Connected components of the moduli spaces of Abelian differentials with prescribed singularities

48 Pages·2003·0.69 MB·English

Invent.math.153,631–678(2003)
DOI:10.1007/s00222-003-0303-x
Connected components of the moduli spaces of
Abelian differentials with prescribed singularities
MaximKontsevich1,AntonZorich2
1 InstitutdesHautesE´tudesScientifiques,LeBois-Marie,35RoutedeChartres,F-91440
Bures-sur-Yvette,France(e-mail:[email protected])
2 Institut Mathe´matique de Rennes, Universite´ Rennes-1, Campus de Beaulieu, 35042
Rennes,cedex,France(e-mail:[email protected])
Oblatum29-XI-2002&24-II-2003
Publishedonline:6June2003–Springer-Verlag2003
Abstract. Consider themoduli space ofpairs (C,ω)whereC isasmooth
compact complex curve of a given genus and ω is a holomorphic 1-form
on C with a given list of multiplicities of zeroes. We describe connected
components of this space. This classification is important in the study of
dynamics of interval exchange transformations and billiards in rational
polygons, andinthestudyofgeometryoftranslation surfaces.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
2 Formulationofresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
3 SpinstructuredeterminedbyanAbeliandifferential . . . . . . . . . . . . . . . . 641
4 Preparationofasurgerytoolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
5 Connectedcomponentsofthestrata . . . . . . . . . . . . . . . . . . . . . . . . 654
AppendixA. Rauzyclassesandzipperedrectangles . . . . . . . . . . . . . . . . . 670
AppendixB. Abeliandifferentialsonhyperellipticcurves . . . . . . . . . . . . . . 676
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
1. Introduction
1.1. StratificationofthemodulispaceofAbeliandifferentials. Forinte-
gerg ≥2wedefinethespaceH asthemodulispaceofpairs(C,ω)where
g
C is a smooth compact complex curve of genus g and ω is a holomorphic
1-form on C (i.e. an Abelian differential) which is not equal identically
to zero. Obviously, H is a complex algebraic orbifold (in other words,
g
asmoothstack)ofdimension4g−3.ItisfiberedoverthemodulispaceM
g
632 M.Kontsevich,A.Zorich
ofcurveswiththefiberover[C]∈M equaltothepunctured vectorspace
g
Γ(C,Ω1)\{0}(modulotheactionofafinitegroup Aut(C)).
C
Orbifold Hg is naturally stratified by the multiplicities of zeroes(cid:1)of ω.
Letk ,...,k beasequenceofpositiveintegers,n ≥ 1withthesum k
1 n i i
equalto2g−2.WedenotebyH(k ,...,k )thesubspaceofH consisting
1 n
of equivalence classes of pairs (C,ω) where ω has exactly n zeroes and
their multiplicities are equal to k ,...,k (for some ordering of zeroes).
1 n
Our notation is symmetric, H(k1,...,kn) = H(kπ(1),...,kπ(n)) for any
permutation π ∈ S .Onehasthen
n
(cid:2)
H = H(k ,...,k ).
g 1 n
n,(k1,...,kn)
k1≤···≤kn
k1+···+kn=2g−2
Thus,wehaveastratificationofthemodulispaceH .Itiswell-knownthat
g
eachstratumH(k ,...,k )isanalgebraic orbifoldofdimension
1 n
(1) dimCH(k1,...,kn)=2g+n−1
(see [11], [17], [19]). Moreover, it carries a natural holomorphic affine
structure. Hereisthedescription ofthisstructure.
Withanypair(C,ω)weassociateanelement[ω]∈H1(C,Zeroes(ω);C),
thecohomologyclassofpair(C,Zeroes(ω))representedbyclosedcomplex-
valued1-formω.Locallyneareachpoint x ofH(k ,...,k )wecaniden-
1 n
tify cohomology spaces H1(C,Zeroes(ω);C) with each other using the
Gauss–Maninconnection.(Forpointsx =(C,ω)withnontrivialsymmetry
we would need to pass first to a finite covering of the neighborhood of x).
Thus,weobtain(locally)aperiodmappingfromH(k ,...,k )toadomain
1 n
in a complex vector space. It is well known that this mapping is holomor-
phicandlocallyone-to-one.Thepullbackofthetautologicalaffinestructure
on H1(C,Zeroes(ω);C) gives an affine structure on H(k ,...,k ). (See
1 n
also[6]forarelatedconstruction concerning smoothclosed1-forms.)
Ingeneral,thestrataH(k ,...,k )arenotfiberbundlesoverthemoduli
1 n
spaceofcurvesM .Forexample,thedimensionofthestratumH(2g−2)
g
for g ≥ 2 equals 2g, while dimension of the moduli space of curves M
g
equals3g−3whichisstrictly largerthan2gfor g ≥ 4.
The goal of this paper is to describe the set of connected components
ofall strata H(k ,...,k ).Surprisingly, wefound that the answer isquite
1 n
complicated, some strata have up to 3 connected components. The full
descriptionoftheconnectedcomponentsofstrataisgiveninSect.2.3.This
resultwasannounced inthepaper[7].
Rema(cid:1)rk1. For any sequence (k1,...,kn) of positive integers ki ≥ 1 such
that k = 2g − 2 we define Hnum(k ,...,k ) the moduli space of
i i 1 n
Abelian differentials on curves with numbered zeroes such that the first
zerohasmultiplicity k etc.Orbifold Hnum(k ,...,k )isafinitecovering
1 1 n
ofH(k ,...,k ).Onecanshowthatpreimageofanyconnectedcomponent
1 n
ConnectedcomponentsofthemodulispacesofAbeliandifferentials 633
of H(k ,...,k ) in Hnum(k ,...,k ) is connected, i.e. the classification
1 n 1 n
of connected components is essentially identical in both cases, no matter
whetherthezeroesarenumberedornot.
1.2. Applicationstointervalexchangetransformations. Themotivation
forourstudy camefrom dynamical systems, namely from thetheory ofso
calledintervalexchangetransformations.
Firstofall,thereisanalternativedescriptionofH intermsofdifferential
g
geometry. Outside of zeroes of an Abelian differential ω one can chose
locally a complex coordinate z in such way that ω = dz. This coordinate
is defined up to a constant, z = z(cid:4) +const, so it determines a natural flat
metric |dz2| on the Riemann surface C punctured at zeroes of ω. At zero
of ω of multiplicity k the metric has a conical singularity with the cone
i
angle2π(k +1).Thisflatmetrichastrivialholonomyinthegroup SO(2):
i
a parallel transport of a vector tangent to the Riemann surface C along
any closed path avoiding conical singularities brings the vector back to
itself. Thus, choosing a tangent direction at any nonsingular point we can
extenditusingtheparalleltransporttoallothernonsingular points,getting
a smooth distribution on the punctured Riemann surface. This distribution
is integrable: it defines a foliation with singularities at the conical points.
Theoriented foliation definedbythepositiverealdirection x incoordinate
z = x+iyiscalledhorizontal;theorientedfoliationdefinedbythepositive
purelyimaginarydirectionyiscalledvertical.Ataconicalpointwithacone
angle2π(k +1)onegetsk +1horizontal (vertical) directions.
i i
Conversely, aflatstructure withtrivial SO(2)-holonomyhaving several
cone type singularities plus a choice of, say, horizontal direction uniquely
determines a complex structure on the surface, and an Abelian differential
inthiscomplexstructure.
An Abelian differential ω defines also two smooth closed real-valued
1-forms ωv = Re(ω) and ωh = Im(ω) on C considered as a smooth
orientedtwo-dimensionalsurface M2.Theverticalandhorizontalfoliations
described above are the kernel foliations of the 1-forms ωv and ωh corres-
pondingly.
Conversely, let M2 be a compact smooth oriented surface of genus g
with a pair of closed 1-forms ωv,ωh such that ωv ∧ ωh > 0 everywhere
on M2 outside of a finite set. Then there is a unique point [(C,ω)] ∈ H
g
producing such M2 withformsωv,ωh.
There is a non-holomorphic continuous action on H of the group
g
GL(2,R)+ (the group of matrices with positive determinants). In terms
ofpairs of 1-forms (ωv,ωh) = (Re(ω),Im(ω))this action is given simply
bylineartransformations
(ωv,ωh)(cid:6)→(aωv+bωh,cωv+dωh).
Later in this text we shall use all descriptions of H : the algebro-
g
geometricone,theoneintermsofflatsurfaceswithachoiceofthehorizontal
direction, andtheoneintermsofpairsofmeasuredorientedfoliations.
634 M.Kontsevich,A.Zorich
It wasproved by H.Masur, see [11] and by W.A.Veech (see [17]) that
for a generic (with respect to the Lebesgue measure) point of any stratum
H(k ,...,k )thehorizontalfoliation(andalsotheverticalone)isuniquely
1 n
ergodic. Letustakeanyinterval I onthesurface M2 transversal tothever-
tical foliation, with the canonical induced length element. The first return
mapT : I −→ I (definedalmosteverywhereon I)isaninterval exchange
map, i.e. a one-to-one map with finitely many discontinuity points such
that the derivative of T is equal almost everywhere to +1. The interval
exchangemapisparametrizedbythenumberm ofmaximalopensubinter-
vals (Ii)i=1,...,m of continuity of the transformation T, by the sequence of
lengths of these subintervals λ ,...,λ where λ > 0, i = 1,...,m, and
1 m i
byapermutation π ∈ S describing theorder inwhichintervals T(I )are
m i
placed in I: the k-th interval is sent to the place π(k). It follows from the
uniqueergodicitythatthepermutationπ isirreducible,whichmeansinour
contextthat∀k =1,...,m −1wehaveπ({1,...,k})(cid:9)={1,...,k}.
Conversely, for any interval exchange map T one can construct an
Abelian differential ω and a horizontal interval I on a complex curve C
such that the first return map to I along the vertical foliation of ω is the
given map T, see [11], [17]. Though the Abelian differential ω is not
uniquelydeterminedbytheintervalexchangemap,thecollection ofmulti-
plicities of zeroes (k ,...,k ) of ω and even the connected component of
1 n
themodulispaceH(k ,...,k )containing point[(C,ω)]areuniquely de-
1 n
terminedbythepermutationπ,see[11],[17].Thusonemaydecomposethe
set of irreducible permutations into groups called extended Rauzy classes
corresponding toconnected components ofthestrataH(k ,...,k ).
1 n
The application of our result to the theory of interval exchange maps
is based on the corollary of the fundamental theorem of H. Masur [11]
and W. Veech [17] which we present in the next section. The corollary
is as follows: dynamical properties of a generic interval exchange map
dependonlyontheextendedRauzyclassofthepermutationofsubintervals.
Genericity isunderstood herewithrespecttotheLebesguemeasureonthe
spaceRm+parameterizinglengths(λi)1≤i≤m ofsubintervalsunderexchange.
Actually, the extended Rauzy classes can be defined in purely com-
binatorial terms, see Appendix 5.4 for details. Thus the problem of the
description of the extended Rauzy classes, and hence, of the description
of connected components of the strata of Abelian differentials, is purely
combinatorial. However, it seems to be very hard to solve it directly. Still,
forsmallgeneratheproblemistractable. W.Veechshowedin[19]thatthe
stratum H(4) has two connected components. P. Arnoux proved that the
stratumH(6)hasthreeconnected components.
In the present paper wegive a classification of extended Rauzy classes
usingnotonlycombinatoricsbutalsotoolsofalgebraicgeometry,topology
andofdynamical systems.
1.3. Ergodic components of the Teichmüller geodesic flow. There is
a natural immersion of the moduli space of Abelian differentials into the
ConnectedcomponentsofthemodulispacesofAbeliandifferentials 635
moduli space of holomorphic quadratic differentials: we associate to an
Abelian differential its square. With every quadratic differential we can
againassociateaflatmetricwithconicalsingularities.ThegroupGL(2,R)+
acts naturally on this larger moduli space as well; this action leaves the
immersed moduli space of Abelian differentials invariant, moreover, on
theimmersed subspace itcoincides withthe action defined inthe previous
section. Thisaction preserves the natural stratification ofthe moduli space
ofquadratic differentials bymultiplicities ofzeroes.
The action of the diagonal subgroup of SL(2,R) ⊂ GL(2,R)+ on the
moduli space of quadratic differentials can be naturally identified withthe
geodesicflowonthemodulispaceofcurvesforTeichmüllermetric(which
ispiecewisereal-analyticFinslermetriconM ).GroupSL(2,R)preserves
g
the hypersurface in the moduli space of quadratic differentials consisting
of those ones for which the associated flat metric has the total area equal
to1.
Numerous important results in the theory of interval exchange maps,
of measured foliations, of billiards in rational polygons, of dynamics on
translationsurfacesarebasedonthefollowingfundamentalobservation by
H.Masur[11]andW.Veech[17]:
Theorem (H.Masur;W.Veech). TheTeichmüllergeodesicflowactsergod-
ically on every connected component ofevery stratum of the moduli space
of quadratic differentials with total area equal to 1; the corresponding
invariant measureonthestratumisafiniteLebesgueequivalent measure.
ThusourclassificationofconnectedcomponentsofthestrataofAbelian
differentials gives the classification of ergodic components of the Teich-
müllergeodesicflowonthestrataofsquaresofAbeliandifferentials inthe
modulispaceofquadratic differentials.
Thecompleteclassification ofconnected components ofstrataofquad-
raticdifferentialsisinprogress(seeanannouncementin[8]).Forexample,
thestratumofthosequadraticdifferentialsonacurveofgenusg =4,which
cannotberepresentedasasquareofanAbeliandifferential,andwhichhave
asinglezeroofdegree12,hastwoconnectedcomponents,butatthemoment
atopologicalinvariantwhichwoulddistinguishrepresentativesofthesetwo
connected components isnotknownyet.
Ingeneral,itseemstobeveryinterestingtodescribeinvariantsubmani-
folds (closures of orbits, invariant measures) for the action of GL(2,R)+
on the moduli spaces. Connected components of the strata are only the
simplest invariant submanifolds, there are many others. For example the
Teichmüller disks of Veech curves form the smallest possible invariant
submanifolds.
One can use a submanifold invariant under the action of GL(2,R)+
to produce other invariant submanifolds in higher genera applying some
fixed ramified covering construction to all pairs (C,ω) constituting the
initial invariant submanifold. In Sect. 2.1 we use a particular case of this
construction todefinesomespecialconnected components ofsomestrata.
636 M.Kontsevich,A.Zorich
2. Formulationofresults
2.1. Hyperelliptic components. First of all, we introduce the moduli
spacesofmeromorphic quadratic differentials.
Definition1. Forintegerg ≥(cid:1)0andcollection(l1,...,ln),n ≥ 1suchthat
l ≥ −1,l (cid:9)= 0forall j and l = 4g−4,denote by Q(l ,...,l )the
j j j j 1 n
moduli space of pairs (C,φ) where C is asmooth compact complex curve
ofgenus gand φ isameromorphic quadratic differential onC withzeroes
ofordersl (simplepolesifl =−1)suchthatφisnotequaltothesquare
j j
ofanAbeliandifferential.
Itisknown(see[19])thatQ(l ,...,l )isacomplexalgebraic orbifold
1 n
ofdimension
(2) dimCQ(l1,...,ln)=2g+n−2.
Sometimesweshalluse“exponential”notationtodenotemultiplezeroes
(simple poles) of the same degree, for example Q(−15,1) :=
Q(−1,−1,−1,−1,−1,1). The condition that φ is not a square is auto-
maticallysatisfiedifatleastoneofparametersl isodd.
j
One can canonically associate with every meromorphic quadratic dif-
ferential (C,φ) another connected curve C(cid:4) with an Abelian differential ω
on it. Namely, C(cid:4) is the unique double covering of C (maybe ramified at
singularitiesofφ),suchthatthepullbackofφisasquareofanAbeliandif-
ferential ω.Wehave automatically σ∗(ω) = −ω where σ isthe involution
onC(cid:4)interchangingpointsinthegenericfiberoverC.CurveC(cid:4)isconnected
becauseofthecondition thatφisnotasquareofanAbeliandifferential.
Thus, weobtain amap from the stratum Q(l ,...,l )of meromorphic
1 n
quadraticdifferentialstothestratumH(k ,...,k )ofAbeliandifferentials,
1 m
where numbers (k ) are obtained from (l ) by the following rule: to each
i j
evenl > 0 we associate a pair of zeroes of ω of orders (l /2,l /2) in the
j j j
list (k ), to each odd l > 0 we associate one zero of order l + 1, and
i j j
associate nothingtosimplepoles(e.g.tol =−1).
j
Lemma1. Thecanonical mapdescribed above
Q(l ,...,l )→H(k ,...,k )
1 n 1 m
isanimmersion.
Proof. Denote as above by C(cid:4) the double covering of C with Abelian
differential ωandinvolution σ.
Considertheinduced involution
σ∗ : H1(C(cid:4),Zeroes(ω);C) → H1(C(cid:4),Zeroes(ω);C).
It defines decomposition H1(C(cid:4),Zeroes(ω);C) (cid:12) V1 ⊕ V−1 of the first
cohomology into the direct sum of subspaces invariant and anti invariant
ConnectedcomponentsofthemodulispacesofAbeliandifferentials 637
under the involution σ∗. By construction [ω] ∈ V−1. Thus, we obtain (lo-
cally)amappingfromQ(l ,...,l )toadomaininthecomplexvectorspace
1 n
V−1 ⊆ H1(C(cid:4),Zeroes(ω);C). It is well known that this mapping is holo-
morphic and locally one-to-one. Since the space H(k ,...,k ) is locally
1 m
identifiedwith H1(C(cid:4),Zeroes(ω);C)bymeansoftheperiodmapping,this
completestheproofoflemma. (cid:15)(cid:16)
Thefollowingtwoseriesofmapsofthiskindwouldplayaspecialrole
forus:
Q(−12g(cid:4)+1,2g(cid:4)−3) →H(2g(cid:4)−2)
(3) Q(−12g(cid:4)+2,2g(cid:4)−2) →H(g(cid:4)−1,g(cid:4)−1),
where g(cid:4) ≥ 2 in both cases. In both cases curve C is rational (i.e. g = 0),
and hence curve C(cid:4) is hyperelliptic of genus g(cid:4). In these two cases the
dimension oftheimagestratum ofAbeliandifferentials coincides withthe
dimension of the original stratum of meromorphic quadratic differentials.
Indeed, formula(2)gives
dimCQ(−12g(cid:4)+1,2g(cid:4)−3)=2·0+(2g(cid:4)+2)−2 =2g(cid:4)
dimCQ(−12g(cid:4)+2,2g(cid:4)−2)=2·0+(2g(cid:4)+3)−2 =2g(cid:4)+1,
whileformula(1)givesthefollowingdimensionsoftheimagestrata:
dimCH(2g(cid:4)−2) =2g(cid:4)+1−1=2g(cid:4)
dimCH(g(cid:4)−1,g(cid:4) −1)= 2g(cid:4)+2−1=2g(cid:4)+1.
Remark2. We have constructed a map Q(l ,...,l ) → H(k ,...,k )
1 n 1 m
using certain canonical double covering C(cid:4) → C. Choosing some other
(ramified)coveringofsomefixedtypeonecanconstructsomeother(local)
maps between moduli spaces of quadratic or Abelian differentials. The
readercanfindadetaileddescriptionofallmapsofthiskindbetweenmoduli
spacesofquadratic differentials, whichgivecoincidence ofdimensions, in
paper[8].
Beforereturningtomaps(3)whichareofaparticularinterestforuswe
needtoprovethefollowingstatement.
Proposition1. Inthecase g = 0everystratum Q(l ,...,l )ofmeromor-
1 n
phicquadratic differentials isnonempty andconnected.
Proof. ForanydivisoronCP1 withgivenmultiplicities thecorresponding
meromorphic quadratic differential exists and is unique up to a non-zero
scalar.Thus,wehave
Q(l ,...,l )/C∗ ∼=
1(cid:3) n (cid:4) (cid:3) (cid:4)
(CP1)n\diagonals / PSL(2,C)×(finitesymmetrygroup) .
ThereforetheorbifoldQ(l ,...,l )isnonemptyandconnected. (cid:15)(cid:16)
1 n
638 M.Kontsevich,A.Zorich
Lemma 1, the observation on coincidence of dimensions of the cor-
responding strata in (3), together with Proposition 1 justify the following
definition.
Definition2. Byhyperellipticcomponentswecallthefollowingconnected
components of the following strata of Abelian differentials on compact
complexcurvesofgenera g ≥2:
The connected component Hhyp(2g − 2) of the stratum H(2g − 2)
consisting of Abelian differentials on hyperelliptic curves of genus g cor-
responding totheorbifoldQ(−12g+1,2g−3);
The connected component Hhyp(g − 1,g − 1) of H(g − 1,g − 1)
corresponding totheorbifold Q(−12g+2,2g−2).
Remark3. PointsofHhyp(2g−2)(respectivelyofHhyp(g−1,g−1))are
Abeliandifferentialsonhyperellipticcurvesofgenusgwhichhaveasingle
zero of multiplicity 2g − 2 invariant under the hyperelliptic involution
(respectively apairofzeroes oforders g−1symmetric toeachotherwith
respecttothehyperelliptic involution).
Note that if an Abelian differential on a hyperelliptic curve has a sin-
gle zero of order 2g −2 then this zero is necessarily invariant under the
hyperelliptic involution σ, because σ∗(ω) = −ω for any Abelian differ-
ential ω. Therefore, this Abelian differential belongs to the component
Hhyp(2g − 2). However, if an Abelian differential ω has two zeroes of
degrees g−1,therearetwopossibilities: thezeroesmightbeinterchanged
bythehyperellipticinvolution,andtheymightbeinvariantunderthehyper-
elliptic involution. In the first case the Abelian differential belongs to the
component Hhyp(g−1,g−1),whileinthesecondcaseitdoesnot.
2.2. Parityofaspinstructure:adefinition.
Definition3. A spin structure on a smooth compact complex curve C is
achoiceofahalfofthecanonicalclass,i.e.ofanelementα ∈ Pic(C)such
that
2α = K :=−c (T ).
C 1 C
Theparityofthespinstructure istheresiduemodulo2ofthedimension
dimΓ(C,L)=dim H0(C,L)
forlinebundle L withc (L)=α.
1
Onacurveofgenus g ≥ 1thereare22g differentspinstructuresamong
which 22g−1 + 2g−1 are even and 22g−1 − 2g−1 are odd. It follows from
the results of M. Atiyah [1] and D. Mumford [14] that the parity of a spin
structure isinvariant undercontinuous deformations.
LetωbeanAbeliandifferential withevenmultiplicities ofzeroes, k =
i
2l foralli,i = 1,...,n.Thedivisorofzeroesofω
i
Zeroes(ω)=2l P +···+2l P
1 1 n n
ConnectedcomponentsofthemodulispacesofAbeliandifferentials 639
representsthecanonicalclass K .Thus,wehaveacanonicalspinstructure
C
onC definedby
αω :=[l1P1+···+lnPn] ∈ Pic(C).
Bycontinuitytheparityofthisspinstructureisconstantoneachconnected
componentofstratumH(2l ,...,2l ).
1 n
Definition4. We say that a connected component of H(2l ,...,2l ) has
1 n
evenoroddspinstructure depending onwhether αω isevenorodd, where
ωbelongs tothecorresponding connected component.
In Sect. 3.1 we present an equivalent definition of the parity of spin
structure intermsofelementary differential topology.
2.3. Mainresults. Firstofall,wedescribeconnectedcomponentsofstrata
inthe“stablerange”whenthegenusofthecurveissufficiently large.
Theorem1. Allconnected componentsofanystratumofAbeliandifferen-
tialsonacurveofgenus g ≥ 4aredescribed bythefollowing list:
The stratum H(2g−2) has three connected components: the hyperel-
liptic one, Hhyp(2g−2), and two other components: Heven(2g−2) and
Hodd(2g−2)corresponding toevenandoddspinstructures.
The stratum H(2l,2l), l ≥ 2 has three connected components: the
hyperelliptic one, Hhyp(2l,2l), and two other components: Heven(2l,2l)
andHodd(2l,2l).
All the other strata of the form H(2l ,...,2l ), where alll ≥ 1, have
1 n i
two connected components: Heven(2l ,...,2l ) and Hodd(2l ,...,2l ),
1 n 1 n
corresponding toevenandoddspinstructures.
ThestrataH(2l−1,2l−1),l ≥2,havetwoconnectedcomponents;one
ofthem:Hhyp(2l−1,2l−1)ishyperelliptic;theotherHnonhyp(2l−1,2l−1)
isnot.
AlltheotherstrataofAbeliandifferentialsonthecurvesofgenerag ≥ 4
arenonemptyandconnected.
Finallyweconsiderthelistofconnectedcomponentsinthecaseofsmall
genera1≤ g ≤3,wheresomecomponentsaremissingincomparisonwith
thegeneralcase.
Theorem2. ThemodulispaceofAbeliandifferentials onacurveofgenus
g = 2 contains two strata: H(1,1) and H(2). Each of them is connected
andcoincides withitshyperelliptic component.
Each of the strata H(2,2), H(4) of the moduli space of Abelian dif-
ferentials on a curve of genus g = 3 has two connected components: the
hyperelliptic one, and one having odd spin structure. The other strata are
connected forgenus g =3.
Parities of spin structures for hyperelliptic strata are calculated in the
AppendixA.4,Corollary5.
Theorems1and2wereannounced in[7].
640 M.Kontsevich,A.Zorich
2.4. Plan of the proof. We possess two invariants of connected compo-
nents: the components could be either hyperelliptic or not, and in the case
of even multiplicities the associated spin structure could be either even or
odd.Weshowthattheseinvariantsclassifytheconnectedcomponents.The
maximal number of connected components is 3, and it is achieved for the
strataH(2g−2)for g ≥4.WecallthestratumH(2g−2)minimal.
Ourplanoftheproofisthefollowing:
In Sect. 3 we give an alternative description of the parity of the spin
structure defined byanAbelian differential having zeroes ofeven degrees.
For a special class of Abelian differentials introduced in Sect. 4 this de-
scriptionintermsofdifferentialtopologywillmakethecomputationofthe
parityofthespinstructure especially easy.
Thesubsetofpoints[(C,ω)]whosehorizontalfoliationhasonlyclosed
leaves, is dense in every stratum. In Sect. 4.1 we consider Abelian dif-
ferentials only of this type. We propose a combinatorial way to represent
suchAbeliandifferentials bydiagrams,anditisparticularlyconvenient for
the minimal stratum. In Sect. 4.1 we establish a criterion for diagrams se-
lecting the ones associated to Abelian differentials. Wecall corresponding
diagramsrealizable. AlsoinSect.4.1wedescribediagramscorresponding
tohyperelliptic Abeliandifferentials.
We complete Sect. 4 by introducing a surgery (“bubbling a handle”)
whichallowsustoconstructanAbeliandifferential intheminimalstratum
in genus g +1 from an Abelian differential from the minimal stratum in
genus g. This surgery can be applied to any Abelian differential; however,
when the horizontal foliation of an Abelian differential has only closed
leaves, one can apply the surgery in such way that the horizontal folia-
tion of the resulting Abelian differential also has only closed leaves. In
this particular case the surgery can be described in terms of diagrams.
Also we describe how the parity of the spin structure changes under the
surgery.
In Sect. 5 we prove the classification theorem. First we prove it for
the minimal stratum. In Sect. 5.1 we study possible transformations of
realizable diagrams representing points in the minimal stratum preserving
the connected component. We prove by induction in genus g ≥ 2 that the
classification ofconnected components oftheminimalstratum H(2g−2)
is as in Theorems 1 and 2. We have to note that a surgery used in the step
of induction (“tearing off a handle”) is based on combinatorial Lemma 20
fromAppendix A.3concerning extendedRauzyclasses.
InSect.5.2westudythetopology oftheadjacency ofstrata, andprove
that the number of connected components in every stratum adjacent to the
minimal stratum is bounded above by the number of connected compo-
nents of the minimal stratum. More precisely, we identify the set of such
components withaquotient ofthesetπ (H(2g−2)).
0
InSect.5.3weprovethatanyAbeliandifferentialwhichdoesnotbelong
totheminimalstratum,canbedegeneratedtoadifferentialwithlesszeroes.
Thus,byinduction weprovethatanyconnected componentofanystratum