgediamondwithoutHodgesymmetry . . . . . . . . . . . . . . . . . . . . . . 4
2.2 obstructeddeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 characteristic0 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Ellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.4 CY n-folds(n ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 HodgedecompositionandKodaira-Akizuki-Nakanovanishing . . . . . . . . . . . 7
2.4 Supersingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Prof. Dr. YukihideTakayama
DepartmentofMathematical Sciences, RitsumeikanUniversity
1-1-1Nojihigashi, Kusatsu,Shiga,525-8577, Japan
[email protected]
2 YukihideTakayama
3 Deformationtheory ofCY3-folds inpositivecharacteristic 9
3.1 InfinitesimalLifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 formal spectrumSpf(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 formal liftingviainfinitesimallifting . . . . . . . . . . . . . . . . . . . . 10
3.1.3 obstructiontoformal lifting . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.4 obstructionsto projectivelifting . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 W -liftabilityofordinaryCY n-folds . . . . . . . . . . . . . . . . . . . . . . . . . 12
2
3.2.1 Cartier operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 iterated Cartier operatorand quasiFrobenius splitting . . . . . . . . . . . . 14
3.2.3 Mehta-Srinivasdeformationtheory . . . . . . . . . . . . . . . . . . . . . 17
3.2.4 Proof ofTheorem26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Construction ofnon-liftable CY 3-folds 18
4.1 whatcauses non-liftability? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1 non-liftabilitybyb (X) = 0(dueto Hirokado) . . . . . . . . . . . . . . . 19
n
4.1.2 non-liftabilitybymodp reduction(dueto Cynk andvan Straten) . . . . . 19
4.2 construction(I): quotientbyfoliation . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 foliationinpositivecharacteristic . . . . . . . . . . . . . . . . . . . . . . 20
4.2.2 constructionofHirokado99variety . . . . . . . . . . . . . . . . . . . . . 22
4.3 construction(II): supersingularK3 penciloverP1 . . . . . . . . . . . . . . . . . . 23
4.3.1 constructionofSchröervarieties . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 construction(III): Schoen typeexamples . . . . . . . . . . . . . . . . . . . . . . . 24
4.4.1 examplesby Hirokado-Ito-Saito . . . . . . . . . . . . . . . . . . . . . . . 24
4.4.2 examplesby Cynk,van Straten and Schütt . . . . . . . . . . . . . . . . . 24
4.5 construction(IV): doublecoverofP3–examplesby Cynkand vanStraten . . . . . 24
4.6 Raynaud-Mukaiconstructioncannot produceCY 3-folds . . . . . . . . . . . . . . 25
Calabi-Yau threefolds inpositivecharacteristic 3
1. Introduction
Calabi-Yau varieties, in particular Calabi-Yau 3-folds, over complex numbers have been exten-
sively studied over the last decades mostly from the interest in mathematical physics. The most
notableproperty ofcomplexCalabi-Yau varietiesis thattheyareunobstructedindeformation.
However, for Calabi-Yau varieties of dimension ≥ 3 in positivecharacteristic, the situation is
quitedifferent.Inlate1990’s,M.HirokadofoundanexampleofCalabi-Yau3-foldincharacteristic
3thatcannotbeliftedtocharacteristic0[16].Afterthatotherexamplesofnon-liftableCalabi-Yau
3-fods have been found [17, 37, 18, 19]. They are all in characteristic p = 2 and p = 3. However,
D. van Straten, S.Cynk and M. Schütt [5, 6] found a large number of examples of non-liftable
Calabi-Yau algebraic spaces in characteristic p ≥ 5, which are not schemes anymore. It is still an
open problemwhetherthereexistnon-liftableCalabi-Yau varietiesincharacteristic≥ 5.
On the other hand, by the cerebrated theorem by P. Deligne and L. Illusie (and M. Raynaud)
[8], for a projective variety X over an algebraically closed field k of char(k) = p ≥ dimX, if
X can be lifted to the ring W (k) of second Witt vectors, Hodge-to-de Rham spectral sequence
2
degenerates at E and Kodaira-Akizuki-Nakano vanishing of cohomologies holds. We note that
1
W -liftability is not a necessary condition for Kodaira type vanishing and there are examples of
2
varietiesthatare W -non-liftablebut Kodairavanishingholds.
2
Thus, for non-liftable Calabi-Yau varieties, even if they are not liftable over the ring W(k) of
Witt vectors, which implies liftabilityto characteristic 0, it is an interesting questionwhether they
are liftable over W (k). The above mentioned Hirokado’s example and the Schröer’s examples
2
are known to be non-liftable over W (k) [10] and W -liftability is still open for other examples.
2 2
So Kodaira type vanishing for non-liftable Calabi-Yau varieties in positive characteristic is still a
mysteriousproblem.
Inthisnote,wegiveanoverviewoftheresearchonCalabi-Yau3-foldsinpositivecharacteristic
inthelastdecades. Theconfigurationofthisnoteisas follows.In section2,wesummarizebriefly
what is different from the geometry of Calabi-Yau varieties in characteristic 0. Among the unique
features of geometry in positiv characteristic, we will focus on obstructedness of deformation,
whichwewillelaborateinsection3.Finalsectionisdevotedtoconstructionofnon-liftableCalabi-
Yau 3-folds. We first summarize the known reasons for non-liftabilityand overview the examples
that have been found so far. We also mention what is known about Kodaira type vanishing for
non-liftableCalabi-Yau varieties.
2. Unique features of CY 3-folds in positiv characteristic
In this section, we overview unique features of Calabi-Yau 3-folds in positive characteristic as
comparedtocharacteristic0case.Someofthefeaturewillbeconsideredindetailinthesubsequent
sections.
Definition 1 (Calabi-Yau n-fold). A Calabi-Yau n-fold X is a smooth projective variety over an
algebraically closed field k such that Hi(X,O ) = 0 for 0 < i < n = dimX and the canonical
X
4 YukihideTakayama
sheaf is trivial ω ∼= O . In particular, a Calabi-Yau 1-fold is an elliptic curve and a Calabi-Yau
X X
2-foldis aK3surface.
In the following, we will mostly consider the case of n = 3 and char(k) = p > 0. Some
authors assume only properness instead of projectivity for Calabi-Yau varieties. But in this note
wewillalways assumeprojectivityforaCalabi-Yau variety.
2.1. Hodge diamond without Hodge symmetry
BySerreduality,weknowthattheHodgenumbershij := dim Hk(X,Ωi )ofaCalabi-Yau3-fold
k X
X areas follows:
h00 1
h10 h01 h10 0
h20 h11 h02 h20 h11 0
h30 h21 h12 h03 = 1 h12 h12 1
h31 h22 h13 0 h11 h20
h32 h23 0 h10
h33 1
In positivecharacteristic, Hodgesymmetry(hij = hji)does notholdingeneral and wehave
Proposition 2 (Hodge symmetry). For a Calabi-Yau 3-foldX, Hodgesymmetry holdsif and only
if h10 = h20 = 0, namely H0(X,Ω1 ) = H0(X,T ) = 0, where T = Hom (Ω1 ,O ) is the
X X X OX X X
tangentbundle.
Proof. Thelastpart uses theisomorphismΩ2 ∼= Ω3 ∧Ω−1 = O ∧Ω−1 ∼= T .
X X X X X X
2.2. obstructed deformation
2.2.1. characteristic 0case
Wefirst recall thenotionofuniversaldeformation.
Definition3 (deformation). LetX beacomplexmanifold.
1. A deformation of X is a smoot proper morphism ψ : X → (S,0), where X and S are
connected complexspaces and 0 ∈ S adistinguishedpoint,such thatX ∼= X := ψ−1(0),
0
2. A deformation X → (S,0) is called universal if any other deformation ψ′ : X′ → (S′,0′)
is isomorphic to the pull-back under a uniquely determined morphism ϕ : S′ −→ S with
ϕ(0′) = 0. Wedenoteauniversaldeformationby X −→ Def(X)
Theorem 4 (Bogomolov-Tian-Todorov). Let X be a Calabi-Yau manifold of any dimension over
algebraically closed field k of char(k) = 0. Then Def(X) is a germ of a smooth manifold with
tangentspaceH1(X,T ).
X
Calabi-Yau threefolds inpositivecharacteristic 5
Proof. By Lefschetz principle and GAGA, we may consider X as a compact complex Kähler
manifold.ThenweprovesmoothnessofDef(X)byacomplexanalyticmethod.Forthedetail,see
Theorem 14.10[11] andthereferences cited there.
Accordingtodeformationtheory,obstructioniscontainedinH2(X,T ).SoifH2(X,T ) = 0
X X
we can say that X is unobstructed. But Theorem 4 claims that even if H2(X,T ) 6= 0, which is
X
actuallypossibleindimension≥ 3, itselementsare notan obstructiontodeformation.
Remark5. Inthealgebraicsetting,Theorem4meansthatanydeformationofX isunobstructedin
the sense that for any small extension ϕ : B −→ A, namely, for any surjective homomorphismϕ
of local Artinian C-algebras with m Kerϕ = 0, any variety X over Spec(A) can be lifted over
B A
SpecB.
2.2.2. Ellipticcurves
For curves, we have H2(X,T ) = 0 for the dimensional reason so that smooth projective curves
X
inany characteristicareunobstructed.In particularellipticcurvesare unobstructed.
2.2.3. K3surfaces
ForaK3 surfaceX, wehaveH2(X,T ) = 0 bythefollowingtheorem,sothat itisunobstructed.
X
Theorem6(Deligne[7]). LetX beaK3surfaceoveranalgebraicallyclosedfieldk ofchar(k) =
p > 0. Then
1. Hodgeto deRhamspectralsequence
Epq = Hq(X,Ωp ) ⇒ Hp+q(X/k)
1 X DR
degeneratesatE . Inparticular,we havetheHodgedecomposition
1
k
Hk (X/k) ∼= Hk−i(X,Ωi )
DR X
i=0
X
2. theHodgediamondis
h00 1
h10 h01 0 0
h20 h11 h02 = 1 20 1
0 h12 0 0
h22 1
6 YukihideTakayama
Unobstructedness H2(X,T ) = 0 in Theorem 6 implies that a K3 surface over k can be
X
formallyliftedovertheringW(k)ofWittvectors,i.e.,thereexistsaformalschemeXˆ overW(k)
suchthatXˆ× k ∼= X.Moreover,dim H1(X,T ) = 20meansthatthemodulispaceofformal
W(k) k X
K3 surfaces is20 dimensional.
Ontheotherhand,ifwewantanalgebraicliftingofaK3surface,thesituationisalittlesubtler.
Projectivelifting problem: Let X be a projective variety over a field k of positive
characteristic. Then, find a projective scheme X over S = SpecR, where R is a ring
ofmixedcharacteristic, such thatX ∼= X × k.
S
Recall that a ring R of mixed characteristic means an integral domain in char(R) = 0 with a
maximalidealm ⊂ Rsatisfyingchar(R/m) > 0.Here,wecanconsiderR = W(k),forexample.
In this case, we also need to lift an ample line bundle on X to the formal lifting Xˆ and apply
Grothendieck’salgebraizationtheorem(seeThéor`eme(5.4.5)[12]orTheorem21.2[14])toobtain
aprojectiveschemeX˜ whoseformal completionat theclosedfiberX is Xˆ.
The obstruction to lifting an invertible sheaf is contained in H2(X,O ) (see Theorem 24 be-
X
low) which is 1-dimensionalfor a K3 surface. Thus, the modulispace of algebraicK3 surfaces is
smallerby onedimension,namely 19dimensional.
ProjectiveliftingproblemforK3surfaces hasbeen solvedexceptthecasep = 2.
Theorem 7 (Ogus [31]). Forp > 2, a K3 surfaceX canbeprojectivelyliftedover W(k).
Proof. By Corollary 2.3 [31], a K3 surface X can be lifted overW(k) if X is not “superspecial”.
By Remark 2.3 [31], if Tateconjecture for smoothpropersurfaces [45] holds,theonly “superspe-
cial”K3surfaceistheKummersurfaceassociatedtoaproductofsupersingularellipticcurvesand
we can show that this can be lifted over W(k). Finally, Tate’s conjecture has been established for
p ≥ 3by severalauthors.
See,forexample[24],forthedetailofdeformationtheoryofK3surfacesinpositivecharacter-
istic.
2.2.4. CY n-folds(n ≥ 3)
Apart from the cases of char(k) = 0 or char(k) = p > 0 with dim ≤ 2, which we have seen so
far, thesituationis quitedifferentfordim ≥ 3.Namely,
Theorem 8 (Hirokado, Schröer, Ito, Saito, Ekedahl, Cynk, van Straten, Schütt). There are exam-
ples of Calabi-Yau 3-folds over an algebraically closed field k of characteristic p = 2,3, that
cannotbe(formally)liftedtocharacteristic0.
Question:Aretherenon-liftableCalabi-Yau 3-fold inp ≥ 5?
By the time when the author writes this article, we only know non-liftable 3-dimensional
Calabi-Yau spaces, (i.e., algebraic spaces which are not schemes) in the case p ≥ 5. Moreover,
whetherthereexistnon-liftableCY n-foldsforn ≥ 4 isunclear.
Calabi-Yau threefolds inpositivecharacteristic 7
2.3. Hodge decomposition and Kodaira-Akizuki-Nakano vanishing
Let
F : X −→ X(p)
be a relative Frobenius morphism. For a complex L• and n ∈ Z we denote by T• := τ L• the
<n
complexsuchthat
Li fori ≤ n−2
Ti = Ker(d : Ln−1 −→ Ln) fori = n−1

0 fori ≥ n

Also,we denoteby W thering ofsecond Wittvectors W(k)/p2W(k), whereW(k) is thering of
2 
Wittvectorsoverk.
Theorem 9 (Deligne-Illusie [8]). Let k be a perfect field of char(k) = p > 0 and X a smooth
schemeover k.If X isliftableoverW , thenwe have
2
∼
ϕ : Ωi [−i] −=→ τ F Ω•
X(p)/k <p ∗ X/S
i<p
M
which is an isomorphism in the derived category D(X(p)) of O -modules with F action such
X(p)
that
Hiϕ = C−1 : Ωi −→ HiF Ω•
X(p)/k ∗ X/k
i i
M M
fori < p,where C istheCartieroperator.
From thistheorem,weobtainthefollowingconsequences.
Corollary 10 (Hodge decomposition). Let X be a smooth proper scheme over a perfect field k of
char(k) = p > 0anddim ≤ p.IfX isliftableoverW ,thenHodgetodeRhamspectralsequence
2
Epq = Hq(X,Ωp) ⇒ Hp+q(X/k)
1 DR
degeneratesatE . In particular,we haveHodgedecomposition:
1
n
Hn (X/k) ∼= Hn−i(X,Ωi)
DR
i=0
M
forn ∈ Z.
Corollary 11 (Kodaira-Akizuki-Nakano vanishing). Let X be a smooth projective scheme over a
perfectfieldk ofchar(k) = p > 0andLanamplelinebundleonX.IfX isliftableoverW ,then
2
we have
Hj(X,Ωi ⊗L−1) = 0 fori+j < inf(dimX,p).
In particular,ifdimX ≤ p, wehaveKodairavanishingHi(X,L−1) = 0for i < dimX.
ForCalabi-Yau 3-folds,thefollowingquestionis widelyopen,apart from partialanswers.
Question: For a non-liftable Calabi-Yau 3-fold X, is it liftable over W ? If not, does
2
Kodairavanishinghold?
8 YukihideTakayama
2.4. Supersingularity
ForaCalabi-Yau n-fold X (n ≥ 2), Artin-Mazurfunctor[2]
Φn : Art −→ Abgr
X
from thecategory ofArtinianlocalringsArt to thecategory ofabeliangroupsAbgr is defined by
Φn(S) := Ker(Hn(X × S,G ) −→ Hn(X,G ))
X et k m et m
and thisispro-representableby a1-dimensionalformalgroupschemeM. Namely,we have
Φn (−) = Hom(−,M).
X
It is known that a 1-dimensional formal group scheme M is the formal additive group scheme
Gˆ or a p-divisible formal group scheme. The group operation of a formal group scheme can be
a
described by a formal group law F(X,Y) ∈ k[[X,Y]] and we can define the height of the formal
group law. By definition, height h(X) of a Calabi-Yau variety X is the heightof theformal group
law.See[15]forthedetailofformal grouplawforthegroupschemeM.
The height ht(X) has convenient description in terms of Serre cohomologies H∗(X,WO )
X
[38] and crystalline cohomologies H∗ (X/W) [3, 4], from which we deduce characterization of
cris
supersingularCalabi-Yau varieties.
We willdenoteby K thequotientfield ofW := W(k).
Proposition12. ForaCalabi-YauvarietyX ofdimensionn = dimX overanalgebraicallyclosed
fieldk, we have
ht(X)
dim Hn(X,WO )⊗ K
K X W
= dim (Hn (X/W)⊗ K) (< ∞) ifHn(X,WO )⊗ K 6= 0
=  K cris W [0,1) X W


 ∞ ifHn(X,WO )⊗ K = 0
X W
where (−) denotes theK-vector subspaceofslopesbetween 0 and1.
[0,1)
Definition13(ordinary/supersingular). LetX beaCalabi-YauvarietyofdimX = n(≥ 2).Then,
wecall X isordinaryifh(X) < ∞and supersingularifh(X) = ∞.
Proposition14 (cf.[1]). If n = 2,i.e., X isa K3 surface,we have1 ≤ h(X) ≤ 10 or h(X) = ∞.
Thesupersingularcaseh(X) = ∞exists onlyinpositivecharacteristic.
OnonlyK3surfacesbutalsoforhigherdimensionalCalabi-Yauvarieties,supersingularityhas
closerelationwithuniquefeatureofgeometry inpositivecharacteristic.
ForaCalabi-Yau varietyX incharacteristic0, theBettinumberb (X),n = dimX,can never
n
be trivial since by Hodge decomposition b (X) ≥ dim H0(X,Ωn) = dim H0(X,O ) = 1. In
n k X k X
positivecharacteristic, weconsidertheétaleBetti numbers
b (X) := dim Hi (X,Q ) := limHi (X,Z/ℓrZ)⊗ Q
i Qℓ et ℓ ←− et Zℓ ℓ
foraprimenumberℓ(6= p) (seeforexample[27]).
Calabi-Yau threefolds inpositivecharacteristic 9
Proposition15(cf.Prop. 8.1[37]). Calabi-Yaun-foldwith b (X) = 0 issupersingular.
n
Calabi-Yau 3-folds with trivial3rd Betti numberdo exist as will be presented in section 4. We
notethattheconverseofProp. 15does not holdsincewehaveb (X) = 22 forasupersingularK3
2
surface X.
Recall that a variety X is called uniruled (or unirational) if there exists a dominant rational
morphismϕ : Y ×P1 → X (orϕ : Pn → X)forsomevarietyY.Incharacteristic0,aCalabi-Yau
varietycannot beuniruledsinceotherwiseω cannot betirivialbecauseofthefollowingfact:
X
Theorem 16 ([28]). Let X be a smooth projectivevarietyover C. Then X is unuruledif and only
if there exists a non-empty open subset U ⊂ X such that for all x ∈ U there exists an irreducible
curveC throughxwith (K ,C) < 0.
X
However,uniruledCalabi-Yau 3-foldsdo existas willbepresented insection4.
Proposition 17 (Theorem 1.3 [16], Theorem 3.1 [17]). A uniruled Calabi-Yau n-fold X is super-
singular.
TheconversetoProp. 17isopen forn ≥ 3. Forn = 2,C. Liedtkeshowedthatasupersingular
K3 surfaceis unirational[23].
Question: Is there any relation between non-liftability to characteristic 0 and super-
singularity?
3. Deformation theory of CY 3-folds in positive characteristic
3.1. Infinitesimal Lifting
Let X be a smooth projective variety over a perfect field k of char(k) = p > 0 and (A,m) a
complete Noetherian local domain of mixed characteristic, i.e., A = limA/mn, char(A) = 0 and
←−
k = A/m. We denotethequotientfield ofAby K.
Example 18. A typical situation we have mostly in mind is that A is the ring of Witt vectors
(W(k),pW(k))overk orfiniteextensionofW(k).
3.1.1. formalspectrum Spf(A)
Set A := A/mn+1 for n ≥ 0. They are Artinian local rings and in particular A = k. Also set
n 0
S := SpecA . Then we havean increasing sequence of infinitesimalneighborhoodsS ⊂ S ,
n n n n+1
butalltheS havethesameunderlyingspace,whichwedenotebySpf(A).Wedefineitsstructure
n
sheafas
O := limO
Spf(A) ←− Sn
WehaveΓ(Spf(A),O ) = A.
S
10 YukihideTakayama
3.1.2. formalliftingvia infinitesimal lifting
AliftX ofX overAisaschemeflatoverA,X → Spec(A),suchthatX ∼= X× k.Itsclosedfiber
A
(or special fiber) X in characteristic char(k) = p > 0 is lifted to thegeneric fiber X := X × K
η A
inchar(K) = 0.
X ֒→ X ←֓ X
η
↓ ↓ ↓
Spec(k) ֒→ Spec(A) ←֓ Spec(K)
One way to construct such a lifting X is infinitesimal lifting of X. Consider the short exact
sequence:
0 −→ mn/mn+1 −→ A −→ A −→ 0 (1)
n n−1
We note that mn/mn+1 ∼= m ⊗ k is a finite k-vector space. Infinitesimal lifting is to try to lift X
overA = k toaschemeX flatoverA = A/m2,thenliftX toaschemeX flat overA andso
0 1 1 1 2 2
ontoobtainafamily{X } ofschemessuchthatX ⊗ A = X (n ≥ 1).Suchafamily
n n≥0 n An n−1 n−1
{X } is called aformalfamilyofdeformationsof X over A. From thisfamily,we obtainaformal
n
lifting:
Proposition19. Givensuchafamily{X } ,weobtainaNoetherianformalschemeX˜ flatover
n n≥0
Spf(A) suchthatX ∼= X˜ × A for all0 ≤ n ∈ Z.
n A n
Proof. DefineX˜ tobethelocallyringedspaceformedbytakingthetopologicalspaceX together
0
withthesheafofringsO := limO .See,forexample,Prop.21.1[14]fortherestofproof.
X˜ ←− Xn
The formal scheme X˜ as in Prop. 19 is called a formal lifting of X. A formal lifting scheme
is locally isomorphicto the completion of the closed fiber, but it is not always so globally. On the
other hand, we say that X can be projectively lifted over the field K of characteristic 0 if there
existsaprojectiveschemeX flat overAwithX ∼= X ⊗ k.
A
Then wehavequestions:givenaprojectivevarietyX overk,
Q1: when do wehaveaformal liftingX˜?
Q2: given a formal lifting X˜, when do we have a projective lifting X, whose completion along
theclosedfiberis X˜?
3.1.3. obstruction to formallifting
Definition20(torsor/principalhomogeneous-space). SupposethatagroupGactsonanon-empty
set S. Then S is called a torsor or a principal homogeneous space under the action of G if there
existsone(and henceall)elements ∈ S such thatG ∼= S viag 7→ g(s )
0 0
Now the answer to the question Q1 is that if H2(X,T ) = 0 we always have a formal lifting
X
ofX: