Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties

65 Pages·2017·0.41 MB·English

J.Eur.Math.Soc. 19,659–723 (cid:13)c EuropeanMathematicalSociety2017
DOI10.4171/JEMS/678
SebastianCasalaina-Martin·SamuelGrushevsky
KlausHulek·RaduLaza
Extending the Prym map to toroidal
compactifications of the moduli space of abelian varieties
(withanappendixbyMathieuDutourSikiric´)
ReceivedMay1,2014andinrevisedformDecember15,2014
Abstract. Themainpurposeofthispaperistopresentaconceptualapproachtounderstandingthe
extensionofthePrymmapfromthespaceofadmissibledoublecoversofstablecurvestodifferent
toroidalcompactificationsofthemodulispaceofprincipallypolarizedabelianvarieties.Bysepa-
ratingthecombinatorialproblemsfromthegeometricaspectswecanreducethistothecomputation
ofcertainmonodromycones.Inthiswaywenotonlyshednewlightontheextensionresultsof
Alexeev,Birkenhake,Hulek,andVologodskyforthesecondVoronoitoroidalcompactification,but
wealsoapplythistoothertoroidalcompactifications,inparticulartheperfectconecompactifica-
tion,forwhichweobtainacombinatorialcharacterizationoftheindeterminacylocus,aswellasa
geometricdescriptionuptocodimensionsix,andanexplicittoroidalresolutionofthePrymmap
uptocodimensionfour.
Keywords. Moduli,Prymvarieties,periodmaps,abelianvarieties
Introduction
AfundamentaltoolinthestudyofalgebraiccurvesisthetheoryofJacobians.Assigning
to a curve its principally polarized Jacobian defines the Torelli period map M → A
g g
from the coarse moduli space of curves of genus g to the coarse moduli space of
principally polarized abelian varieties (ppav) of dimension g. It is a well known fact,
due to Mumford and Namikawa [Nam80], that the Torelli map extends to a morphism
M → A¯V fromtheDeligne–MumfordcompactificationtothesecondVoronoitoroidal
g g
S.Casalaina-Martin:DepartmentofMathematics,UniversityofColorado,
Boulder,CO80309,USA;e-mail:[email protected]
S.Grushevsky:DepartmentofMathematics,StonyBrookUniversity,
StonyBrook,NY11794,USA;e-mail:[email protected]
K.Hulek:Institutfu¨rAlgebraischeGeometrie,LeibnizUniversita¨tHannover,
30060Hannover,Germany;e-mail:[email protected]
R.Laza:DepartmentofMathematics,StonyBrookUniversity,StonyBrook,NY11794,USA;
e-mail:[email protected]
M.DutourSikiric´:RudjerBoskovic´Institute,Bijenicka54,10000Zagreb,Croatia;
e-mail:[email protected]
MathematicsSubjectClassification(2010):Primary14H40,14K10,14H10
660 SebastianCasalaina-Martinetal.
compactification.Morerecently,AlexeevandBrunyate[AB12]havestudiedextensions
oftheTorellimaptoothertoroidalcompactificationsandhaveshownthattheperiodmap
extendstoamorphismtotheperfectconecompactificationA¯P,butnottoamorphismto
g
thecentralconecompactificationA¯C forg ≥9,disprovingaconjectureofNamikawa.
g
WhiletheTorellimapisinjectiveforallg,forg ≥4itisnotdominant.Onegeometric
approach to understanding higher-dimensional ppavs is via Prym varieties, which are
ppavs associated to connected e´tale double covers of curves. Associating to a cover its
principally polarized Prym variety defines the Prym period map Rg+1 → Ag, where
Rg+1 is the coarse moduli space of connected e´tale double covers of curves of genus
g + 1. The Prym period map is dominant for g ≤ 5, and has been used to provide a
geometric approach to the Schottky problem for g = 4,5, to study the rationality of
threefolds,andtogiveabetterunderstandingofthegeometryofA andA .
4 5
IncontrasttothecaseofJacobians,ithasbeenknownsincetheworkofFriedmanand
Smith[FS86]thatthePrymperiodmapdoesnotextendtoamorphismfromBeauville’s
modulispaceRg+1ofadmissibledoublecoverstoanyofthestandardtoroidalcompacti-
fications.SubsequentworkofAlexeev,Birkenhake,andHulek[ABH02]andVologodsky
[Vol02] identifies the indeterminacy locus of the rational map Rg+1 (cid:57)(cid:57)(cid:75) A¯Vg; it is the
closureofthelocusofso-calledFriedman–Smithcoverswithatleastfournodes(see§6).
In this paper, we investigate the problem of extending the Prym map to other toroidal
compactifications.Ourmainresultsare:
• A complete combinatorial characterization of the indeterminacy locus of the Prym
maptotheperfectandcentralconecompactifications(Theorem5.6).Thetechniques
alsogiveacompletecombinatorialcharacterizationoftheindeterminacylocusofthe
PrymmaptothesecondVoronoicompactification,providinganotherproofof[ABH02,
Thm.3.2].
• A geometric characterization of the indeterminacy locus of the Prym map
Rg+1 (cid:57)(cid:57)(cid:75) A¯Pg to the perfect cone compactification up to codimension 6 in Rg+1 in
termsofFriedman–Smithcovers(Theorem7.1).
• An explicit resolution of the Prym map Rg+1 (cid:57)(cid:57)(cid:75) A¯Pg up to codimension 4 (Theo-
rem8.1).ThisalsoresolvesthePrymmaptoA¯V andA¯C uptocodimension4.
g g
InAppendixE,MathieuDutourSikiric´ alsoprovesanextensionresultforthePrymmap
tothecentralconecompactification(TheoremE.1).
In this paper, we approach the extension problem for the Prym map in terms of the
Hodge-theoretic framework of a general period map M → D/(cid:48) from a moduli space
toaclassicalperioddomain.Thisallowsustodeterminetheconditionsforextensionsof
periodmapstomodulispacesthatarecompactifiedsothatthemonodromytransforma-
tions are of Picard–Lefschetz type (i.e. given by rank 1 forms). In this way we separate
the geometric aspects of the problem from the combinatorial issues involved in dealing
withvariousadmissibleconedecompositions.
In particular, the approach unifies the arguments for Jacobians and Pryms, and we
discuss the Torelli map throughout for motivation. As a result, we also get a new proof
oftheextensionresultsof[ABH02]forRg+1 (cid:57)(cid:57)(cid:75)A¯Vg.In[ABH02],theauthorshavethe
additional goal of determining compactified Pryms as stable semiabelic pairs; focusing
ExtendingthePrymmap 661
here on the extension condition allows us to give a more direct, Hodge-theoretic argu-
ment. With the work in [ABH02], translating from our results to the language of stable
semiabelicpairsisstraightforward(§2.4,§9).Inaddition,oneofouroriginalmotivations
for this work was investigating the extension of the period map for cubic threefolds to
amorphismfromasuitableGITcompactificationofthemodulispaceofthreefoldstoa
suitable compactification of A , stemming from our work [CML09] and [CML13], and
5
usingsomeoftheresultsofourwork[GH12].Themethodsweuseinthispaperapplyin
thatsettingalso,andwewillreturntothestudyoftheperiodmapforcubicthreefoldsin
subsequentwork.
A few words about the structure of the paper. We start in Section 1 by reviewing
some basic facts about the toroidal compactifications (second Voronoi, perfect, central)
thatweconsiderinourpaper.Wethendiscuss(Section2)thegeneralframeworkofde-
generations of Hodge structures and the connection to toroidal compactifications. This
is mostly standard (see e.g. [Cat84] for an exposition), but we find it convenient to in-
cludeashortdiscussionofthis,adaptedtoourneeds.InSection3,webrieflyreviewthe
standardcompactificationofthemoduliofPrymvarietiesbyadmissiblecovers[Bea77]
andtheassociatedcombinatorialdata(graphswithaninvolution,etc.).InSection4,we
specializethediscussionofSection2tocurvesandPrymvarietiesanddiscussthecom-
putationofthemonodromyconesintermsofthedualgraph.Themonodromyconefor
Jacobiansisclassical(e.g.[Nam76])andthatofPrymsisessentiallycontainedin[FS86]
and[ABH02].Nonetheless,webelievethatourpresentationunifies,simplifies,andclar-
ifiessomeoftheargumentsintheliterature.Ourgoalwillbetoapplysimilartechniques
tothestudyofothermodulispacesviaHodgetheoryinthefuture.
Withthesepreliminaries,newresultsstartinSection5,wherewerecasttheextension
criteria for the Torelli map, and then prove combinatorial criteria, in terms of the dual
graph,fortheextensionofthePrymmaptovarioustoroidalcompactificationsofA ,ob-
g
tainingTheorem5.6andthusgivinginadditionanewproofof[ABH02,Thm.3.2].We
thenproceedtorelatethesecombinatorialconditionstogeometricconditionsonadmis-
siblecovers.Theso-calledFriedman–Smithcoversarecentraltothisdiscussionandwe
describeinSection6theirmonodromyindetail:inSubsection6.2wecomputethemon-
odromy cones, and in Theorem 6.4 we discuss their properties with respect to the fans
definingdifferenttoroidalcompactifications.InSection7,weusethesecomputationsto
describe the indeterminacy locus of the Prym map geometrically, and it is interesting
to note that this behavior for the perfect cone compactification is quite different from
that for the second Voronoi compactification. We are able to give a complete geometric
characterizationoftheindeterminacylocusofthePrymmaptotheperfectconecompact-
ificationA¯P uptocodimension6(Theorem7.1),utilizingtherecentresultsofMeloand
g
Viviani[MV12].
Thecomputationsalsoallowustodescribetheresolutionoftheperiodmapintermsof
explicit,toroidalmodificationsofthemodulispaceofadmissiblecovers.InSection8we
describetheresolutionoftheperiodmaptotheperfectconecompactificationcompletely
uptocodimension4(Theorem8.1).InSection9westartadiscussiononthefibersofthe
Prym map. More precisely, we discuss which types of admissible covers are mapped to
whichstrata.Thisalsoprovidesanotherlinkto[ABH02]sincewediscusstherelationship
662 SebastianCasalaina-Martinetal.
betweenthemonodromyconesandthedegenerationdataof1-parameterfamilies,which
inturndeterminesemiabelicvarietieswhicharelimitsofPryms.
ManyoftheargumentsinthepaperregardingthePrymmapinlowcodimensionrely
on working through a number of examples, and explicit computations of monodromy
cones. These are somewhat lengthy and technical, and to maintain the structural unity
of the argument we collect these explicit computations in the appendices. Appendix A
treats the combinatorics of the Friedman–Smith cones and relates these to various cone
decompositions.InAppendixBwediscusssomeexampleswherethePrymmapextends;
this comes down to proving that certain monodromy cones belong to either the second
Voronoi,perfectconeorcentralconedecomposition.AppendixCcontainssomelengthy
calculations where we discuss further degenerations of Friedman–Smith examples. In
particularwecomputetheirmonodromyconesanddiscusstowhich,ifany,conedecom-
positionsthesebelong.Finally,inAppendixDwediscussamethodwhichallowsusto
simplifycertainmonodromyconesandthustoreducetopreviouscalculations.
Notation
Wewillusecalligraphicletterstorefertomodulistacks(e.g.Ag,Rg+1,etc.),anditalic
lettersfortheassociatedcoarsemodulispaces(e.g.Ag,Rg+1,etc.).Sinceallthespaces
occurring here (with the exception of Alexeev’s stack of stable semiabelic pairs) are
Deligne–Mumford stacks, all the period maps are assumed to be locally liftable, and
the extensions are insensitive to finite covers, there is essentially no difference between
usingstacksortheassociatedcoarsemodulispace.Infact,wewilltypicallysticktothe
coarsemodulispace,exceptforthesituationswherewewanttoemphasizethemodular
meaning.
1. Briefreviewoftoroidalcompactifications
In this section, we briefly review the theory of toroidal compactifications of A (see
g
[AM+10],[Nam80]and[FC90]formoredetails),focusingonthethreeclassicallyknown
toroidalcompactifications(uptorefinementoffans,i.e.blow-ups),thatistheperfectcone
(alsoknownasfirstVoronoi),secondVoronoi,andcentralconecompactification.Primar-
ilythepurposehereistofixthenotationandterminologyneededlater.
Notation1.1. Asiscustomary,whennecessary,wewillusesubscripts(e.g.HZ)toindi-
catethecoefficientsformodulesandalgebraicgroups.Unlessspecified,thecoefficients
areeitherQorR.
1.1. TheSatake–Baily–Borelcompactification
Fix a free abelian group H of rank 2g, and a non-degenerate, skew-symmetric, bilinear
formQonH.WeletD betheclassifyingspaceofpolarizedweight1Hodgestructures
onH:
D :={F ∈Grass(g,HC):Q(F,F)=0, iQ(F,F)>0}∼=GR/K,
ExtendingthePrymmap 663
where GR ∼= Sp(2g,R) and K = U(r) is the maximal compact subgroup. By taking
Qtobethestandardsymplecticform,D canbe(canonically)identifiedwiththeSiegel
upperhalf-spaceH ,thespaceofsymmetricg×gcomplexmatriceswithpositivedefinite
g
imaginarypart.ThefractionallineartransformationsgiveanactionofGZ = Sp(2g,Z)
∼
onD =H ,andweset
g
A :=H /Sp(2g,Z).
g g
The Satake–Baily–Borel (SBB) compactification A∗ is a normal, projective compactifi-
g
cationofA thatadmitsastratification
g
A∗g =Ag (cid:116)Ag−1(cid:116)···(cid:116)A0.
We recall that A∗ and the above stratification are obtained (set-theoretically) by
g
adding to D the so-called rational boundary components F , and then taking the quo-
W0
tient with respectto the natural GZ = Sp(2g,Z) action. Namely, therational boundary
componentsF ofDcorrespondtothechoiceofrationalmaximalparabolicsubgroups
W0
PW0 ⊂Sp(Q,HQ),whichinturncorrespondtothechoiceofatotallyisotropicsubspace
W0 ⊆HQ(ofwhichPW0 isthenthestabilizer).NotethatsinceSp(2g,Z)actstransitively
onthesetofisotropicsubspacesW0ofHQoffixeddimension,thesetofrationalbound-
arycomponentsisessentiallyindexedbyν (= dimW ) ∈ {0,...,g}.Furthermore,the
0
choiceofW0definesaweightfiltrationonHQ:
W−1 :={0}⊆W0 ⊆W1 :=(W0)⊥Q ⊆W2 :=HQ. (1.1)
ThepolarizationQinducesapolarization(non-degeneratesymplecticform)Q¯ onGrW =
1
W /W . It is then standard (e.g. [Cat84, p. 84]) that the boundary component F is
the1 cla0ssifying space Dg(cid:48) (with g(cid:48) = g −ν) of Q¯-polarized Hodge structures on GW0rW1 ,
givingthecomponentAg(cid:48) =FW0/GZ ofA∗g (N.B.F{0} =D,andaftertheidentification
FW0 =Dg(cid:48) =Hg(cid:48),theactionofGZrestrictstotheactionofSp(2g(cid:48),Z).)
1.2. Toroidalcompactifications
Toroidal compactifications are certain refinements of the SBB compactification A∗, de-
g
pending on a choice of a compatible collection of admissible cone decompositions, (cid:54).
EachsuchchoicegivesacompactificationA¯(cid:54) withacanonicalmapA¯(cid:54) →A∗.Herewe
g g g
reviewafewpointsabouttheconstructionfromtheperspectiveofHodgetheory(essen-
tiallyfollowing[Cat84]).
TheconstructionisrelativeoverA∗,andonestartsbyconsideringatotallyisotropic
g
subspace W0 ⊆ HQ of dimension ν ≤ g and the corresponding boundary component
ofA∗g.ConsiderthentherealLiesubalgebraofsp(Q,HR)preservingW0:
n(W0):={N ∈sp(Q,HR):Im(N)⊆W0}.
ThenforanyN ∈ n(W )wehaveN2 = 0,andthusN definesaweightfiltrationcom-
0
patiblewiththatinducedbyW (see(1.1)).Inotherwords,wehave
0
Im(N)=W (N)⊆W ⊆W =W⊥ ⊆W (N)=ker(N)=Im(N)⊥,
0 0 1 0 1
664 SebastianCasalaina-Martinetal.
andinparticularanaturalsurjection
GrW (:=W /W )(cid:16)Gr (N)(:=W (N)/W (N)). (1.2)
2 2 1 2 2 1
Furthermore,sinceN isanilpotentsymplecticendomorphism,wegetanaturalisomor-
phism
Gr (N)−→N Gr (N)−Q−(−N−(·−),→·) Gr (N)∨,
2 0 2 (1.3)
v (cid:55)→N(v)(cid:55)→Q(N(·),v),
which can be interpreted as giving a non-degenerate bilinear form Q on Gr (N). The
N 2
formQ turnsouttobesymmetric,andbypull-backcanbeviewedasaformonGrW;
N 2
thusthereisanaturalmap(definedoverQ)
n(W )−→∼ Hom(Sym2GrW,R), (1.4)
0 2
which(asisnothardtosee)isanisomorphism.
Asdescribedabove,n(W )iscanonicallyidentifiedwiththeLiealgebraofsymmetric
0
bilinearforms(orequivalentlysymmetricg(cid:48)×g(cid:48)matriceswithg(cid:48) =g−ν)onGrW.With
2
thisidentification,weconsidertheconeofpositivedefiniteg(cid:48)×g(cid:48)symmetricmatrices
n(W )+ :={N ∈n(W ):Q ispositivedefinite}.
0 0 N
Let (cid:54) be a compatible collection of admissible cone decompositions (see §1.3). Now
for each cone σ ∈ (cid:54) , there is an associated space B(σ ) together with a map
W0 W0 W0
B(σ ) → F , where F is the rational boundary component associated to W (see
W0 W0 W0 0
e.g.[Cat84,p.91]).Thesemapsarecompatibleinthesensethatifτ ≤ σ isaface,
W0 W0
thenthereisacommutativediagram
(cid:47)(cid:47)
B(τ ) B(σ )
W0 W0
(cid:36)(cid:36) (cid:122)(cid:122)
F
W0
OannaectthioennosentsDD(cid:54)(cid:54),an=d(cid:83)theWn0((cid:83)seσtW-t0h∈e(cid:54)oWre0tBic(aσllWy)0)A.¯(cid:54)gTh=eaDct(cid:54)io/nGoZf,GinZdu=cinSgp(a2lsgo,Zan)aetxutreanldmsatop
A¯(cid:54) →A∗.
g g
1.3. Admissibleconedecompositionsforquadraticforms
We now review some basic terminology and results about cone decompositions. Let (cid:51)
beafreeZ-moduleofrankg.Thespaceofquadraticformson(cid:51)is(Sym2(cid:51))∨,which
comesequippedwithanaturaldiagonalactionofGL((cid:51))=AutZ((cid:51)).Oneconsidersthe
opencone
C ⊂(Sym2(cid:51))∨⊗ZR,
Q
ofpositivedefinitequadraticforms,andthenletsC beitsrationalclosure.Obviously,
C and CQ are GL((cid:51))-invariant. For any subgroup (cid:48) ⊆ GL((cid:51)) (typically we will be
ExtendingthePrymmap 665
interested in (cid:48) = GL((cid:51))), a (cid:48)-admissible rational polyhedral decomposition (cid:54) (for
short,anadmissibledecomposition)ofC isa(cid:48)-invariantcollectionof(rational,convex,
Q
polyhedral) subcones covering C which satisfies certain natural axioms (see [Nam80]
or[FC90,Ch.IV,Def.2.2,p.96]fordetails),mostnotablytherequirementthatthereare
onlyfinitelymanyorbitsofconesof(cid:54)modulotheactionof(cid:48).
FortheconstructionofthetoroidalcompactificationsA¯(cid:54) onerequiresanadmissible
g
decompositionforthespaceofquadraticformsassociatedtoeachisotropicsubspaceW
0
(see (1.4)). As discussed, all isotropic subspaces W of fixed dimension are conjugate,
0
andthuswhatoneneedsisanadmissibledecompositionforeachlattice(cid:51)(cid:48) ofrank0 ≤
g(cid:48) ≤ g,compatibleinthefollowingsense.Wesaythat(cid:54)(cid:48) and(cid:54) arecompatibleifthere
existsasurjection(cid:51) (cid:16) (cid:51)(cid:48) suchthat(cid:54)(cid:48) isobtainedfrom(cid:54) viapull-backbythenatural
inclusion CQ((cid:51)(cid:48)) ⊆ CQ((cid:51)). If this is the case for one surjection (cid:51) (cid:16) (cid:51)(cid:48), it will be
trueforallsurjections.Inparticular,specifyinganadmissibledecompositionfor(cid:51)then
uniquely specifies compatible admissible decompositions for all lattices (cid:51)(cid:48) of smaller
rank.Inshort,allweneedtodefineatoroidalcompactificationA¯(cid:54) isanadmissiblecone
g
decompositionfortherankglattice.
ThreeadmissibledecompositionsareclassicallyknownforA ,namelytheso-called
g
secondVoronoidecomposition,theperfectcone(orfirstVoronoi)decomposition,andthe
centralconedecomposition(thesecan,ofcourse,befurthersubdivided).Thesedecom-
positions are discussed in [Nam80, §8, §9]. We shall address all three decompositions
andtheassociatedtoroidalcompactifications.Thoughwewillnotreviewtheirdefinitions
(theinterestedreadershouldsee[Nam80]),wewilldiscusstherelevantfactsaboutthem
inthefollowingsubsection.Thereisalsoanotheradmissibledecomposition,namelythat
into C-types [RB78], which is less known to algebraic geometers. This coincides with
thesecondVoronoidecompositionforg ≤ 4,butforg ≥ 5secondVoronoiisaproper
refinement of the C-type decomposition. To our knowledge no geometric interpretation
ofthecorrespondingtoroidalcompactificationisknown.
Finally, we recall some terminology. A cone σ ⊆ CQ is called basic if the integral
generatorsofits1-dimensionalfacescanbecompletedtoaZ-basisof(Sym2(cid:51))∨.Itis
calledsimplicialifthesegeneratorscanbecompletedtoaQ-basis,i.e.iftheyarelinearly
independent.
1.4. Admissibleconedecompositionsandrank1quadrics
In the geometric context of our paper, we will only be interested in cones spanned by
rank 1 quadrics (i.e. squares of linear forms), since our (log of) monodromy operators
willberank1.Forsuchconesitisessentiallyacombinatorialproblemtodecideifthey
belongtothesecondVoronoi,perfect,orcentralconedecompositions.Theseresultsare
wellknownandwewillreferthereaderto[AB12]and[MV12]forfurtherdetails.
For(cid:96)1,...,(cid:96)n ∈(cid:51)∨R\{0},letσ :=R≥0(cid:104)(cid:96)2i(cid:105)ni=1bethecorrespondingconegenerated
byrank1quadricsinSym2(cid:51)∨.Givenabasisfor(cid:51),wewilloftenrefertotheconeσ by
R
writingthematrixwhosei-throwistheexpressionfor(cid:96) intermsofthedualbasistothe
i
givenbasis,andtoanysuchmatrixwewillassociatesuchacone.
666 SebastianCasalaina-Martinetal.
Inthisset-up,wethenhavethefollowingcombinatorialresultsthatdeterminewhether
asetoflinearformsin(cid:51)∨generateaconecontainedinaconeofoneofthethreestandard
admissibledecompositions.
Lemma1.2 (SecondVoronoi). Let(cid:51)beafreeZ-moduleofrankg.Suppose(cid:96) ,...,(cid:96)
1 n
∈(cid:51)∨areprimitivenon-zerolinearforms.Thefollowingareequivalent:
(1) {(cid:96)2,...,(cid:96)2}lieinacommonconeofthesecondVoronoidecomposition.
1 n
(2) R≥0(cid:104)(cid:96)21,...,(cid:96)2n(cid:105)isaconeinthesecondVoronoidecomposition.
(3) Any R-linearly independent subset {(cid:96)j}j∈J ⊆ {(cid:96)1,...,(cid:96)n} is a Z-basis of the
Z-moduleR(cid:104)(cid:96)j(cid:105)j∈J ∩(cid:51)∨.
(4) Any R-linearly independent subset {(cid:96)j}j∈J ⊆ {(cid:96)1,...,(cid:96)n} of maximal rank is a
Z-basisoftheZ-moduleR(cid:104)(cid:96)j(cid:105)j∈J ∩(cid:51)∨.
Proof. Thisiswellknown.Wedirectthereaderto[AB12,Lem.4.5]andthereferences
therein. (cid:117)(cid:116)
OnemaytakeasadefinitionthatamatroidalconeisasecondVoronoiconegeneratedby
rank1quadrics(thisisessentiallythecontentofLemma1.2).Itfollowsfromthelemma
thatafaceofamatroidalconeismatroidal,andthatmatroidalconesaresimplicial.We
denoteby(cid:54) ⊆(cid:54) thecollectionofmatroidalcones.
mat V
Toconnectthediscussionwiththatof[ABH02],werecallthenotionofadicing.Fix
a collection of codimension 1 affine spaces {Hi}i∈I in (cid:51)R. Let H = (cid:83)i∈I Hi be the
associatedarrangementofaffinespaces.ThearrangementH isstratifiedbytheintersec-
tions of the H . We say that H defines a dicing of (cid:51) if the union of the 0-dimensional
i
strataofH isexactlythelattice(cid:51).
Lemma1.3. Let (cid:51) be a free Z-module of rank g. Suppose that (cid:96) ,...,(cid:96) ∈ (cid:51)∨ are
1 n
R-linearlyindependent.Then(cid:96) ,...,(cid:96) formaZ-basisfor(cid:51)∨ ifandonlyiftheydeter-
1 n
mineadicingof(cid:51)R.Moreprecisely,thismeansthatthecollectionofhyperplanes
Hi,m :={x ∈(cid:51)R :(cid:96)i(x)=m}
withi =1,...,nandm∈Zdefinesadicingof(cid:51).
Proof. Thisfollowsfromthedefinitionsandislefttothereader. (cid:117)(cid:116)
Remark1.4. Associated to a quadratic form q ∈ C is the so-called Delaunay decom-
positionof(cid:51)⊗ZR.ThesecondVoronoidecompositionisdefinedsothattheDelaunay
decompositionofaquadricremainsunchangedforallquadricsinagiven(open)second
Voronoi cone. We will only be interested in quadratic forms that lie in second Voronoi
conesgeneratedbyrank1quadrics.Inthiscase,theDelaunaydecompositionhasawell
known and simple description (see [ER94, Thm. 3.2] or [ABH02, proof of Lem. 3.1]):
If(cid:96)1,...,(cid:96)n ∈ (cid:51)∨ span(cid:51)∨R,andσ = R≥0(cid:104)(cid:96)21,...,(cid:96)2n(cid:105)isasecondVoronoicone,then
the Delaunay decomposition for any (positive definite) quadric q ∈ σ◦ is given by the
(dicing)hyperplanearrangementassociatedto(cid:96) ,...,(cid:96) .
1 n
ExtendingthePrymmap 667
Lemma1.5 (Perfect cone). Let (cid:51) be a free Z-module of rank g. Suppose (cid:96) ,...,(cid:96)
1 n
∈(cid:51)∨areprimitivenon-zerolinearforms.Thefollowingareequivalent:
(1) {(cid:96)2,...,(cid:96)2}lieinthesameconeoftheperfectconedecomposition.
1 n
(2) ThereexistsaquadraticformQon(cid:51)∨ suchthat
R
(a) Q((cid:96))>0forall(cid:96)∈(cid:51)∨\{0},i.e.Qispositivedefinite.
R
(b) Q((cid:96))≥1forall(cid:96)∈(cid:51)∨\{0}.
(c) Q((cid:96) )=1,i =1,...,n.
i
Proof. This follows from the definition of the perfect cone decomposition in [Nam80].
(Seealsotheproof of[AB12,Thm.4.7].) (cid:117)(cid:116)
Remark1.6. Since cones in the perfect cone decomposition are generated by rank 1
quadrics,aconeintheperfectconedecompositionisasecondVoronoiconeifandonly
if it is matroidal (i.e. (cid:54) ∩(cid:54) ⊆ (cid:54) ). Recently Melo and Viviani [MV12, Thm. A]
P V mat
showed that matroidal cones are in the perfect cone decomposition (i.e. (cid:54) ⊆ (cid:54) ),
mat P
establishing that (cid:54) ∩(cid:54) = (cid:54) . Note in particular that the following special case
P V mat
of[MV12,Thm.A]followsdirectlyfromthedefinitionsandLemma1.5:ifσ ∈ (cid:54) is
mat
generatedbyatmostgrank1quadraticforms,thenσ ∈(cid:54) .Inparticular,ifq ∈σ ∈(cid:54)
P P
isarank1quadric,thenR≥0(cid:104)q(cid:105)isafaceof σ.
Lemma1.7 (Central cone). Let (cid:51) be a free Z-module of rank g. Suppose (cid:96) ,...,(cid:96)
1 n
∈(cid:51)∨areprimitive,non-zero,linearforms.Thefollowingareequivalent:
(1) {(cid:96)2,...,(cid:96)2}lieinthesameconeofthecentralconedecomposition.
1 n
(2) ThereexistsaquadraticformQon(cid:51)∨ suchthat
R
(a) Q((cid:96))>0 forall(cid:96)∈(cid:51)∨\{0},i.e.Qispositivedefinite.
R
(b) Q((cid:96))≥1 forall(cid:96)∈(cid:51)∨\{0}.
(c) Q((cid:96) )=1,i =1,...,n.
i
(d) Q((cid:96))∈Z forall(cid:96)∈(cid:51)∨.
Proof. This follows from the definition of the central cone decomposition in [Nam80].
(Seealsotheproof of[AB12,Thm.4.8].) (cid:117)(cid:116)
Remark1.8. We note that all but the last condition above are the same as for the per-
fect cone compactification, and thus it turns out that if a collection of rank 1 quadratic
forms lies in a central cone, they also lie in a perfect cone, but not vice versa (see also
Remarks5.2and5.3below).
Given an admissible cone decomposition (cid:54), we will denote by (cid:54)(1) the collection of
conesthataregeneratedbyrank1quadrics.Notethatifσ ∈ (cid:54)(1) andτ isafaceofσ,
thenτ ∈(cid:54)(1).Notealsothatbydefinition(cid:54) =(cid:54)(1).Wecansummarizethediscussion
P P
aboveasfollows:
σ ∈(cid:54)(1)(=(cid:54) ) orσ ∈(cid:54)(1) ⇒ σ ∈(cid:54) (=(cid:54)(1)).
V mat C P P
668 SebastianCasalaina-Martinetal.
Remark1.9. Themetrics
(cid:88) (cid:88)
Q (x):= x x , Q (x):= x x (1.5)
A i j D i j
1≤i≤j≤n 1≤i≤j≤n,(i,j)(cid:54)=(1,2)
defineconesoftypeAandDrespectivelyintheperfectconedecomposition(infactalso
in the central cone decomposition). Cones of type A are matroidal, whereas for n ≥ 4,
typeDconesarenot(andalsofailtobesimplicial).
Remark1.10. At this point we recall the relation between the three known admissible
decompositions. For g = rank(cid:51), g ≤ 3, all three decompositions (namely the second
Voronoi,perfectconeandcentralcone)coincide.Forg =4itisstilltruethattheperfect
coneandthecentralconedecompositionscoincide,andthesecondVoronoidecomposi-
tionisarefinementofthese.Moreprecisely,theonlynon-basicconeoftheperfectcone
decomposition,namelytheD cone,issubdividedintobasicconesinthesecondVoronoi
4
decomposition(see[RE88]fordetails).Forg = 5thesecondVoronoidecompositionis
stillarefinementoftheperfectconedecomposition[RB78],butthisisnolongerthecase
forg ≥6[ER01].Ingeneral,allthreedecompositionsaredifferentinthesensethatnone
isarefinementofother.
2. Monodromyconesandextensionstotoroidalcompactifications
ThecentralquestionaddressedinthispaperisthatofextendingtheperiodmapforPrym
varieties to toroidal compactifications. The basic set-up for such a problem is that of a
locally liftable map P : B◦ → D/(cid:48) from a smooth base B◦ to a locally symmetric
variety(e.g.mapsarisingfromweight1variationsofHodgestructure(VHS)associated
tofamiliesofvarietiesX◦/B◦).Wethenconsiderapartialsimplenormalcrossingsmooth
compactificationB◦ ⊂ B andweareaskingaboutextensionsofthemapP fromB toa
(cid:54)
given(fixed)toroidalcompactificationD/(cid:48) .Sincetheproblemisessentiallylocal,we
mayassumewithoutlossofgeneralitythatB◦isapolycylinder(i.e.B0 =(S◦)k×Sn−k ⊂
B = Sn,whereS◦ = S \{0}andS istheunitdisk),andthatthemonodromyoperators
aroundtheboundarydivisorsareunipotent.
With this set-up the extension question has an elegant answer. Namely, one defines
a monodromy cone associated to the period map P, and then P extends if and only if
the monodromy cone is compatible with the cones of the admissible decomposition (cid:54).
Wereviewthisbelow,followingCattani[Cat84],withafocusonweight1variationsof
Hodgestructures(althoughsomeoftheconsiderationsapplymoregenerally).
2.1. Degenerationsofweight1Hodgestructures
ThemonodromyconeforavariationofHodgestructuresisabasictoolinunderstanding
extensions of period maps. Here we review the definition of the log of monodromy, the
monodromycone,andtheconnectionwithquadraticforms.