Table Of ContentTHE FIELD OF DEFINITION OF AFFINE INVARIANT
SUBMANIFOLDS OF THE MODULI SPACE OF
ABELIAN DIFFERENTIALS
2
1
0 ALEX WRIGHT
2
t
c Abstract. The field of definition of an affine invariant subman-
O ifold is the smallest subfield of R such that can be defined
M M
in local period coordinates by linear equations with coefficients in
7
1 this field. We show that the field of definition is equal to the in-
tersectionofthe holonomyfields oftranslationsurfacesin , and
] is a real number field of degree at most the genus. M
S
We show that the projection of the tangent bundle of to
D
M
absolute cohomology H1 is simple, and give a direct sum decom-
.
h position of H1.
t Applicationsincludeexplicitfullmeasuresetsoftranslationsur-
a
m faceswhoseorbitclosuresareaslargeaspossible,andevidencefor
finiteness of algebraically primitive Teichmu¨ller curves.
[
TheproofsuserecentresultsofArturAvila,AlexEskin,Maryam
1
Mirzakhani, Amir Mohammadi, and Martin Mo¨ller.
v
6
0
8
4
1. Introduction
.
0
1 1.1. Background. During the past three decades, it has been discov-
2
ered that many properties of a translation surface are determined to
1
: a certain extent by its SL(2,R)–orbit closure. The orbit closure is
v
relevant for:
i
X
The dynamics of the straight line flow, for example the Veech
r •
a dichotomy and deviations of ergodic averages [Vee89,For02,
EKZ];
Counting problems (the Siegel-Veech formula) [Vee98,EM01];
•
Flat geometry, for example which translation surfaces can be
•
written as a convex polygon with edge identifications [Vee95].
RecentlyEskin-Mirzakhani-Mohammadihaveannouncedaproofthat
the SL(2,R)–orbit closure of every unit area translation surface is the
set of unit area translation surfaces in some affine invariant submani-
fold [EM,EMM] (definitions are recalled in Section 2).
1.2. Statement of results. We show that each affine invariant sub-
manifold has a well-defined field of definition k( ), which is the
M M
1
2 A.WRIGHT
smallest subfield of R such that can be defined in local period coor-
M
dinates by linear equations with coefficients in this field. Furthermore,
Theorem 1.1. The field of definition k( ) of an affine invariant
M
submanifold is a real number field of degree at most the genus. It
M
is equal to the intersection of the holonomy fields of all translations
surfaces in .
M
Generic translation surfaces. A unit area translation surface is
said to be –generic if its orbit closure is equal to 1, the set of
M M
unit area translation surfaces within . Work of Masur and Veech
M
[Mas82,Vee82] gives the ergodicity of the SL(2,R)–action on 1,
M
and ergodicity guarantees that almost every point (with respect to a
natural invariant smooth measure) in 1 is –generic. (Masur and
M M
Veech worked in a less general setting, but their proofs apply equally
well to affine invariant submanifolds.) However, prior to this work not
many examples of explicit generic translation surfaces were known, see
below for a summary.
Wesaythatatranslationsurface(X,ω) 1 has –specialperiods
∈ M M
if there is some subfield k of the holonomy field of (X,ω), such that
(1) k has degree at most the genus g of (X,ω), and
(2) k( ) k, and
M ⊂
(3) there is some linear equation on local period coordinates with
coefficients in k which does not hold identically on , but
M
nonetheless holds at (X,ω).
All linear equations in this paper are assumed to be homogeneous, i.e.
have constant term 0.
A translation surface is said to have –typical periods if it does not
M
have –special periods. In other words, (X,ω) has –typical periods
M M
if its periods do not satisfy any linear equation which might, accord-
ing to Theorem 1.1, define an affine invariant submanifold properly
contained in .
M
Corollary 1.2. Let be any affine invariant submanifold. Let be
M G
the set of translation surfaces (X,ω) 1 with –typical periods.
∈ M M
Then
(1) has full measure in 1, and
G M
(2) every translation surface in is –generic.
G M
Let Q denotethealgebraic closure ofQ, andset Q = Q R. The set
R
∩
includes in particular the translation surfaces in 1 whose period
G M
coordinates span a Q –vector space of dimension dim .
R C
M
FIELD OF DEFINITION 3
Example 1.3. Consider a translation surface (X,ω) in a stratum ,
H
and assume that the period coordinates of (X,ω) are contained in
Q[π,π2,π3,...] and are linearly independent over Q. Then the pe-
riod coordinates of (X,ω) are automatically linearly independent over
Q , and so (X,ω) has –typical periods. Corollary 1.2 gives that such
R
H
(X,ω) are –generic.
H
The set also contains a dense set of translation surfaces whose
G
periods lie in a number field.
1
Connection to monodromy. Let H denote the flat bundle over
whose fiber over (X,ω) is H1(X,C), and let H1 denote
Mthe flat bundle whose fiber ov∈erM(X,ω) is H1(X,Σ,C), wherreelΣ is the
1 1
set of singularities of (X,ω). Let p : H H denote the natural
rel →
projection from relative to absolute cohomology. Note that T( ) is a
1 M
flat subbundle of H .
rel
1
The field of definition of a flat subbundle E H is the small-
est subfield of R so that locally the linear subsp⊂ace E of H1(X,C)
can be defined by linear equations (with respect to an integer basis of
H1(X,Z)) with coefficients in this field. The trace field of a flat bundle
over is defined as the field generated by traces of the corresponding
M
representation of π1( ).
M
Theorem 1.4. Let be an affine invariant submanifold. The field
M
of definition of p(T( )) and trace field of p(T( )) are both equal to
M M
k( ).
M
Set VId = p( ( )). There is a semisimple flat bundle W, and for
T M
each field embedding ρ : k( ) C there is a flat simple bundle V
ρ
M →
which is Galois conjugate to VId, so that
1
H = V W.
ρ
⊕
!
ρ
M
The bundle W does not contain any subbundles isomorphic to any
V . Both W and V are defined over Q.
ρ ρ
⊕
In particular,
dim p(T( )) deg k( ) 2g.
C Q
M · M ≤
1
The direct sum decomposition of H in Theorem 1.4 was previously
establishedinthecaseofTeichmu¨llercurvesbyMartinM¨oller[Mo¨l06].
1
In the Teichmu¨ller curve case, M¨oller showed that the splitting of H is
compatible with the Hodge decomposition; this is not currently known
for general affine invariant submanifolds.
4 A.WRIGHT
Evidence for finiteness of algebraically primitive Teichmu¨ller
curves. Algebraically primitive Teichmu¨ller curves correspond to two
complex dimensional affine invariant submanifolds where thetrace field
of p(T( )) has degree equal to the genus.
M
TheequidistributionresultsofEskin-Mirzakhani-Mohammadi [EMM]
and Theorem 1.4 allow us to show
Theorem 1.5. Let be a connected componentof the minimalstratum
H
in prime genus. If contains infinitely many algebraically primitive
H
Teichmu¨ller curves, then some subsequence equidistributes towards .
H
We hope that Theorem 1.5 will eventually lead to a proof that there
are only finitely many algebraically primitive Teichmu¨ller curves in the
minimal stratum in prime genus greater than two.
Bainbridge and M¨oller have recently informed the author that they
have recently established finiteness in (4) using different methods.
H
Countability of affine invariant submanifolds. Theorem 1.1 also
provides a new proof of a result of Eskin-Mirzakhani-Mohammadi.
Corollary 1.6 ([EMM], Prop. 2.16). There are only countably many
affine invariant submanifolds in the moduli of translation surfaces.
Proof. There areonly countably many systems ofreal linear equations
all of whose coefficients lie in a number field.
1.3. Previous results. Here we list the previous results which form
the context for our applications to problems predating [EM,EMM].
We do not use any of these results, and so omit some definitions which
the reader can find in the references.
Generic translation surfaces. In the case that is a connected
M
component of a stratum, we simply call an -generic translation sur-
M
face generic.
McMullen has classified orbit closures in genus 2 (strata of abelian
differentials).
Theorem 1.7 ([McM07], Thm. 1.2). The SL(2,R)–orbit closure of
any unit area translation surface in (2) or (1,1) is either
H H
(1) equal to a closed orbit, or
(2) equal to a locus of (X,ω) of unit area, where Jac(X) admits real
multiplication with ω as an eigenform, or
(3) equal to the whole stratum.
All of the possible orbit closures (affine invariant manifolds) listed
had been previously studied by both McMullen and Calta [McM03a,
FIELD OF DEFINITION 5
Cal04]. CaltadescribestheseorbitclosuresintermsoftheJ–invariant,
and gives explicit linear equations defining the affine invariant mani-
folds [Cal04].
It follows directly from this theorem of McMullen that in particular
any translation surface in (2) or (1,1) whose absolute periods do
H H
not satisfy a linear relation with coefficients in Q[√d] for some d 1 is
≥
generic. In genus 2, McMullen’s result is of course much stronger than
our results, and shows that Corollary 1.2 is not sharp even in genus 2.
Let be the locus of hyperelliptic translation surfaces in (2,2),
L H
where both of the singularities are fixed by the hyperelliptic involution.
Examples of generic translation surfaces have been constructed in
L
by Hubert-Lanneau-M¨oller [HLM09a,HLM12,HLM09b] and also
in hyp(4) by Nguyen [Ngu11]. These results in and hyp(4) are
H L H
complementary to ours; they do not provide a full measure set, but
do provide many especially interesting and important examples not
covered by our results.
Theorem 1.8 ([HLM09b], Thm 0.2). Suppose (X,ω) is obtained
∈ L
by the Thurston-Veech construction, has cubic trace field, and has a
completely periodic direction which is not parabolic. Then (X,ω) is
–generic.
L
In particular, there are –generic translation surfaces whose periods
L
liein a cubic field (that is, there is a real cubic field kso that theperiod
coordinates lie in k[i]).
Theorem 1.9 ([Ngu11], Thms 1.1, 1.2, Cor. 1.3). Every translation
surface in hyp(4) admits a specific decomposition into parallelograms
H
and cylinders. When two edges are assumed to be parallel, then for
an explicit generic subset of the remaining parameters, the translation
surface is generic. In particular, in hyp(4) there are generic sur-
H
faces arising from the Thurston-Veech construction, and generic sur-
faces with period coordinates in a quadratic field.
Anysurface in Hhyp(4) with acompletelyperiodicdirectionconsisting
of three cylinders whose moduli are independent over Q is generic.
These results of McMullen, Hubert-Lanneau-M¨oller and Nguyen all
rely on explicit decompositions of the translation surface into simple
pieces such as tori. Sufficiently simple decompositions are not available
for the generic translation surface in high genus.
In the case that is two complex dimensional (i.e., corresponds to
M
a closed SL(2,R)–orbit), then every translation surface in is –
M M
generic. See[Wri]foralistofknownclosedSL(2,R)–orbits; additional
examples arise from covering constructions.
6 A.WRIGHT
In the case that arises from certain covering constructions based
M
on closed SL(2,R)–orbits, then a classification of –generic trans-
M
lation surfaces is contained in the work of Eskin-Masur-Schmoll and
Eskin-Marklof-Witte-Morris[EMS03,EMWM06]. Infact,[EMS03,
EMWM06] classify unipotent flow invariant measures for these .
M
Algebraically primitive Teichmu¨ller curves. There are infinitely
many algebraically primitive Teichmu¨ller curves in (2), which were
H
constructed by McMullen and Calta [Cal04,McM03a]. McMullen
showed that there is only one algebraically primitive Teichmu¨ller curve
in (1,1) [McM06].
H
Finiteness of algebraically primitive Teichmu¨ller curves is known in
hyp(g 1,g 1) by work of M¨oller [Mo¨l08], and in (3,1) by work
H − − H
of Bainbridge-Mo¨ller [BM].
1.4. Tools and motivation. That affine invariant submanifolds are
defined over number fields is easier than the other statements of The-
orem 1.1. This can be proved using only a Closing Lemma for Te-
ichmu¨ller geodesic flow. We sketch this approach briefly in Section 4.
The result also follows, together with the bound on the degree of the
field of definition, from the inequality in Theorem 1.4.
BothHamensta¨dt[Ham10]andEskin-Mirzakhani-Rafi[EMR]have
proven Closing Lemmas for Teichmu¨ller geodesic flow. The version we
require is extremely close to that given in [EMR]. The Closing Lemma
is also used in the proof of Theorem 1.4.
Theorem 1.4 also uses a result of Avila-Eskin-M¨oller.
1
Theorem 1.10 ([AEM], Theorem 1.5). The bundle H over is
M
semisimple. That is, every flat subbundle has a flat complement.
1
In fact, Avila-Eskin-M¨oller prove this for the bundle H whose fiber
R
over (X,ω) is H1(X,R). The result follows for H1 = H1 C from
R R
⊗
general principles, see Section 5.
That affine invariant submanifolds are defined over a number field
is motivated by the expectation that they have a fairly rigid algebro-
geometric structure. The use of the Closing Lemma in the proof is
motivated by the fact that, up to scaling, the real and imaginary parts
of a translation surface on a closed orbit for the Teichmu¨ller geodesic
flow have period coordinates in a number field.
That the field of definition of p(T( )) is equal to that of T( ) is
M M
motivatedby the fact that for any (X,ω) having a hyperbolic affine dif-
feomorphism, and in particular any (X,ω) lying on a closed SL(2,R)–
orbit, the absolute and relative periods span the same Q–vector sub-
space of C [McM03b, Thm. 9.4].
FIELD OF DEFINITION 7
Organization. Section 2 contains definitions, and Section 3 proves
Corollary 1.2 using Theorem 1.1. Section 4 discusses the Closing
Lemma, and is used to prove the “Simplicity Theorem” in Section 5.
The Simplicity Theorem includes the statement that p(T( )) is sim-
M
ple from Theorem 1.4. The remainder of Theorem 1.4 is established
in Section 6, using the Simplicity Theorem and the Closing Lemma.
Theorem 1.1 is proven in Section 7, using Theorem 1.4. Theorem 1.5
is covered in the final section.
Acknowledgements. Specialthanksgofirstofalltotheauthor’sthe-
sisadvisor Alex Eskin, who generously contributed ideas tothis project
at several stages. The author is also grateful to Jayadev Athreya, Jon
Chaika, Ursula Hamensta¨dt, Ronen Mukamel, Kasra Rafi, John Smil-
lie, and many other members of the translation surfaces community for
helpful and enjoyable conversations. The author thanks David Auli-
cino, Simion Filip, Howard Masur, and again Alex Eskin for comments
on earlier drafts.
2. Period coordinates and field of definition
Suppose g 1 and let α be a partition of 2g 2. The stratum
≥ −
(α) is defined to be the set of (X,ω) where X is a genus g closed
H
Riemann surface, and ω is a holomorphic 1-form on X whose zeroes
have multiplicities given by α. In fact, we immediately replace (α)
H
by a finite cover which is a manifold instead of an orbifold.
H
Given a translation surface (X,ω), let Σ X denote the set of
⊂
zeros of ω. Pick any basis ξ1,...,ξn for the relative homology group
{ }
H1(X,Σ;Z). The map Φ : Cn defined by
H →
Φ(X,ω) = ω,..., ω
(cid:18)Zξ1 Zξn (cid:19)
defines local period coordinates on a neighborhood (X,ω) . We
∈ H
mayalsorefertotheabsolute periodcoordinates ofatranslationsurface:
these are the integrals of ω over a basis of absolute homology H1(X,Z).
Period coordinates provide with a system of coordinate charts
H
with values in Cn and transition maps in SL(n,Z). An affine invariant
submanifold of is a closed connected subset which is defined
H M ⊂ H
by real linear equations on period coordinates in each coordinate chart.
Given any subspace V Cn, one can define its field of definition as
⊂
the unique subfield k C so that V can be defined by linear equations
⊂
with coefficients in k but V cannot be defined by linear equations with
coefficients in a proper subfield of k. We require that the variables
defining these linear equations are the coordinates of Cn.
8 A.WRIGHT
Lemma 2.1. The field of definition of any subspace V Cn is well-
⊂
defined, and furthermore it is the smallest subfield of C such that V is
spanned by vectors with coordinates in this subfield.
Proof. Left to the reader. In fact the field of definition is the field
generated by the coefficients in the row reduced echelon form of any
system of linear equations defining V.
Given an affine invariant submanifold , we define its field of def-
M
inition k( ) to be the field of definition of the linear subspace of Cn
M
which defines in period coordinates. Since the transition maps are
M
in SL(n,Z), this does not depend on which coordinate chart is used.
Given a translation surface (X,ω), its holonomy is the subset Λ C
⊂
of the holonomies of all saddle connections. (A saddle connection is
a straight line between two singularities on the translation surface,
and the holonomy of a saddle connection is the integral of ω over the
relative homology class of this line.) Similarly the absolute holonomy
is the subset Λ C of the holonomies of all saddle connections
abs
⊂
representing absolute homology classes.
Given e1,e2 Λabs linearly independent over R, the holonomy field
∈
of (X,ω) is the smallest subfield k R so that Λabs ke1 ke2. This
⊂ ⊂ ⊕
does not depend on e1,e2. The definition of holonomy field is due to
Kenyon-Smillie [KS00].
Note in particular, that for some A GL(2,R), we have Ae1 = 1
∈
and Ae2 = i. Hence AΛabs k[i], and in particular all of the absolute
⊂
period coordinates of A(X,ω) lie in k[i].
3. Generic translation surfaces
In this section we prove Corollary 1.2 assuming Theorem 1.1.
Proof of Corollary 1.2. Inlocalperiodcoordinates, issomedim –
C
M M
dimensional subspace V of Cn. The unit area translation surfaces in V
form a hypersurface, and the natural invariant volume on 1 is up to
M
scaling the disintegration of Lebesgue measure on V [EM].
There are only countably many linear equations on V which are
nonzero on V and which have coefficients in a number field of degree at
mostg. Theirunionismeasurezero, andtheircomplement iscontained
in in local period coordinates. Hence has full measure.
G G
Consider anytranslationsurface(X,ω)in . Itsorbitclosureissome
G
affine invariant manifold , which is defined in local period
N ⊂ M
coordinates by linear equations with coefficients in k = k( ). By
N
Theorem 1.1, k( ) is contained in the holonomy field of (X,ω) ,
N ∈ N
and is a field of degree at most g. Furthermore, since k( ) is the
M
FIELD OF DEFINITION 9
intersection of the holonomy fields of surfaces in , and k = k( ) is
M N
the intersection of the holonomy field of surfaces in , we see
N ⊂ M
that k k( ).
⊃ M
However, the coordinates of (X,ω) by assumption do not satisfy any
linear equations with coefficients in such a field k which do not hold
identically on . Hence = .
M N M
4. Closed orbits for the Teichmu¨ller geodesic flow
The Teichmu¨ller geodesic flow is given by
et 0
g = SL(2,R).
t 0 e−t ⊂
(cid:18) (cid:19)
4.1. Closed orbits are abundant. The following Closing Lemma is
due to Eskin-Mirzakhani-Rafi [EMR, Section 9] using results of Forni.
See also the Closing Lemma of Hamenstadt [Ham10].
Lemma 4.1 (Closing Lemma). Let be a compact subset of an affine
K ′
invariantsubmanifold . Given anyopensetU which intersects
M ′ ⊂ M
, there is a smaller open set U U and a constant L0 > 0 with the
K ⊂
following property.
Suppose that g : [0,L] is a segmentof a g –orbit (parameterized
t
→ M
by t) such that
(1) g(0),g(L) U,
∈
(2) L > L0,
(3) t [0,L] g(t) > L/2.
|{ ∈ | ∈ K}|
′
Then there exists a closed g –orbit which intersects U .
t
Proof. The proof follows as in [EMR, Section 9]. They discuss only
strata of quadratic differentials, but the key feature of local product
structure (expanding and contracting foliations) is guaranteed by real
linearity for any affine invariant submanifold.
In fact stronger statements are true; see the references. We have
stated the Closing Lemma as is because we only need the existence of
closed orbits.
4.2. The action on (relative) cohomology. If g (X,ω) = (X,ω),
T
then there is an induced pseudo-Anosov diffeomorphism on X. The
∗
action g on the relative and absolute cohomology of X is via this
T
pseudo-Anosov.
∗
The action of g on absolute cohomology is related to its action on
T
relative cohomology H1(X,Σ,C), where Σ is the set of singularities.
After replacing the pseudo-Anosov with a power, we may assume it
10 A.WRIGHT
fixes Σ pointwise. In this case the pseudo-Anosov acts trivially on
ker(p), and thus the action on H1(X,Σ,C) has block triangular form,
where the diagonal blocks are the identity (on ker(p)) and the action
on absolute cohomology.
Both e−T and eT are simple eigenvalues for g∗ [Fri85, FLP79,
T
Pen91], and the eigenvectors are Re(ω) and Im(ω). The abelian dif-
ferential defines both a cohomology class and a relative cohomology
class, and this statement is true in both cohomology and relative coho-
mology. Furthermore, we emphasize that the generalized eigenspaces
of e−T and eT are one-dimensional in both cohomology and relative
cohomology.
Remark 4.2. We may now sketch a proof that every affine invariant
submanifold is defined over a number field. Details are left to the
M
reader, as this will be reproved in a stronger form later.
Pick any period coordinate chart for . It is not hard using the
M
Closing Lemma and ergodicity to show that closed orbits are dense in
1. If (X,ω) lies on a closed orbit, then the real and imaginary parts
M
of the period coordinates lie in a number field up to scaling, because
they are eigenvectors of simple eigenvalues of an integer matrix. The
integer matrix is g∗ : H1(X,Σ,Z) H1(X,Σ,Z).
T →
TheresultnowfollowsfromthefactthatanysubspaceV Cn which
⊂
is spanned by points with coordinates in a number field is defined over
a number field.
5. The Simplicity Theorem
In this section we prove that p(T( )) is simple, using Avila-Eskin-
M
M¨oller’s Theorem (Theorem 1.10). First we address the difference be-
tween the statements of Theorem 1.10 and [AEM, Theorem 1.5].
Proof of Theorem 1.10. The difference between the statements is
simply the difference between using real and complex cohomology.
Flat bundles correspond to representations of the fundamental group
of the base. [AEM, Theorem 1.5] gives that the representation to real
cohomology is the sum of real representations without real invariant
subspaces. The complexification of any real representation without
invariant subspaces is either simple or the direct sum of two complex
conjugate simple representations.
Theorem 5.1 (TheSimplicityTheorem). The bundle p(T( )) is sim-
M
ple. Furthermore T( ) has no direct sum decompositions, and any
M
proper flat subbundle of T( ) is contained in ker(p).
M
