the example of a geodesi ow on SO(4)
Peter Bueken
?;y
and Pol Vanhaeke
Abstrat. We introdue the moduliproblem for (algebrai ompletely) integrable sys-
tems. ThisproblemonsistsinonstrutingamodulispaeofaÆnealgebraivarietiesand
expliitlydesribingamapwhihassoiatestoageneriaÆnevariety,whihappearsasa
levelsetoftherstintegralsofthesystem(or,equivalently,ageneriaÆne varietywhih
is preservedbythe ows of theintegrable vetor elds),a point inthismodulispae. As
an illustration,we work outtheexampleofan integrable geodesiowon SO(4). Inthis
ase, thegeneri invariant variety is an aÆne part of theJaobian of a Riemann surfae
of genustwo. Our onstrution relies heavilyon thefat that these aÆne partshave the
additionalpropertyof being4:1unramiedovers of Abeliansurfaesof type (1;4).
?
PostdotoralResearheroftheFundforSientiResearh{Flanders(FWO{Vlaanderen)
y
DepartmentofMathematis,KatholiekeUniversiteitLeuven,Celestijnenlaan200B,B-3001Heverlee(Leuven),
Belgium.E-mailaddress: Peter.Buekenwis.kuleuven.a.be
UniversitedePoitiers,Mathematiques,SP2MI,Boulevard3{Teleport2{BP179,86960FuturosopeCedex,
Frane. E-mailaddress: Pol.Vanhaekemathlabo.univ-poitiers.fr
1991MathematisSubjetClassiation. 14H40,14J10,58F07.
Keywordsand phrases. Integrablesystems,Abeliansurfaes,modulispaes.
Animportantpropertywhihdistinguishesintegrablesystemsfromgeneridynamial
systems is that theow of an integrable vetor eld,starting from an arbitrary point, is
notdense inthe orresponding energy levelbutis onstrainedto asubvarietyof halfthe
dimension (or less) of phase spae. These subvarieties are the level sets of the Poisson
ommuting funtions thatmake up theintegrable systemand theabove propertyfollows
from the fat that X
G
F = fF;Gg =0, where F and G areany two Poisson ommuting
funtions (suh astheHamiltonianand anyofits rst integrals).
In a real setting the relevane of these invariant manifolds for understanding and
desribing the mehanis of the integrable system an be seen from the Arnold-Liouville
theorem andfrom theexistene ofation-angle variables: ifsuh aninvariantmanifoldis
ompatthenitisdieomorphito arealtorusandtheowisalinearowonthistorus;
moreover, suha linearizingdieomorphism an beonstruted semi-loally (ona neigh-
borhood of a ompat invariant manifold)onto a produt of a toruswith a linear spae,
thelatterbeingequippedwiththestandardsympletiorPoissonstruture. Inpartiular,
itentailsquasi-periodiityofthesolutions,heneofthemotionofthemehanialsystem.
See[4℄ or[5℄.
In a omplex analyti or a omplex algebrai setting the struture of the invariant
manifolds is more ompliated. First, while two real tori are dieomorphi if and only
if they have the same dimension, the onditions for two omplex tori (quotients of C n
byalattie) to be biholomorphi(or, inan algebraisetting,isomorphi)are muhmore
involved, as beomesalready apparent inthe one-dimensionalase (Riemann surfaes of
genus 1; ellipti urves). Seond, sine omplex toriare ompat they annot live inan
aÆne variety and the omplex invariant manifolds are at best aÆne parts of omplex
algebrai tori and this fat puts an extra ondition on the possibility of two invariant
manifolds being biholomorphi or isomorphi: suh an isomorphism must preserve the
divisor at innity, i.e., the divisor to be glued to the aÆne variety in order to omplete
it into a ompatomplex torus. Bothaspets have theirrelevane formehanis. First,
while the solutions orresponding to non-isomorphi omplex tori an in both ases be
written in terms of theta funtions, the harateristis of these quasi-periodi funtions
willbe dierent; and seond, even when the toriare isomorphi, iftheir aÆne parts are
notisomorphithenthebehaviourofthesystemfornite(omplex)timewillbedierent
beause in an a..i. system on an aÆne variety every (omplex) integral urve hits the
divisorat innityaftera nitetime(see [2℄ or[3℄).
Theseonsiderationsleadustowhatweallthemoduliproblemforintegrablesystems.
In order to give a preise denition we will restrit ourselves to algebrai ompletely
integrablesystems(a..i.systems)onanaÆnevariety;thedenitionaneasilybeadapted
to other situations when needed. We reall from [14℄ that the generi invariantmanifold
of ana..i. system isan aÆne partof an Abelianvariety(a omplexalgebraitorus)and
that a large amount of expliit information about these tori (suh as equations for an
aÆne part, embeddings inprojetive spae and equations for the divisorat innity)an
beobtainedbystudyingtheLaurentsolutionstothedierentialequations,whihdesribe
thevetor eld denedbytheHamiltonian. The divisorwhih isneeded to ompletethe
aÆne part into an Abelianvariety indues a polarizationon the orresponding torus; in
allknownases thepolarizationtype,whihisadisreteinvariant oftheAbelianvariety,
is the same for all these invariant manifolds. We are thus led to onsider, on one side,
the family of aÆne parts of Abelian varieties that appear as invariant manifolds in the
a..i. system, on the other side, themodulispae 1
of Abelianvarieties witha presribed
1
In order to have a good moduli spae (e.g. one that admits an algebrai struture),
sends an invariant manifold to its isomorphism lass. The moduli problem for an a..i.
systemonsistsin expliitlyonstruting the modulispae and theanonial map.
To illustrate our point, we will treat a non-trivial example, namely an integrable
system that appears in Adler and van Moerbeke's lassiation of integrable geodesi
owsonSO(4)astheaseofmetriII(see[1℄). Ithasthefollowinggeometridesription
(for further details, see Setion 2). Phase spae is C 6
and there are four independent
quadratipolynomialfuntionsH
1
;:::;H
4
whihPoissonommute;infatwewillexhibit
a tri-Hamiltonianstruture forthisintegrable system. TheaÆne surfaes thatappearas
thebersof themap
:C 6
!C 4
:z=(z
1
;:::;z
6
)7!(H
1 (z);H
2 (z);H
3 (z);H
4 (z))
are invariant for the ow of the two ommuting vetor elds X
1
and X
2
and for generi
h 2C 4
the invariant surfae 1
(h) is isomorphi to an aÆne part of the Jaobian of a
(ompat)Riemannsurfae
h
. Wewillonstrutsuhanisomorphism;inlassialtermi-
nologywe separatethevariablesof theintegrablesystem(leadingto expliitsolutions;as
faraswe ouldhekaseparationof variablesforthisintegrablesystemwasnotknown).
From thisisomorphismwean reado, intermsof theWeierstrass pointson
h
,therel-
ativepositionofthefoururveswhiharemissingintheaÆnepartA
h
. Ontheonehand
we an dedue fromit thatthequotient A
h
=T
h
isan aÆne partof anAbelian surfaeof
type (1;4). Onthe other handit willallow usto set up a basiorrespondene between
theaÆnevarietiesA
h
(modulo isomorphism)and thespaeofRiemann surfaesofgenus
two, equipped with a deomposition of their Weierstrass points (modulo isomorphism).
Asthe latter spae isin turnisomorphi to a modulispae of polarizedAbeliansurfaes
of type (1;4) (see [13℄)thisredues themoduliproblemto a questionof determiningthe
moduli (i.e., the moduli spae and the orresponding map) for the underlying Abelian
surfaes oftype (1;4). This redutionwillbedone inSetion3.
Abeliansurfaes oftype(1;4) admit a holomorphimapto projetive spae P 3
and
the image is (in general) an oti surfae with four singular points of order four, as was
shown in [6℄. In order to nd an equation of this oti we rst ompute the Laurent
solutions to the integrable vetor eld X
1
and dedue from it fourindependent setions,
invariantforthegroupation,ofthelinebundle[D
h
℄,whihorrespondsto thedivisorat
innity. When the four singularpoints of order fourare taken as base pointsin P 3
, the
oeÆients of theequation for theoti surfaeare losely relatedto moduliforAbelian
surfaesoftype(1;4). Thusweneedtoreduetheequationoftheotithatwehavefound
to its more symmetri form. This omputationwill be feasiblethanks to therihness of
theunderlyinggeometry.
WhentheotiisreduedtoitssymmetriformitsoeÆientsareexpressedinterms
of the oordinates of the Weierstrass points of
h
rather than h, an expliitdependene
whih disappears in the nal step of the omputation. Indeed, it remains then to pass
from the oeÆients of this oti to the modulispae of Abelian surfaes of type (1;4).
Inidently this moduli spae was onstruted by the seond author when studying the
Garnier system, another a..i. system, whose invariant manifolds are Abeliansurfaes of
type (1;4) (see [13℄). The modulispae is desribed there as a Zariski open subset of a
oneinweightedprojetivespae andwean usetheexpliitlygivenmapfrom thespae
of parameters of the oti to this one to omplete our example: we end up with ve
expliit polynomials in the values h = (h
1
;h
2
;h
3
;h
4
) of the onstants of motion whih
take thesame value(asan elementof weightedprojetivespae) on two setsof onstants
ingeneralone also has to throwout afew badelements.
nalstep willbedoneinSetion 4.
It would be interesting to study the moduli problem for other integrable systems,
suh astheToda lattiesand the lassialintegrabletops.
2. An integrable geodesi ow on SO(4)
ItwasshownbyAdlerand vanMoerbeke(see [1, Theorem4℄)that there existthree
lassesofleft-invariantmetrisonSO(4)forwhihthegeodesiowreduestoanalgebrai
ompletely integrable system (a..i. system) on its Lie algebra so(4). In the sequel, we
willonsidertheseond ase, known asthease of metriII. Insuitableoordinates, the
rst vetor eldX
1
ofthisa..i. system isgiven by thedierentialequations
_ z
1
=2z
5 z
6
;
_ z
4
=2z
2 z
3
; _ z
2
=2z
3 z
4
;
_ z
5
=z
3 (z
1 +z
4 );
_ z
3
=z
5 (z
1 +z
4 );
_ z
6
=2z
1 z
5 :
(1)
The seondvetor eldX
2
,ommuting withX
1
,is given bythedierentialequations
_ z
1
=z
2 z
6
;
_ z
4
=z
2 (2z
3 z
6 );
_ z
2
=z
4 (2z
3 z
6 );
_ z
5
=z
3 z
4
;
_ z
3
=z
4 z
5
;
_ z
6
=z
1 z
2
;
(2)
the vetor elds X
1
and X
2
admit four independent quadrati invariants, given by the
followingexpressions:
H
1
=z 2
3 z
2
5
;
H
2
=z 2
1 z
2
6
;
H
3
=z 2
2 z
2
4
;
H
4
=(z
1 +z
4 )
2
+4(z 2
3 z
2 z
5 z
3 z
6 ):
(3)
ItiseasytoverifythatthereexistpreiselythreelinearlyindependentlinearPoissonstru-
tureson C 6
with respet to whih X
1
and X
2
are Hamiltonian; moreover, these Poisson
strutures are ompatible, implyingthat the integrable system admits a tri-Hamiltonian
struture. Expliitly,forany(;;)2C 3
,thematrix
0
B
B
B
B
B
0 z
6
z
5
0 z
3 2z
6 (z
2 2z
5 )
z
6
0 2z
4
(z
6 2z
3
) 0 z
1 z
4
z
5
2z
4
0 z
5 2z
2
(z
1 +z
4
) 0
0 (2z
3 z
6
) z
5 +2z
2
0 z
3
z
2
z
3 +2z
6
0 (z
1 +z
4
) z
3
0 2z
1
(2z
5 z
2
) z
1 +z
4
0 z
2
2z
1
0
1
C
C
C
C
C
A
isthePoissonmatrixof aPoissonstrutureP
onC
6
. If(;;)6=(0;0;0) thenP
generates the Hamiltonian vetor elds X
1
and X
2
as desribed in the following table;
generatorsforthealgebraofCasimirs ofthese struturesP
alsofollowfrom thetable.
H
1
H
2
H
3
H
4
P
100
0 0 2X
2
2X
1
P
010
0 2(X
1 X
2
) 0 2X
1
P
001
2X
1
0 0 8X
2
Table1
1 2 3 4
whih belongsto some 2
Zariskiopen subsetHof C 4
,theaÆne surfae
A
h
=fz2C 6
jH
i
(z)=h
i
; i=1;:::;4g
isisomorphito anaÆne partoftheJaobian ofaompat Riemannsurfae
h
ofgenus
two (whihdepends onh2H ), A
h
=Ja (
h )nD
h
and thatthevetor eldsX
1
and X
2
are linear 3
when restrited to these surfaes A
h
, thereby proving that the above system
is algebraiompletelyintegrable (see [2, Setion4℄). TheaÆne part A
h
,thedivisor D
h
andtheRiemannsurfae
h
anbedesribedasfollows. FirstnotiethatthegroupT of
involutions, generatedby
1 (z
1
;:::;z
6
)=( z
1
; z
2
;z
3
; z
4
; z
5
;z
6 );
2 (z
1
;:::;z
6
)=( z
1
;z
2
; z
3
; z
4
;z
5
; z
6 );
(4)
ommuteswiththevetor eldsX
1
and X
2
andleavestheaÆnesurfaes A
h
invariant;in
fatthey generate, foranyh2H ,a groupT
h
oftranslationsoverhalfperiodsinthetori
Ja (
h
). Asaonsequene, thedivisorsD
h
arealso stableunderthesetranslations. Fora
morepreisedesriptionofthedivisorsD
h
oneappliesPainleveanalysistothevetoreld
X
1
(or any ombination ofX
1
and X
2
). To do thisone searhes Laurent solutionsto the
dierential equations (1), dependingon ve free parameters (prinipalbalanes). There
arepreiselyfoursuhfamilies,labeledby
1
=1;
2
=1,andthey areexpliitlygiven
asfollows (a;b;:::;e arethefreeparameters).
z
1
=
(a 1)
1
t
1 bt+(b 2
d e)t 2
+O(t 3
)
;
z
2
=
1
2
t
a abt+((a 1)(ae ab 2
)+a 2
d)t 2
+O(t 3
)
;
z
3
=
2
2t
1+bt ((a 1)e+ad ab 2
)t 2
+O(t 3
)
;
z
4
=
1
t
a+abt+t 2
+O(t 3
)
;
z
5
=
1
2
2t
1+bt+dt 2
+O(t 3
)
;
z
6
=
(a 1)
2
t
1+bt et 2
+O(t 3
)
:
(5)
WhenanyofthesefamiliesofLaurentsolutionsissubstitutedintheequationsH
i
(z)=h
i ,
i =1;:::;4, the resulting expressions are independent of t. This leadsto four algebrai
equationsinthevefreeparameters,givingexpliitequationsforanaÆnepart
h of
h .
Eah ofthese equationsis easilyrewrittenas
y 2
=x(1 x)
4x 3
h
1 (4h
1 +h
4 )x
2
+(h
4 h
3 h
2 )x+h
3
: (6)
In what follows, we willrefer to theurve in C 2
,given by(6), as theurve
h
. Inorder
to reover the Riemann surfae
h
from it one hasto adjoin one point whih we denote
by1
h
. Sine there are four familiesof Laurent solutions (5), thedivisor D
h
onsists of
fouropies
h (
1
;
2 ),
2
1
= 2
2
=1,ofthe urve
h ,i.e.,
D
h
=
h
(1;1)+
h
(1; 1)+
h
( 1;1)+
h
( 1; 1):
2
Expliitequations forHwillbegiven inthenext setion.
3
Realle.g. from [7,Chapter 2.7℄ or[8, Chapter11℄ thattheJaobian ofan algebrai
urve isa omplextorus.
Ja (
h ) inP
15
: thesetions ofthe linebundleon Ja(
h
), denedby D
h
,orrespond to
themeromorphifuntionsonJa (
h
)withasimplepole(atworst)atthedivisorD
h and,
in turn, these are found by onstruting those polynomials on C 6
whih have a simple
polein t (atworst)when any of thefour familiesof Laurent solutionsare substituted in
them(see[14, ChapterV℄).Apartfromtheonstant funtionz
0
=1and thefuntionsz
i ,
i=1;:::;6,one easilyndsthe followingindependentfuntions withthisproperty:
z
7
=z
5 (2z
3 z
6 ) z
2 z
3
;
z
8
=z
1 (2z
3 z
6 ) z
4 z
6
;
z
9
=z
4 (2z
5 z
2 ) z
1 z
2
;
z
10
=(2z
5 z
2 )
2
z 2
6
;
z
11
=(2z
3 z
6 )
2
z 2
2
;
z
12
=z
1 z
2 z
3 z
4 z
5 z
6
;
z
13
=z
2 z
3 z
6 z
1 z
4 z
5
;
z
14
=z
2 z
5 z
6 z
1 z
3 z
4
;
z
15
=z
1 z
2 z
5 z
3 z
4 z
6 :
(7)
The embeddingof Ja (
h ) inP
15
is given on theaÆne partA
h
bythemap
:A
h
!P 15
:P =(z
1
;:::;z
6
)7!(1:z
1
(P)::z
15 (P)):
Thesesetionswillbeused laterto onstrut twomaps whih aresimilarto and whih
maptwo dierent quotientsof A
h
birationallyinto P 3
.
3. Linearizing variables
In this setion we show that from the point of view of moduli, the family of aÆne
surfaes A
h
, h 2 H , an be replaed by a family of polarized Abelian surfaes of type
(1;4). Inorder todothiswewillrstonstrutanexpliitmapfromtheaÆnesurfaeA
h
(h 2H ) to an aÆne part of Ja (
h
). We do this byfollowingan algorithm, outlined in
[12, Setion 3℄, whih leadsto linearizing variables forany two-dimensionala..i. system.
We note that although the haraterization of the aÆne surfaes A
h
as aÆne parts of
hyperellipti Jaobians was already given by Adler and van Moerbeke in [2, Setion 4℄,
neitheran expliitmapnorlinearizing variablesfollowfromtheir results.
We dene H to be the set of those h =(h
1
;h
2
;h
3
;h
4 )2 C
4
for whih the urve (6)
is a non-singular urve of genus two, i.e., that its right handside is of degree 5 and has
no multipleroots; notiethat thisentails inpartiular that h
1 h
2 h
3
6=0 forall h 2H . It
willfollowfromouronstrution that, forevery h2H ,A
h
is indeedan aÆnepart ofthe
Jaobian, thereby justifying the notation H . In order to apply the proedure desribed
in [12, Setion 3℄, we x an arbitrary element h 2H and we hoose one omponent, say
C =
h
(1; 1), of the divisor D
h
on Ja (
h
). The meromorphi funtions on Ja (
h )
whih have at worst a doublepole along the divisor C an be obtained by onstruting
thosepolynomialsonC 6
whihhaveatworstadoublepoleintwhentheLaurentsolutions
(5) orresponding to
1
=1,
2
= 1 are substituted into them (and no poles when the
other solutionsaresubstituted). It iseasilyomputedthat thespaeof suhpolynomials
isspannedby
0
=1;
1
=(z
2 +z
4 )(z
3 +z
5 );
2
=(z
3 +z
5 )(z
1 +z
6 );
3
=(z
1 +z
6 )(z
2 +z
4 ); (8)
wherewethink ofthese polynomialsasbeing restritedto A
h
. The mapping, givenon
Ja (
h
)nC by
:Ja (
h
)nC !P 3
:P =(z
1
;z
2
;:::;z
6 )7!(
0 (P):
1 (P):
2 (P):
3 (P))
maps the surfaeJa (
h
) to its Kummer surfae, whih is a singular quartiin P 3
. An
equation for thisquartisurfae an be omputed by eliminating thevariablesz
1
;:::;z
6
(3)forthevariablesz
1
;z
2
;:::;z
6
and substitutingthese valuesintheremainingequation,
theequation fortheKummer surfaeofJa (
h
) an bewritten intheform
2
3 ((
1 +
2 2h
1 )
2
+8h
1
1 )+f
3 (
1
;
2 )
3 +f
4 (
1
;
2
)=0; (9)
wheref
3
(respetivelyf
4
)isa polynomialofdegreethree (respetivelyfour)in
1 and
2 .
Itfollowsfrom(9)(see[12,Theorem9℄)thatasystemoflinearizingvariables(x
1
;x
2 )
isgiven bytheequations
2h
1 (x
1 +x
2 )=
1 +
2 2h
1
; 2h
1 x
1 x
2
=
1
: (10)
Thisishekedinthepresentaseasfollows. Firstmakeuseof(8),torewritetheequations
(10) as
(z
3 +z
5 )(z
2 +z
4
)= 2h
1 x
1 x
2
; (z
3 +z
5 )(z
1 +z
6 )=2h
1 (x
1
1)(x
2
1): (11)
Sine h 2H the variables x
1 and x
2
are bothdierent from 1 and from 0 so that below
we an dividebyx
i
and by x
i
1 asneessary. Deriving theequations(11) withrespet
to thevetor eld X
1
given by (1)we ndthat
_ x
1 x
1
1 +x_
2 x
1
2
=z
1 +z
4 +2z
3
;
_ x
1 (x
1 1)
1
+x_
2 (x
2 1)
1
=z
1 +z
4 +2z
5 :
(12)
Thenwe ansolvetherst threeequationsof(3),togetherwith(11)andthediereneof
thetwoequations in(12) forz
1
;:::;z
6
. Substitutingthese valuesintheseond equation
of (12)we ndthat
_ x
1
x
1 (x
1 1)
2
_ x
2
x
2 (x
2 1)
2
= 1
x
1 x
2
4h
1 +
h
2
(x
1
1)(x
2 1)
+ h
3
x
1 x
2
: (13)
Notie that this equation is linear in x_ 2
1
and x_ 2
2
: Finally we substitute the values for
z
1
;:::;z
6
inthefourth equation of (3)to ndanotherequation whih islinear inx_ 2
1 and
_ x 2
2
,leadingto
_ x 2
i
=
f(x
i )
(x
1 x
2 )
2
; i=1;2;
where
f(x)=x(1 x)[4h
1 x
3
(4h
1 +h
4 )x
2
+(h
4 h
2 h
3 )x+h
3
℄:
(Wenotethattheurvey 2
=f(x)ispreiselytheurve
h
givenby(6).) It followsthat,
in terms of the oordinates x
1
;x
2
given by (10), the dierential equations (1) redue to
theJaobiform
_ x
1
p
f(x
1 )
+ _ x
2
p
f(x
2 )
=0;
x
1 _ x
1
p
f(x
1 )
+ x
2 _ x
2
p
f(x
2 )
=1; (14)
sothat x
1 andx
2
areindeedlinearizingvariables.
Theonstrutionofthese linearizingvariablesleadsto anexpliitmapinto theJao-
bian Ja (
h
) as follows. Realling thatwedenote by1
h
thepoint whih is addedto
h
inordertoompleteitintoaompatRiemannsurfae,themapP 7