ORBITS IN PRYM EIGENFORM LOCI For any g ≥ 1 and any integer partition κ

51 Pages·2015·0.36 MB·English

GL+(2,R)-ORBITS IN PRYM EIGENFORM LOCI
ERWANLANNEAUANDDUC-MANHNGUYEN
ABSTRACT. ThispaperisdevotedtotheclassificationofGL+(2,R)-orbitclosuresofsurfacesinthe
intersectionofthePrymeigenformlocuswithvariousstrataofAbeliandifferentials. Weshowthatthe
followingdichotomyholds: anorbitiseitherclosedordenseinaconnectedcomponentofthePrym
eigenformlocus.
The proof uses several topological properties of Prym eigenforms, in particular the tools and the
proofareindependentoftherecentresultsofEskin-Mirzakhani-Mohammadi.
AsanapplicationweobtainafinitenessresultforthenumberofclosedGL+(2,R)-orbits(notneces-
sarilyprimitive)inthePrymeigenformlocusΩE (2,2)foranyfixedDthatisnotasquare.
D
1. INTRODUCTION
For any g ≥ 1 and any integer partition κ = (κ ,...,κ ) of 2g−2 we denote by H(κ) a stratum of
1 r
the moduli space of pairs (X,ω), where X is a Riemann surface of genus g and ω is a holomorphic
1-form having r zeros with prescribed multiplicities κ ,...,κ . Analogously, one defines the strata
1 r
of the moduli space of quadratic differentials Q(κ(cid:48)) having zeros and simple poles of multiplicities
κ(cid:48),...,κ(cid:48) with(cid:80)s κ(cid:48) = 4g−4(simplepolescorrespondtozerosofmultiplicity−1).
1 s i=1 s
The 1-form ω defines a canonical flat metric on X with conical singularities at the zeros of ω.
Therefore we will refer to points of H(κ) as flat surfaces or translation surfaces. The strata admit a
naturalactionofthegroupGL+(2,R)thatcanbeviewedasageneralizationoftheGL+(2,R)actionon
thespaceGL+(2,R)/SL(2,Z)offlattori. Foranintroductiontothissubject,werefertotheexcellent
surveys[MT02,Zor06].
Ithasbeendiscoveredthatmanytopologicalanddynamicalpropertiesofatranslationsurfacecan
berevealedbyitsGL+(2,R)−orbitclosure. Themostspectacularexampleofthisphenomenonisthe
caseofVeechsurfaces,orlatticesurfaces,thatissurfaceswhoseGL+(2,R)-orbitisaclosedsubsetin
its stratum; for such surfaces, the famous Veech dichotomy holds: the linear flow in any direction is
eitherperiodicoruniquelyergodic.
ItfollowsfromthefoundationresultsofMasurandVeechthatmostofGL+(2,R)orbitsaredense
in their stratum. However, in any stratum there always exist surfaces whose orbits are closed: e.g.
coveringsofthestandardflattorusandarecommonlyknownassquare-tiledsurfaces.
During the past three decades, much effort has been made in order to obtain the list of possible
GL+(2,R)-orbit closures and to understand their structure as subsets of strata. So far, such a list is
onlyknowningenustwobytheworkofMcMullen[McM07],buttheproblemiswideopeninhigher
genus,eventhoughsomebreakthroughshavebeenachievedrecently(seebelow).
Date:April23,2015.
Keywordsandphrases. Realmultiplication,Prymlocus,Translationsurface.
1
2 ERWANLANNEAUANDDUC-MANHNGUYEN
IngenustwothecomplexdimensionsoftheconnectedstrataH(2)andH(1,1)are,respectively,4
and 5. In this situation, McMullen proved that if a GL+(2,R)-orbit is not dense, then it belongs to a
Prymeigenformlocus,whichisasubmanifoldofcomplexdimension3. Inthiscase,theorbitiseither
closed or dense in the whole Prym eigenform locus. These (closed) invariant submanifolds, that we
denotebyΩE ,whereDisadiscriminant(thatisD ∈ N, D ≡ 0,1 mod 4),arecharacterizedbythe
D
followingproperties:
(1) Everysurface(X,ω) ∈ ΩE hasaholomorphicinvolutionτ : X → X,and
D
(2) The Prym variety Prym(X,τ) = (Ω−(X,τ))∗/H (X,Z)− admits a real multiplication by some
1
quadraticorderO := Z[x]/(x2+bx+c), b,c ∈ Z, b2−4c = D.
D
(whereΩ−(X,τ) = {η ∈ Ω(X) : τ∗η = −η}).
Later,thesepropertieswereextendedtohighergenera(uptogenusfive)byMcMullen(see[McM03a,
McM06,LN13]formoredetails).
Recently,Eskin-Mirzakhani-Mohammadi[EMi13,EMiMo13]haveannouncedaproofofthecon-
jecture that any GL+(2,R)-orbit closure is an affine invariant submanifold of H(κ). This result is
of great importance in view of the classification of orbit closures as it provides some very important
characterizationsofsuchsubsets. Howeverapriorithisresultdoesnotallowustoconstructexplicitly
suchinvariantsubmanifolds.
Sofar,mostofGL+(2,R)-invariantsubmanifoldsofastratumareobtainedfromcoveringsoftrans-
lationsurfacesoflowergenera. Theonlyknownexamplesofinvariantsubmanifoldsnotarisingfrom
thisconstructionbelongtooneofthefollowingfamilies:
(1) PrimitiveTeichmüllercurves(closedorbits),and
(2) Prymeigenforms.
This paper is concerned with the classification of GL+(2,R)−orbit closures in the space of Prym
eigenforms. To be more precise, for any non empty stratum Q(κ(cid:48)), there is a (local) affine map φ :
Qg(cid:48)(κ(cid:48)) → Hg(κ) given by the orientating double covering (the indices g and g(cid:48) are the genus of
the corresponding Riemann surfaces). When g−g(cid:48) = 2, following McMullen [McM06] we call the
image of φ a Prym locus and denote it by Prym(κ). Those Prym loci contain GL+(2,R)-invariant
suborbifoldsdenotedbyΩE (κ)(seeSection2formoreprecisedefinitions). Wewillinvestigatethe
D
GL+(2,R)-orbitclosuresinΩE (κ). Thefirstmaintheoremofthispaperisthefollowing.
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Theorem 1.1. Let (X,ω) ∈ ΩE (κ) be a Prym eigenform, where ΩE (κ) has complex dimension 3
D D
(i.e. ΩE (κ)iscontainedinoneofthePrymlociinTable1). WedenotebyOitsorbitunderGL+(2,R).
D
Then
(1) EitherOisclosed(i.e. (X,ω)isaVeechsurface),or
(2) OisaconnectedcomponentofΩE (κ).
D
Observe that the case κ = (1,1) is part of McMullen’s classification in genus two, which is
obtained via decompositions of translation surfaces of genus two into connected sums of two tori
(see[McM07]).
Remark 1.2. The classification of connected components of ΩE (2,2) and ΩE (1,1,2) will be ad-
D D
dressed in a forthcoming paper [LN13c] (see also [LN13] for related work). The statement is the
following: foranydiscriminant D ≥ 8andκ ∈ {(2,2),(1,1,2)},thelocusΩE (κ)isnon-emptyifand
D
GL+(2,R)-ORBITSOFPRYMEIGENFORMS 3
Q(κ(cid:48)) Prym(κ) g(X) Q(κ(cid:48)) Prym(κ) g(X)
Q (−16,2) Prym(1,1) (cid:39) H(1,1) 2 Q (12,2) Prym(12,22) (cid:39) H(02,2) 4
0 2
Q (−13,1,2) Prym(1,1,2) 3 Q (−1,2,3) Prym(1,1,4) 4
1 2
Q (−14,4) Prym(2,2)odd 3 Q (−1,1,4) Prym(2,2,2)even 4
1 2
Q (−12,6) Prym(3,3) (cid:39) H(1,1) 4 Q (8) Prym(4,4)even 5
2 3
TABLE 1. Prym loci for which the corresponding stratum of quadratic differentials
has(complex)dimension5. ThePrymeigenformlocusΩE (κ)hascomplexdimen-
D
sion3. ObservethatthestratumH(1,1)ingenus2isaparticularcaseofPrymlocus.
onlyifD ≡ 0,1,4 mod 8,anditisconnectedifD ≡ 0,4 mod 8,andhastwoconnectedcomponents
otherwise.
EventhoughTheorem1.1isaparticularcaseoftherecentresultsofEskin-MirzakhaniandEskin-
Mirzakhani-Mohammadi[EMi13,EMiMo13],ourproofisindependentfromtheseworks. Itisbased
on the geometry of the kernel foliation on the space of Prym eigenforms. It is also likely to us that
the method introduced here can be generalized to yield Eskin-Mirzakhani-Mohammadi’s result in
invariantsubmanifoldswhichpossessthecompleteperiodicproperty(seeSection2.3).
We will also prove a finiteness result for Teichmüller curves in the locus ΩE (2,2)odd; this is our
D
secondmainresult:
Theorem 1.3. If D is not a square then there exist only finitely many closed GL+(2,R)-orbits in
ΩE (2,2)odd.
D
WeendwithafewremarksonTheorem1.3.
Remark1.4.
• To the authors’ knowledge, such finiteness result is not a direct consequence of the work by
Eskin-Mirzakhani-Mohammadi.
• InPrym(1,1)astrongerstatementholds: thereexistonlyfinitelymanyGL+(2,R)-closedor-
bitsin (cid:116) ΩE (1,1)(see[McM05b,McM06a]). ThesameresultholdsforPrym(1,1,2):
D
Dnotasquare
this is proved in a forthcoming paper by the first author and M. Möller [LMöl13]. However,
thisisnolongertrueinPrym(2,2)odd aswewillseeinTheoremA.1.
• Other finiteness results on Teichmüller curves have been obtained in other situations by dif-
ferentmethods,seeforinstance[Möl08,BaMöl12,MaWri13].
Outline of the paper. We end this section with a sketch of the proofs of Theorem 1.1 and The-
orem 1.3. Before going into the details, we single out the relevant properties of ΩE (κ) for our
D
purpose. In what follows (X,ω) will denote a surface in ΩE (κ) (sometimes we will simply use X
D
whenthereisnoconfusion).
(1) Each locus is preserved by the kernel foliation, that is, (X,ω) + v is well defined for any
sufficientlysmallvectorv ∈ R2 (seeSection3). UptoactionofGL+(2,R),thereexistsε > 0
suchthataneighborhoodof(X,ω)inΩE (κ)canbeidentifiedwiththeset
D
{(X,ω)+v, |v| < ε}.
4 ERWANLANNEAUANDDUC-MANHNGUYEN
(2) EverysurfaceinΩE (κ)iscompletelyperiodicinthesenseofCalta: anydirectionofasimple
D
closedgeodesicisactuallycompletelyperiodic,whichmeansthatthesurfaceisdecomposed
intocylindersinthisdirection. Thenumberofcylindersisboundedfromabovebyg+|κ|−1,
where|κ|isthenumberofzerosofω(seeSection2).
(3) Assumethat(X,ω)decomposesintocylindersinthehorizontaldirection, thenthemoduliof
thosecylindersarerelatedbysomeequationswithrationalcoefficients(seeProposition4.12).
(4) The cylinder decomposition in a completely periodic direction is said to be stable if there
is no saddle connection connecting two different zeros in this direction. The stable periodic
directionsaregenericforthekernelfoliationinthefollowingsense: ifthehorizontaldirection
isstablefor(X,ω),thenthereexistsε > 0suchthatforanyv ∈ R2 with|v| < ε,thehorizontal
directionisalsoperiodicandstableonX+v. Ifthehorizontaldirectionisunstablethenthere
exists ε > 0 such that for any v = (x,y) with |v| < ε and y (cid:44) 0 the horizontal direction is
periodicandstableonX+v.
The properties (1)-(2)-(3) are explained in [LN13a] (see Section 3.1 and Corollary 3.2, Theo-
rem1.5,Theorem7.2,respectively). WewillgivemoredetailsonProperty(4)inSection4.
Wenowgiveasketchoftheproofofourresults. Thefirstpartofthepaper(Sections3-6)isdevoted
totheproofofTheorem1.1,whilethesecondpart(Sections7-11)isconcernedwithTheorem1.3.
Sketch of proof of Theorem 1.1. Let (X,ω) ∈ ΩE (κ) be a Prym eigenform and let O := GL+(2,R)·
D
(X,ω)bethecorrespondingGL+(2,R)−orbit. WewillshowthatifOisnotaclosedsubsetinΩE (κ)
D
thenitisdenseinaconnectedcomponentofΩE (κ).
D
WefirstproveaweakerversionofTheorem1.1(seeSection5)undertheadditionalconditionthat
thereexistsacompletelyperiodicdirectionθon(X,ω)thatisnotparabolic. Westartbyapplyingthe
horocycle flow in that periodic direction, and use the classical Kronecker’s theorem to show that the
orbitclosurecontainstheset(X,ω)+x(cid:126)v,where(cid:126)vistheunitvectorindirectionθ,and x ∈ (−ε,ε)with
ε > 0smallenough. Thenweapplythesameargumenttothesurfaces(X,ω)+x(cid:126)vinanotherperiodic
direction that is transverse to θ. It follows that O contains a neighborhood of (X,ω), and hence for
any g ∈ GL+(2,R), O contains a neighborhood of g · (X,ω). Using this fact, we show that for any
(Y,η) ∈ O\O, the closure O also contains a neighborhood of (Y,η), from which we deduce that O is
anopensubsetofΩE (κ). HenceOmustbeaconnectedcomponentofΩE (κ).
D D
In full generality, (see Section 6) we show that if the orbit is not closed and all the periodic di-
rections are parabolic, then it is also dense in a component of ΩE (κ). For this, we consider a
D
surface (Y,η) ∈ O \ O for which the horizontal direction is periodic. From Property (1), we see
that there is a sequence ((Xn,ωn))n∈N of surfaces in O converging to (Y,η) such that we can write
(X ,ω ) = (Y,η) + (x ,y ), where (x ,y ) −→ (0,0). Property (4) then implies that the horizontal
n n n n n n
direction is periodic for (X ,ω ). Moreover, we can assume that the corresponding cylinder decom-
n n
positionin(X ,ω )isstable(fornlargeenough).
n n
For any x ∈ (−ε,ε), where ε > 0 is small enough, we show that (up to taking a subsequence) the
orbitofthehorocycleflowthough(X ,ω )containsasurface(X ,ω )+(x ,0)suchthatthesequence
n n n n n
(x )convergesto x. Asaconsequence,weseethatOcontains(Y,η)+(x,0)forevery x ∈ (−ε,ε). We
n
cannowconcludethatOisacomponentofΩE (κ)bytheweakerversionofTheorem1.1.
D
GL+(2,R)-ORBITSOFPRYMEIGENFORMS 5
Sketch of proof of Theorem 1.3. We first show a finiteness result up to the (real) kernel foliation for
surfaces in ΩE (2,2)odd (see Theorem 11.2): If D is not a square then there exists a finite family
D
P ⊂ ΩE (2,2)odd suchthatforany(X,ω) ∈ ΩE (2,2)odd withanunstablecylinderdecomposition,
D D D
uptorescalingbyGL+(2,R),wehavethefollowing
(X,ω) = (X ,ω )+(x,0) forsome(X ,ω ) ∈ P .
k k k k D
Compareto[McM05a,LN13]whereasimilarresultisestablished.
Nowletusassumethatthereexistsaninfinitefamily, sayY = (cid:83) GL+(2,R)·(X,ω), ofclosed
i∈I i i
GL+(2,R)-orbits,generatedbyVeechsurfaces(X,ω), i ∈ I.
i i
Bypreviousfinitenessresult,uptotakingasubsequence,weassumethat(X,ω) = (X,ω)+(x,0)
i i i
for some (X,ω) ∈ P , where x belongs to a finite open interval (a,b) which is independent of i (see
D i
Theorem8.1). Uptotakingasubsequence,onecanassumethatthesequence(x)convergestosome
i
x ∈ [a,b]. Hencethesequence(X,ω) = (X,ω)+(x,0)convergesto(Y,η) := (X,ω)+(x,0).
i i i
If x ∈ (a,b) then (Y,η) belongs to ΩE (2,2)odd, otherwise, that is x ∈ {a,b}, (Y,η) belongs to one of
D
the following loci ΩED(0,0,0),ΩED(4), or ΩED(cid:48)(2)∗, with D(cid:48) ∈ {D,D/4} (see Section 8). Then by
usingaby-productoftheproofofTheorem1.1,replacingObyY(seeTheorem6.2andTheorem9.4)
we obtain that Y is dense in a component of ΩE (2,2)odd. We conclude with Theorem 10.1 which
D
assertsthatthesetofclosedGL+(2,R)−orbitsisnotdenseinanycomponentofΩE (2,2)odd whenD
D
isnotasquare.
Acknowledgments. WewouldliketothankCorentinBoissy,PascalHubert,JohnSmillie,andBarak
Weiss for useful discussions. We would also like to thank the Université de Bordeaux and Institut
FourierinGrenobleforthehospitalityduringthepreparationofthiswork. Someoftheresearchvisits
which made this collaboration possible were supported by the ANR Project GeoDyM. The authors
arepartiallysupportedbytheANRProjectGeoDyM.
2. BACKGROUND
Foranintroductiontotranslationsurfaces,andanicesurveyonthistopic,seee.g.[MT02,Zor06].
InthissectionwerecallnecessarybackgroundandrelevantpropertiesofΩE (κ)forourpurpose. For
D
ageneralreferenceonPrymeigenforms,see[McM06].
Wewillusethefollowingnotationsalongthepaper:
B(ε) = {v ∈ R2, |v| < ε},B(M,ε) = {A ∈ GL+(2,R), ||A−M|| < ε}and
(cid:82)
ω(γ) := ω,foranyγ ∈ H (X,Z),
γ 1
where|.|istheEuclideannormonR2,and||.||issomenormonM(2,R).
2.1. Prym loci and Prym eigenforms. Let X be a compact Riemann surface, and τ : X → X be a
holomorphicinvolutionofX. WedefinethePrymvarietyofX:
Prym(X,τ) = (Ω−(X,τ))∗/H (X,Z)−,
1
where Ω−(X,τ) = {η ∈ Ω(X) : τ∗η = −η}. It is a sub-Abelian variety of the Jacobian variety
Jac(X) := Ω(X)∗/H (X,Z).
1
For any integer vector κ = (k ,...,k ) with nonnegative entries, we denote by Prym(κ) ⊂ H(κ)
1 n
the subset of pairs (X,ω) such that there exists an involution τ : X → X satisfying τ∗ω = −ω, and
6 ERWANLANNEAUANDDUC-MANHNGUYEN
dimCΩ−(X,τ) = 2. FollowingMcMullen[McM06],wewillcallanelementofPrym(κ)aPrymform.
For instance, in genus two, one has Prym(2) (cid:39) H(2) and Prym(1,1) (cid:39) H(1,1) (the Prym involution
beingthehyperellipticinvolution).
LetY bethequotientof X bythePryminvolution(hereg(Y) = g(X)−2)andπthecorresponding
(possibly ramified) double covering from X to Y. By push forward, there exists a meromorphic qua-
dratic differential q on Y (with at most simple poles) so that π∗q = ω2. Let κ(cid:48) be the integer vector
that records the orders of the zeros and poles of q. Then there is a GL+(2,R)-equivariant bijection
betweenQ(κ(cid:48))andPrym(κ)[L04,p.6].
All the strata of quadratic differentials of dimension 5 are recorded in Table 1. It turns out that
thecorrespondingPrymvarietieshavecomplexdimensiontwo(i.eif(X,ω)istheorientatingdouble
coveringof(Y,q)theng(X)−g(Y) = 2).
WenowgivethedefinitionofPrymeigenforms. Recallthataquadraticorderisaringisomorphic
to O = Z[X]/(X2 + bX + c), where D = b2 − 4c > 0 (quadratic orders being classified by their
D
discriminantD).
Definition2.1(Realmultiplication). LetAbeanAbelianvarietyofdimension2. WesaythatAadmits
a real multiplication by O if there exists an injective homomorphism i : O → End(A), such that
D D
i(O ) is a self-adjoint, proper subring of End(A) (i.e. for any f ∈ End(A), if there exists n ∈ Z\{0}
D
suchthatnf ∈ i(O )then f ∈ i(O )).
D D
Definition 2.2 (Prym eigenform). For any quadratic discriminant D > 0, we denote by ΩE (κ) the
D
setof(X,ω) ∈ Prym(κ)suchthatdimCPrym(X,τ) = 2,Prym(X,τ)admitsamultiplicationbyOD,and
ωisaneigenvectorofO . SurfacesinΩE (κ)arecalledPrymeigenforms.
D D
Prym eigenforms do exist in each Prym locus described in Table 1, as real multiplications arise
naturallywithpseudo-Anosovhomeomorphismscommutingwithτ(see[McM06]).
2.2. PeriodicdirectionsandCylinderdecompositions. Wecollecthereseveralresultsconcerning
surfaceshavingadecompositionintoperiodiccylinders.
Let(X,ω)beatranslationsurface. AcylinderisatopologicalannulusembeddedinX,isometricto
a flat cylinder R/wZ×(0,h). In what follows all cylinders are supposed to be maximal, that is, they
are not properly contained in a larger one. If g ≥ 2, the boundary of a maximal cylinder is a finite
unionofsaddleconnections. IfCisacylinder, wewilldenotebyw(C),h(C),µ(C)thewidth, height,
andmodulusofCrespectively(µ(C) = h(C)/w(C)).
Anotherimportantparameterofacylinderisitstwistt(C). Notethatweonlydefinet(C)whenCisa
horizontalcylinder. Forthat,wefirstmarkapairoforientedsaddleconnectionsonthebottomandthe
top boundaries of C. This allows us to define a saddle connection contained in C joining the origins
of the marked saddle connections. This gives us a twist vector, its vertical component equals h(C)
and its horizontal component is t(C). We emphasis that t(C) depends on the marking (see [HLM06,
Section3]). However,thechoiceofthemarkingisirrelevantforourargumentsthroughoutthispaper.
Therefore,wewillrefertot(C)asthetwistassociatedtoanymarking.
AdirectionθiscompletelyperiodicorsimplyperiodiconXifallregulargeodesicsinthisdirection
areclosed. Thismeansthat X istheclosureofafinitenumberofcylindersindirectionθ,wewillsay
thatX admitsacylinderdecompositioninthisdirection.
GL+(2,R)-ORBITSOFPRYMEIGENFORMS 7
We can associate to any cylinder decomposition a separatrix diagram which encodes the way the
cylinders are glued together, see [KZ03]). Given such a diagram, one can reconstruct the surface
(X,ω)(uptoarotation)fromthewidths,heights,andtwistsofthecylinders(seeSection4).
2.3. Completeperiodicity. Atranslationsurface(X,ω)issaidtobecompletelyperiodicifitsatisfies
thefollowingproperty: letθ ∈ RP1 beadirection,ifthelinearflowF inthedirectionθhasaregular
θ
closedorbiton X,thenθ isaperiodicdirection. Flattoriandtheirramifiedcoveringsarecompletely
periodic,aswellasVeechsurfaces.
Itturnsoutthat,ifthegenusisatleasttwo,thesetofsurfaceshavingthispropertyhasmeasurezero.
Indeed complete periodicity is locally expressed via proportionality of a non-empty set of relative
periods,andthusisdefinedbysomequadraticequationsintheperiodcoordinates. Thispropertyhas
been initiated by Calta [C04] (see also [CS07]) where she proved that any surface in ΩE (2) and
D
ΩE (1,1) is completely periodic. Later the authors extended this property to any Prym eigenform
D
givenbyTable1. ThispropertyisalsoprovedbyA.Wright[Wri13]byadifferentargument.
Theorem2.3([C04,LN13a,Wri13]). AnyPrymeigenforminthelociΩE (κ) ⊂ Prym(κ)ofTable1
D
iscompletelyperiodic.
3. KERNEL FOLIATION ON PRYM LOCI
WebrieflyrecallthekernelfoliationforPrymloci(see[EMZ03,MZ08,C04,MW08]and[Zor06,
§9.6] for related constructions). We refer to [LN13a, Section 3.1] for details. This notion was intro-
ducedbyEskin-Masur-Zorich,andwascertainlyknowntoKontsevich.
Let (X,ω) ∈ H(κ) be a translation surface with several distinct zeros. Using the period mapping,
wecanidentifyaneighborhoodof(X,ω)inH(κ)withanopensubsetU ⊂ Cd,whered = dimH(κ).
We have a foliation of U by subsets consisting of surfaces having the same absolute periods. The
set of surfaces in this neighborhood that have the same absolute coordinates as X corresponds to the
intersection of U with an affine subspace of dimension |κ|−1. Therefore the leaves of this foliation
havedimension|κ|−1. Itisnotdifficulttoseethatthisfoliationisinvariantbythecoordinatechanges
oftheperiodmappings. ThuswehaveafoliationdefinedgloballyinH(κ),thisisthekernelfoliation.
It turns out that the kernel foliation also exists in Prym(κ) and ΩE (κ), for all κ in Table 1. In
D
particular, the leaves of the kernel foliation in ΩE (κ) have dimension one. We refer to [LN13a,
D
Section3.1]foradescriptionofthisfoliationinΩE (κ)withmoredetails.
D
SincetheleavesofthekernelfoliationinΩE (κ)havedimensionone,wehavealocalactionofC
D
on ΩE (κ) as follows: for any Prym eigenform (X,ω) and w ∈ C with |w| small enough, (X(cid:48),ω(cid:48)) :=
D
(X,ω)+wistheuniquesurfaceintheneighborhoodof(X,ω)(inΩE (κ))suchthatω(cid:48) hasthesame
D
absoluteperiodsasω,andforachosenrelativerelativecyclec ∈ H (X,Σ,Z),wehaveω(cid:48)(c) = ω(c)+w
1
(Σisthesetofzerosofω). Anexplicitconstructionfor(X,ω)+wwillbegiveninSection4.3.
have
It is worth noticing that we do not have a global action of C on each leaf of the kernel foliation,
i.e even (X,ω)+w and (X,ω)+w exist, (X,ω)+w +w may not be well defined. Nevertheless,
1 2 1 2
there still exists a local action of C in a neighborhood of (X,ω) on which a local chart (by period
mappings) can be defined. In particular, if |w | and |w | are small enough then (X,ω)+(w +w ) =
1 2 1 2
((X,ω)+w )+w = ((X,ω)+w )+w .
1 2 2 1
8 ERWANLANNEAUANDDUC-MANHNGUYEN
Convention: Throughoutthispaper,weonlyconsidertheintersectionofkernelfoliationleaveswith
a neighborhood of (X,ω) on which this local action of C is well-defined, and by (X,ω)+w we will
meanthesurfaceobtainedfrom(X,ω)bytheconstructiondescribedabove.
Therelativeperiodsof(X(cid:48),ω(cid:48)) := (X,ω)+warecharacterizedbythefollowinglemma(seeFigure1
foranexampleinPrym(1,1,2)).
Lemma 3.1. Let c be a path on X joining two zeros of ω, and c(cid:48) be the corresponding path on X(cid:48).
Then
(1) Ifthetwoendpointsofcareexchangedbyτthenω(cid:48)(c(cid:48))−ω(c) = ±w.
(2) Ifoneendpointofcisfixedbyτ,buttheotherisnot,thenω(cid:48)(c(cid:48))−ω(c) = ±w/2.
Thesignofthedifferenceisdeterminedbytheorientationofc.
B
B
A C C
1 1
A
C
3 C
3
C C
C C
C 2 C 2
τ(C3) τ(C3)
B
τ(C1) B τ(C1)
A
A
(X,ω) (X,ω)+(s,t)
FIGURE 1. Decomposition of a surface (X,ω) ∈ Prym(1,1,2). The cylinder C2 is
fixed by the Prym involution τ, while the cylinders C and τ(C) are exchanged for
i i
i = 1,3. Along a kernel foliation leaf(X,ω)+(s,t) the twists and heights change as
follows: t (s) = t − s, t (s) = t , t (s) = t + s/2 and h (t) = h − t, h (t) = h ,
1 1 2 2 3 3 1 1 2 2
h (t) = h +t/2. Weemphasisthattheformulaforthetwistsdoesnotdependonthe
3 3
choiceofthemarking.
We end this section by a description of a neighborhood of a Prym eigenform: up to the action of
GL+(2,R)aneighborhoodofapoint(X,ω)inΩE (κ)canbeidentifiedwiththeball{(X,ω)+w,|w| < ε}.
D
Proposition3.2([LN13a]). Forany(X,ω) ∈ ΩE (κ),if(X(cid:48),ω(cid:48))isaPrymeigenforminΩE (κ)close
D D
enoughto(X,ω),thenthereexistsauniquepair(g,w),whereg ∈ GL+(2,R)closetoId,andw ∈ R2
with|w|small,suchthat(X(cid:48),ω(cid:48)) = g·((X,ω)+w).
Proof. Forcompletenessweincludetheproofhere(see[LN13a,Section3.2]).
Let(Y,η) = (X,ω)+w,with|w|small,beasurfaceintheleafofthekernelfoliationthrough(X,ω). We
denoteby[ω]and[η]theclassesofωandηinH1(X,Σ;C)−. Letρ : H1(X,Σ;C)− → H1(X,C)−bethe
naturalprojection. Wethenhave[η]−[ω] ∈ kerρ. Ontheotherhand,theactionofg ∈ GL+(2,R)on
GL+(2,R)-ORBITSOFPRYMEIGENFORMS 9
H1(X,Σ;C)− satisfiesρ(g·[ω]) = g·ρ([ω]). Thereforetheleavesofthekernelfoliationandtheorbits
ofGL+(2,R)aretransversal. Sincetheirdimensionsarecomplementary,thepropositionfollows. (cid:3)
4. STABLE AND UNSTABLE CYLINDER DECOMPOSITIONS
4.1. Cylinder decompositions. A separatrix is a geodesic ray emanating from a zero of ω. It is a
well-knownfactthatadirectionisperiodicifandonlyifalltheseparatricesinthisdirectionaresaddle
connections. Inthiscasethesurfacedecomposesintofinitelymanycylindersinthisdirection. Since
the Prym involution τ preserves the set of cylinders, it naturally induces an equivalence relation on
this set. We will often use the term “number of cylinders up to Prym involution” for the number of
τ-equivalenceclassesofcylinders.
Definition4.1. Acylinderdecompositionof(X,ω)issaidtobestableifeveryseparatrixjoinsazero
ofωtoitself. Thedecompositionissaidtobeunstableotherwise.
Lemma4.2. Letθbeaperiodicdirectionfor(X,ω) ∈ H(κ)andgbethegenusofg. IfXhasg+|κ|−1
cylindersinthedirectionθ,thenthecylinderdecompositioninthisdirectionisstable(|κ|isthenumber
ofzerosofω).
Proof. LetC ,...,C be the cylinders inthe direction θ of X. Fori = 1,...,n, letc be a core curve
1 n i
ofC. Cutting X along c we obtain r compact surfaces with boundary denoted by X ,...,X . Note
i i 1 r
that each of X must contain at least a zero of ω. Therefore we have r ≤ |κ|. Let n be the number of
i i
boundary components of X. Remark that we have (cid:80) n = 2n, and χ(X) ≤ 2−n, where χ(.) is
i 1≤i≤r i i i
theEulercharacteristic. Byconstruction,wehave
(cid:88)r (cid:88)r (cid:88)r
2−2g = χ(X) = χ(X) ≤ (2−n) = 2r− n = 2r−2n.
i i i
i=1 i=1 i=1
Itfollowsimmediatelythat
n ≤ g+r−1 ≤ g+|κ|−1.
Fromthepreviousinequalities,weseethattheequalityn = g+|κ|−1isrealizedifandonlyifr = |κ|
andeachX hasgenuszero. Inparticular,ifn = g+|κ|−1,theneachcomponentX containsaunique
i i
zero of ω. If there is a saddle connection joining two distinct zeros of ω, then these two zeros must
belong to the same X, and we have a contradiction. Therefore, the cylinder decomposition must be
i
stable. (cid:3)
Remark4.3. InH(1,1)themaximalnumberofcylindersinacylinderdecompositionisthree,anda
cylinderdecompositionisstableifandonlyifthismaximalnumberisattained. Inhighergenus,there
arestablecylinderdecompositionswithlessthann+|κ|−1cylinders.
Lemma4.4. Let(X,ω) ∈ Prym(κ)beasurfaceinoneofthestratagivenbyTable1. Ifthehorizontal
direction is periodic for (X,ω) then the number n of horizontal cylinders, counted up to the Prym
involution, satisfies n ≤ 3. Moreover, if κ (cid:44) (1,1,2,2) and n = 3 then the cylinder decomposition in
thehorizontaldirectionisstable.
Remark 4.5. Observe that Lemma 4.4 is false for the stratum Prym(1,1,2,2). However, using the
identification Prym(1,1,2,2) (cid:39) H(0,0,2) the statement becomes true with the convention that a
cylinder decomposition of (X,ω) ∈ Prym(1,1,2,2) is stable if and only if the decomposition of the
correspondingsurfaceinH(0,0,2)is.
10 ERWANLANNEAUANDDUC-MANHNGUYEN
Proof. Let us assume that the horizontal direction is completely periodic. We first show that the
number n of horizontal cylinders, counted up to the Prym involution, satisfies n ≤ 3. Let n be the
f
number of fixed cylinders (by the Prym involution) and let 2 · n be the number of non-invariant
p
cylinders. Obviouslyn = n +n .
f p
The next observation is that each fixed cylinder contains exactly two regular fixed points of the
Prym involution, which project to simple poles of the corresponding quadratic differential. Hence if
Prym(κ)isthecoveringofQ(−1p,k ,...,k )wherek ≥ 0thenn ≤ (cid:98)p/2(cid:99). Nowsincethenumberof
1 n i f
cylindersisatmostg+|κ|−1,wegetn ≤ (cid:98)(g+|κ|−1−n )/2(cid:99). Hence
p f
n = n +n ≤ (cid:98)(g+|κ|−1+n )/2(cid:99).
f p f
Thevaluesofg+|κ|−1forthedifferentcasesofTable1arethefollowing:
Q(κ(cid:48)) Prym(κ) g+|κ|−1 Q(κ(cid:48)) Prym(κ) g+|κ|−1
Q (−16,2) Prym(1,1) 3 Q (12,2) Prym(12,22) 7
0 2
Q (−13,1,2) Prym(1,1,2) 5 Q (−1,2,3) Prym(1,1,4) 6
1 2
Q (−14,4) Prym(2,2)odd 4 Q (−1,1,4) Prym(2,2,2)even 6
1 2
Q (−12,6) Prym(3,3) 5 Q (8) Prym(4,4)even 6
2 3
Ontherighttable,theinequality p ≤ 1holdsforallcases,thusn = 0. Thereforen ≤ (cid:98)7/2(cid:99) = 3.
f
Foralltheothercasesonthelefttable,onehas,respectively:
(1) Ifκ = (1,1)thenn ≤ 3andn ≤ (cid:98)(3+n )/2(cid:99) ≤ 3.
f f
(2) Ifκ = (1,1,2)thenn ≤ 1andn ≤ (cid:98)(5+n )/2(cid:99) ≤ 3.
f f
(3) Ifκ = (2,2)thenn ≤ 2andn ≤ (cid:98)(4+n )/2(cid:99) ≤ 3.
f f
(4) Ifκ = (3,3)thenn ≤ 1andn ≤ (cid:98)(5+n )/2(cid:99) ≤ 3.
f f
The first statement of the lemma is proved. Now we notice that if n = 3 then in every case, but
κ = (1,1,2,2), one has n + 2 · n = g + |κ| − 1. Hence by Lemma 4.2 the horizontal direction is
f p
stable. (cid:3)
4.2. Combinatorial data. Let (X,ω) be a surface for which the horizontal direction is completely
periodic. Since each saddle connection is contained in the upper (respectively, lower) boundary of a
uniquecylinder,wecanassociatetothecylinderdecompositionthefollowingdata:
• twopartitionsofthesetofsaddleconnectionsintok subsets,wherek isthenumberofcylin-
ders,eachsubsetinthesepartitionsisequippedwithacyclicordering,and
• apairingofsubsetsinthesetwopartitions.
Wewillcallthesedatathecombinatorialdataortopologicalmodelofthecylinderdecomposition.
Note that while there exists only one topological model for cylinder decompositions with maximal
numberofcylindersinPrym(1,1),ingeneral,thereareseveraltopologicalmodelsforsuchdecompo-
sitionsinotherPrymlociinTable1.
4.3. Kernel foliation and stable decomposition. We will now carefully investigate the kernel foli-
ation leaf nearby a surface (X,ω) for which the horizontal direction is periodic. In what follows, we
onlyconsidertheintersectionofthekernelfoliationleaveswithaneighborhoodof(X,ω)onwhicha
localchartbytheperiodmappingisdefined. Thisrestrictionmeansthatthesurfacesinthesameleaf
as(X,ω)canbewrittenas(X,ω)+v,with|v|smallenough. RemarkthatforallPrymlociinTable1,