Deformations and elements of deformation theory

0.09 MB·

by  Nikolaj Glazunov

Deformations and elements of deformation theory
NIKOLAJGLAZUNOV
NationalAviationUniversity
6 DepartmentofElectronics
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1CosmonautKomarovAvenue,03680Kiev
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UKRAINE
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[email protected]
n
a
J Abstract: This article consisted of an elementary introduction to deformation theory of varieties, schemes and manifolds,
8 with some applications to local and global shtukas and fever to Newton polygons of p-divisible groups . Soft problems
and results mainly are considered. In the framework we give review of some novelresults in the theory of local shtukas,

] Anderson-modules,globalshtukas, Newton polygonsofp-divisiblegroupsandon deformationsof p-divisiblegroupswith
T
givenNewtonpolygons
N
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h Key–Words: Dualnumber,infinitesimaldeformation,Drinfeldmodule,localAnderson-module,localshtuka,globalshtuka,
at modulistack,formalLiegroup,cotangentcomplex,commutativegroupscheme,uniformization,rigidity,Newtonpolygon,
m loopgroupofareductivegroup
[

1 Introduction
1
v
1 This article consisted of an elementaryintroductionto deformationtheory of varieties, schemes and manifolds, with some
9 applicationstolocalandglobalshtukasandfevertoNewtonpolygonsofp-divisiblegroups. Fromthepointofviewofrigid,
9
hardandsoftproblemsandresultsinthispaperweconsidermainlysoftproblemsandresults. Intheframeworkwereview
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some novel results and methods in the theory of local shtukas, Anderson-modules, global shtukas and Newton polygons
0
. of p-divisible groups. These include (but not exhaust) methods and results by V. Drinfeld[1], U. Hartl, E. Viehmann [5],
1
R.Singh[7],A.Rad[9],S.Harashita[6]andothers.M.Gromov[8]inhistalkattheInternationalCongressofMathematicians
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6 in Berkeley have presented problemsand results of soft and hard symplectic geometry. In this connectionwe will present
1 somesoftproblemsandresultsindynamicsandinarithmeticgeometry. ”Soft”problemsandresultsinourtalkarelimited
v: to the frameworkof deformations, infinitesimal deformations, and elements of local Anderson-modules,local shtukas and
i globalshtukas. Review of some novelresults and methodson rigidity in arithmetic geometryand in dynamicsis givenin
X
author’spapers[30,31].
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a
2 Infinitesimal neighborhoods and infinitesimal deformations
Dualnumberswere introducedandwere usedinW. Clifford[13] , E. Study[14], R. von-Mises[15]. Expandfollowingto
[12,11]thedefinitionofdualnumberssothatitwastrueoveranyfield. Letk beafield,andk[ǫ]-theringofpolynomials
overkinthevariableǫ. Factoringk[ǫ]bytheideal(ǫ2)togivethedesiredringofdualnumbersD overk : D = k[ǫ]/(ǫ2).
Like the classical ring of dual numbers, ring D is a nilpotent ring. Further, unless specifically stated, we are under dual
numbersmeanelementsoftheabove-definedringsD. LetSpecDbethecorrespondingaffinescheme. Theschemehasone
geometricpointwhichcorrespondstothemaximalideal(ǫ). Itsstructuresheafhasnilpotentelementsthatdistinguishesthe
schemefromclassicalalgebraicvarieties.
2.1 Infinitesimal neighborhoods
Let X be a scheme over an algebraicallyclosed field k. Here and in the followingsubsection we mean under a pointof a
scheme its geometric(closed) point, unless otherwise stated. We fix, followingto [11]], notations: o - closed point of the
schemeSpeck,o-closedpointschemeSpecD,i:Speck→SpecD-canonicalembedding,underwhicho=i(o).
Lemma1. Anymorphismφ:SpecD→X definesamorphismφ◦i(o):Speck→X,wherex=φ◦i(o)isaclosedpoint
ofX.
Proof.ThehomomorphismD →kwithkernel(ǫ)definesthecanonicalembeddingi,andthemorphismφ◦i(o)defines
aclosedpointx∈X.
LetU beanaffineneighborhoodofthepointx, m bethemaximalidealofthepointxink[U].
x
Lemma 2. Let M be the set of morphisms of schemes SpecD → X such that φ(o) = x. Then M (SpecD,X) =
x x
M (SpecD,U).
x
Theprooffollowsfromlocalpropertiesofschemes.
Definition3. TheschemeT =Speck[U]/m2 iscalledtheinfinitesimalneighborhoodofthefirstorderofthepointx.
x x
Remark4. ThehomomorphismSpeck[U] → Speck[U]/m2 definestheclosureimbeddingT → U andT istheclosed
x x x
subschemeinU.
Usingthesestatementswehave
Proposition5. MorphismsSpecD→X thattransformSpeckinx∈X areinone-to-onecorrespondencewithmorphisms
SpecD→T .
x
2.2 Infinitesimal deformations
LetX →Y beamorphismofschemes. AschemeX isflatoverY ifthesheafO isflatoverO [12,11].
X Y
Definition6. LetX,T beschemesandX →T theflatmorphismwithfixedpointt∈T suchthatX ≃X .Intheconditions
t o
theschemeX iscalledthe(global)deformationoftheschemeX .
o
Definition7. LetX betheschemeoffinitetypeoverfieldk andD theringofdualnumbersoverk.Intheconditionsthe
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′ ′
schemeX whichisflatoverDandsuchthatX ⊗ k ≃X iscalledtheinfinitesimaldeformationoftheschemeX .
D o o
Proposition8. GivenaglobaldeformationoftheschemeX ,thenthereexistsaninfinitesimaldeformationofthescheme
o
X .
o
Proof. ByLemma1thereexiststhemorphismSpecD → T whichisdefinedbysomeelementofthetangentspaceto
′ ′
T inpointt. HencethereisaschemeX flatoverD withtheclosedfiberX sothatX ⊗ k ≃ X . Itgivestherequired
o D o
infinitesimaldeformation.
3 On Local shtukas and divisible local Anderson-modules
In their paper [5] U. Hartl, E. Viehmann have investigated deformationsand moduli spaces of bounded local G−shtukas.
Latest(boundedlocalG−shtukas)arefunctionfieldanalogsforp−divisiblegroupswithextrastructure.
Theauthor[7]investigatesrelationbetweenfiniteshtukasandstrictfiniteflatcommutativegroupschemesandrelation
between divisible local Anderson modules and formal Lie groups. Let NilpFq[[ξ]] be the category of Fq[[ξ]]-schemes on
whichξ islocallynilpotent. LetS ∈ NilpFq[[ξ]]. ThemainresultofthisdissertationbyR. Singhisthe following(section
2.5)interestingresult: itispossibletoassociateaformalLiegrouptoanyz-divisiblelocalAndersonmoduleoverS inthe
casewhenξislocallynilpotentonS.
AgeneralframeworkforthedissertationisthedecenttheorybyA.Grothendieck[3]. andhiscolleagues,itsextensions
andspecializationstofinitecharacteristics.
In Chapter 1 the author of the dissertation [7] defines cotangent complexes as in papers by S. Lichtenbaum and M.
Schlessinger[16],byW.Messing[17],byV.Abrashkin[18]andprovethattheyarehomotopicallyequivalent.
Moregenerallytoanymorphismf : A → B ofcommutativeringobjectsinatoposisassociatedacotangentcomplex
L and to any morphism of commutativering objects in a topos of finite and locally free Spec(A)-group scheme G is
B/A
associatedacotangentcomplexL ashaspresentedinbooksbyL.Illusie[19].
G/Spec(A)
In section 1.5 the author of the dissertation [7] investigates the deformations of affine group schemes follow to the
mentionedpaperofAbrashkinanddefinesstrictfiniteO−moduleschemes.Nextsectionisdevotedtorelationbetweenfinite
shtukasbyV.Drinfeld[1]andstrictfiniteflatcommutativegroupschemes. Thecomparisonbetweencotangentcomplexand
FrobeniusmapoffiniteF -shtukasisgiveninsection1.7.
p
z−divisiblelocalAndersonmodulesbyU.Hartl[21]andlocalschtukasareinvestigatedinChapter2.
Sections2.1,2.2and2.3onformalLiegroups,localshtukasanddivisiblelocalAnderson-modulesdefineandillustrate
notionsforlateruse. Manyofthese,ifnotnew,aresetinanewform,
InSection2.4theequivalencebetweenthecategoryofeffectivelocalshtukasoverSandthecategoryofz-divisiblelocal
AndersonmodulesoverS istreated.
InthelastsectionthetheoremaboutcanonicalF [[z]]-isomorphismofz-adicTate-moduleofz-divisiblelocalAnderson
q
moduleGofrankroverS andTatemoduleoflocalshtukaoverS associatedtoGisgiven.
4 On uniformizing the moduli stacks of global G-shtukas
ThedissertationbyArastehRad [9]isaPh.D.Thesis, writtenunderU.Hartl(Mu¨nster). Thedissertationisdevotedtothe
developmentof the theoryof local P-shtukas with the aim of their relation to the modulistack of globalG-shtukas. Here
PisaparaholicBruhat-TitsgroupschemebyPappas,Rapoport[23]andGisaparahoricBruhat-Titsgroupschemeovera
smoothprojectivecurveoverfinitefieldF withqelementsofcharacteristicp.
q
LetCbeasmoothprojectivegeometricallyirreduciblecurveoverF .AglobalG-shtukaGoveranF -schemeSisatuple
q q
(G,s1,...,sn,τ)consistingofaG-torsorGoverCS :=C×FqS,ann-tupleof(characteristic)sections(s1,...,sn)∈Cn(S)
andaFrobeniusconnectionτ definedoutsidethegraphsofthesectionss byHartl,Rad[22].
i
LocalG-shtukasbyHartl, Viehmann[5]andbyViehmann[10]aregeneralizationstoarbitraryreductivegroupsofthe
localanalogueofDrinfeldshtukas.
DrinfeldShtukas(thespaceFSh ofF-sheaves)wasconsideredbyDrinfeld[1]andbyLafforgue[4].
D,r
For more results concerning local shtukas and Anderson-modulessee dissertation by Singh [7] written also under U.
Hartl.
The mainresultsof the dissertation[9] are the following. The analogueof the Serre-Tatetheoremoverfunctionfields
thatrelatingthedeformationtheoryofglobalG-shtukastothedeformationtheoryoftheassociatedlocalP -shtukasviathe
ν
global-localfunctor (Theorem 4.1). Representablity of the Rappoport-Zinkfunctor (Theorem 6.3.1.). The uniformization
theoremfromSection7. Finally,thediscussionaboutuniformizationandlocalmodelofthemoduliofglobalG-shtukasare
given.
5 On the supremum of Newton polygons of p-divisible groups with a given p-
kernel type
The author of the paper [6] provesthe existence of the supremum of Newton polygonsof p-divisible groupswith a given
p-kerneltype and providesan algorithm determining it. The main results of the paper [6] are the following Theorem1.1.
ξ(w)isthebiggestoneoftheNewtonpolygonsξ withµ(ξ)⊂w.,andCorollary2.2. ThereexiststhesupremumofNewton
polygonsofp-divisiblegroupswiththegivenpm-kerneltype.
Letkbeanalgebraicallyclosedfieldofcharacteristicp> 0,canddbenon-negativeintegerswithr := c+d> 0. Let
W betheWeylgroupofthegenerallineargroupGL ,s ∈W thesimplereflection,S ={s ,...s },J :=S\{s }and
r i 1 r−1 α
letwbeanyelementoftheset(J,∅)-reducedelementsofW byN.Bourbaki[20]. Thetheoremisanunpolarizedanalogue
ofCorollaryIIbyHarashita[24].
Inthepolarizedcase,theexistenceofthesupremumξ(w)followsfromtheresultsbyEkedahlandvanderGeer[25]. In
thecasethereisagoodmodulispaceA ofprinciplepolarizedabelianvarieties. Intheunpolarizedcasethereisnoagood
g
modulispacelikeA .
g
ThedifferenceoftheauthormethodincomparisonwiththeEkedahl-vanderGeerapproachistheusingofT -action
m
by Vasiu [26] which gives that the set of T -orbits is naturally bijective to the set of isomorphism classes of truncated
m
Barsotti-Tategroupsoflevelmoverkwithcodimensioncanddimensiond.
6 On the Newton strata in the loop group of a reductive group
Theauthorofthepaper[10]generalizespurityoftheNewtonstratificationtopurityforasinglebreakpointoftheNewton
pointin the contextof local G-shtukasrespectively of elementsof the loop groupof a reductivegroup. As an application
sheprovesthatelementsoftheloopgroupboundedbyagivendominantcoweightsatisfyageneralizationofGrothendieck‘s
conjectureondeformationsofp-divisiblegroupswithgivenNewtonpolygons.
LetGbeasplitconnectedreductivegroupoverF ,letT beasplitmaximaltorusofGandletLGbetheloopgroupof
p
GbyFaltings[2].
Let R be a F -algebraand K be the sub-groupscheme of LG with K(R) = G(R[[z]]). Let σ be the Frobeniusof k
q
overF and also of k((z)) overF ((z)). For algebraicallyclosed k, the set of σ-conjugacyclasses [b] = {g−1bσ(g)|g ∈
q q
G(k((z)))}ofelementsb∈LG(k)isclassifiedbytwoinvariants,theKottwitzpointκ (b)andtheNewtonpointν.
G
Theauthorofthepaper[10]provesthefollowingtwomainresults. Theorem1: LetS beanintegrallocallynoetherian
schemeandletb∈LG(S). Letj ∈J(ν)beabreakpointoftheNewtonpointν ofbatthegenericpointofS. LetU bethe
j
opensubschemeofS definedbytheconditionthatapointxofS liesinU ifandonlyifpr (ν (x))=pr (ν). ThenU
j (j) b (j) j
isanaffineS-scheme.
Theorem2: Letµ (cid:22) µ ∈ X (T)bedominantcoweights. LetS = S Kzµ‘K. Let[b]beaσ-conjugacy
1 2 ∗ µ1,µ2 µ1(cid:22)µ‘(cid:22)µ2
class with κ (b) = µ = µ as elements of π (G) and with ν (cid:22) µ . Then the Newton stratum N = [b]∩S is
G 1 2 1 b 2 b µ1,µ2
non-emptyandpureofcodimensionhρ,µ2−νbi+ 12def(b)inSµ1,µ2. TheclosureofNbistheunionofallNb‘ for[b‘]with
κG(b‘)=µ1andνb‘ <νb.
Here ρ is the half-sum of the positive roots of G and the defect def(b) is defined as rkG− rkFqJb where Jb is the
reductivegroupoverF withJ (k((z)))={g ∈LG(k)|gb=bσ(g)}foreveryfieldkcontainingF andwithalgebraically
q b q
closedk.
TheproofofTheorem1isbasedonageneralizationofsomeresultsbyVasiu[27].
AninterestingfeatureofhermethodintheproveofTheorem2istheusingofvariousresultsontheNewtonstratification
on loop groups as Theorem 1 and the dimension formula for affine Deligne-Lusztig varieties by Go˝rtz, Haines, Kottwitz,
Reuman[28]togetherwithresultsonlengthsofchainsofNewtonpointsbyChai[29].
7 Conclusion
Deformationsandelementsofdeformationtheoryofmanifolds,varietiesandschemeshavepresented. Intheframeworkwe
have reviewed some novelresults and methodsin the theory of local shtukas, Anderson-modules, globalshtukas, Newton
polygonsofp-divisiblegroupsandondeformationsofp-divisiblegroupswithgivenNewtonpolygons.Inthisconnectionwe
havepresentedsomesoftproblemsandresultsindynamicsandinarithmeticgeometry. ”Soft”problemsandresultsinour
considerationsarelimitedtotheframeworkofdeformations,infinitesimaldeformations,elementsoflocalAnderson-modules,
localshtukas,globalshtukasanddeformationsofp-divisiblegroupswithgivenNewtonpolygons.
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