The Eynard--Orantin recursion for simple singularities

0.2 MB·English

by  Todor Milanov

THE EYNARD–ORANTIN RECURSION FOR SIMPLE SINGULARITIES
TODORMILANOV
5
1
0 Abstract. Accordingto[9]and[19],theancestorcorrelatorsofanysemi-simple
2
cohomologicalfieldtheorysatisfylocalEynard–Orantinrecursion. Inthispaper,

n we prove that for simple singularities, the local recursion can be extended to a
a globalone. Thespectralcurveoftheglobalrecursionisaninterestingfamilyof
J
RiemannsurfacesdefinedbytheinvariantpolynomialsofthecorrespondingWeyl
5
group. Wealsoprovethatforgenus0and1,thefreeenergiesintroducedin[10]
1
coincideuptosomeconstantfactorswithrespectivelythegenus0and1primary

] potentialsofthesimplesingularity.
G
A
.
h
t Contents
a
m
1. Introduction 1
[
2. Analyticextensionofthekernel ofthelocal Eynard–Orantinrecursion 8

1 3. From localto global 11
v
4. Thefree energies andtheprimarypotentialsingenus0 and 1 15
7
7 AppendixA. Thespaceofholomorphic1-forms 21
6
References 23
3
0
.
1
0
5 1. Introduction
1
:
v 1.1. Motivation. According to [9] and [19], the ancestor correlators of any semi-
i
X simple cohomological field theory satisfy local Eynard–Orantin recursion. The
r term local refers to thefact that thespectral curveis justa disjointunionof several
a
discs. If we are interested in computing specific ancestor Gromov–Witten (GW)
invariants in terms of Givental’s R-matrix, then the local recursion is all that we
need. However,ifwewanttounderstandthenatureofthegeneratingfunctionfrom
the point of view of representations of vertex algebras (see [2]) and integrable sys-
tems (see [15]), then it is important to extend the local recursion to a global one,
i.e., extend the spectral curve and the recursion kernel to global objects (see [3]).
The appropriate spectral curve however, looks quite complicated in general, since
2000Math. Subj.Class. 14D05,14N35,17B69.
Keywordsandphrases: periodintegrals,Frobeniusstructure,Gromov–Witteninvariants,vertex
operators.
1
2 TODORMILANOV
it is parametrized by period integrals. In particular, finding whether an appropriate
generalization of the global Eynard–Orantin recursion [10, 3] exists in the settings
of semi-simple cohomological field theories is a very challenging and important
problem.
In this paper we would like to solve the above problem for simple singularities.
Inthiscase,thespectralcurveturnsouttobeaclassicalRiemannsurfacedefinedby
the invariant polynomials of the monodromy group of non-maximal degree, while
theinvariantpolynomialofmaximaldegreedefinesabranchedcoveringofP1. This
brunched covering was also studied by K. Saito (unfortunately he did not write a
text),becauseitisacoveringofwhathecalledaprimitivedirectionin thespaceof
miniversaldeformationsofthesingularity.
I think that the spectral curve for simple singularities is important also in the
representation theory of the corresponding simple Lie algebras. For example, one
canobtainasimpleproofofthewellknownfactthattheorderoftheWeylgroupis
theproductofthedegrees oftheinvariantpolynomials(seeAppendixA).
Finally, after a small modification our argument should work also for all finite
reflection groups. The spectral curve is a certain family of Hurwitz covers of P1
parametrized by an open subset in the space of orbits of the corresponding reflec-
tion group. It would be interesting to obtain the Frobenius structure on the space
of orbits of the reflection group (see [8, 22]) via the construction of a Frobenius
structureon themodulispaceofHurwitzcovers(see [8], Lecture5).
1.2. Singularity theory. Let f C[x ,x ,x ] be a weigthed-homogeneous poly-
1 2 3
∈
nomial that has an isolated critical point at 0 of ADE type. Such polynomials cor-
respond to the ADE Dynkin diagrams and are listed in Table 1, where we have
included also the Coxeter number h and the Coxeter exponents of the correspond-
ingsimpleLiealgebra.
Table1. Simplesingularities
Type f(x) Exponents h
A xN+1+x2+x2 1,2,...,N N+1
N 0 1 2
D xN 1+x x2+x2 1,3,...,2N 3,N 1 2N 2
N 0− 0 1 2 − − −
E x4+x3+x2 1,4,5,7,8,11 12
6 0 1 2
E x3x +x3+x2 1,5,7,9,11,13,17 18
7 0 1 1 2
E x5+x3+x2 1,7,11,13,17,19,23,29 30
8 0 1 2
Wefix aminiversaldeformation
N
F(t,x) = f(x)+ tv(x), t = (t ,...,t ) B := CN, (1)
i i 1 N
X ∈
i=1
THEEYNARD–ORANTINRECURSIONFORADESINGULARITIES 3
where v(x) N isaset ofweighted-homogeneouspolynomialsthatrepresent a ba-
{ i }i=1
sisoftheJacobialgebra
H = C[x ,x ,x ]/(f , f , f ).
1 2 3 x1 x2 x3
Theformω := dx dx dx isprimitiveinthesenseofK.Saito[21,24]andthespace
1 2 3
B inherits a Frobenius structure (see [16, 23]). For somebackground on Frobenius
structureswereferto[8]. TheFrobeniusmultiplicationonT Bisobtainedfromthe
t
multiplicationintheJacobialgebraof F(t, )viatheKodaira-Spencerisomorphism
·
T B (cid:27) C[x ,x ,x ]/(F (t,x),...,F (t,x)), ∂/∂t ∂F/∂t.
t 1 2 3 x1 x3 i 7→ i
WhiletheFrobeniuspairing (, )on TBistheresiduepairing
1 φ (x)φ (x)
(φ (x),φ (x)) := 1 2 dx ...dx ,
1 2 t (2π√−1)3 ZΓ Fx1(t,x)···Fx3(t,x) 1 N
where the cycle Γ is a disjoint union of sufficiently small tori around the critical
points of F defined by equations of the type F = = F = ǫ. In particular,
| x1| ··· | x3|
wehavethefollowingidentifications:
T B (cid:27) TB (cid:27) B T B (cid:27) B H,
∗ 0
× ×
wherethefirstisomorphismisgivenbytheresiduepairing,thesecondbytheLevi–
Civitaconnectionoftheflatresiduepairing,andthelastoneistheKodaira–Spencer
isomorphism
T B (cid:27) H, ∂/∂t ∂ F mod(f ,..., f ). (2)
0 i 7→ ti t=0 x1 x3
(cid:12)
Let B B be the subset of semi-simple(cid:12)points, i.e., points t B such that the
ss (cid:12)
⊂ ∈
critical valuesof F(t, )forma coordinatesysteminaneighborhoodoft. Forevery
·
t B ,usingGivental’shigher-genusreconstructionformalism[12,13],wedefine
ss
∈
ancestorcorrelationfunctionsofthefollowingform (c.f. [19])
a ψk1,...,a ψkn (t), a H, k Z (1 i n). (3)
h 1 1 n n ig,n i ∈ i ∈ ≥0 ≤ ≤
A priory, each correlator depends analytically on t B , but it might have poles
ss
∈
along the divisor B B . According to [20] the correlation functions (3) extend
ss
\
analyticallytotheentiredomain B.
1.3. The period vectors. Put X = B C3 and S = B C. Let Σ S be the
× × ⊂
discriminantofthemap
ϕ : X S, ϕ(t,x) := (t,F(t,x)).
→
RemovingthesingularfibersX = X ϕ 1(Σ)weobtainasmoothfibrationX S ,
′ − ′ ′
where S = S Σ, known as the Mil\norfibration. Let us denote by X = ϕ →1(t,λ)
′ t,λ −
the fiber over (\t,λ) S . The vector spaces H2(X ;C) and H (X ,C) form the so
′ t,λ 2 t,λ
∈
called vanishing cohomology and homology bundles. They are equipped with flat
Gauss–Maninconnections.
4 TODORMILANOV
We fix (0,1) S as a reference point and denote by h := H2(X ,C). The dual
0,1
space h = H (X∈ ,C) is equipped with a non-degenerate intersection pairing and
∗ 2 0,1
wedenoteby( )thenegativeoftheintersectionpairing,sothat(αα) = 2forevery
| |
vanishing cycle α. The set R of all vanishing cycles together with the pairing ( )
|
is a root systemof type ADE. Moreover, according to the Picard–Lefschetz theory
(see[1]) theimageofthemonodromyrepresentation
π (S ) GL(h ) (4)
1 ′ ∗
→
is theWeyl group of R, i.e., the monodromytransformation s along a simpleloop
α
around the discriminant corresponding to a path along which the cycle α vanishes
isthefollowingreflection
s (x) = x (αx)α, α R, x h .
α ∗
− | ∈ ∈
Letusintroducethenotationd ,where x = (x ,...,x )isacoordinatesystemon
x 1 m
some manifold, for the de Rham differential in the coordinates x. This notation is
especiallyusefulwhenwehavetoapplyd tofunctionsthatmightdependonother
x
variablesas well. Themainobject ofourinterestare thefollowingperiodintegrals
I(k)(t,λ) = d (2π) 1∂k+1 d 1ω T B (cid:27) H, (5)
α − t − λ Z −x ∈ t∗
αt,λ
where α h is a cycle from the vanishing homology, α H (X ,C) is the
t,λ 2 t,λ
parallel tra∈nsportofαalongareference path,and d 1ω is any2∈-form η Ω2 such
that d η = ω. The periods are multivaluedanalytic−xfunctions in (t,λ) B∈ CC3 with
x
polesalongthediscriminantΣ. ∈ ×
1.4. The period isomorphism. Let us fix a coordinate system t = (t ,...,t ) on
1 N
B defined by a miniversal unfolding of f of the type (1). We may assume that
v (x) = 1 and denote by t λ1 the point with coordinates (t ,t ,...,t λ). Note
N 1 2 N
that X = X ,so thepe−riodvectorshavethefollowingtranslationsy−mmetry
t,λ t λ1,0
−
I(k)(t,λ) = I(k)(t λ1,0). (6)
α α −
Sometimeswe restrict theperiod integralsto λ = 0 and itwill beconvenientto use
as a reference point 1 B. Note that this choice is compatible with the choice
of the other reference−po∈int (0,1) B C in a sense that the values of the period
∈ ×
vectorsat thesetwopointsare identifiedviathetranslationsymmetry(6).
NowwecanstatethefollowingresultthatgoesbacktoLooijenga[18]andSaito
[21]. Themonodromycoveringspaceof B := S Bisthecovering B of B corre-
′ ′ ′ ′
∩
sponding to the kernel of the monodromy representation (4). It can be constructed
e
as the set of equivalence classes of pairs (t,C), where t B and C is a path in B
′ ′
∈
from the reference point 1 to t and the equivalence relation (t ,C ) (t ,C ) is
1 1 2 2
t = t andC C 1 isint−hekernelofthemonodromyrepresentation(4∼). Notethat
1 2 1◦ 2−
THEEYNARD–ORANTINRECURSIONFORADESINGULARITIES 5
the period integrals are by definition functions on B . In particular we have a well
′
defined periodmap
e
Φ : B h , Φ(C,t),α := (I( 1)(t,0),1),
′ → ′ h i α−
where h is the coempleement in h oef the reflection hyperplanes of the roots R, i.e.,
′
h = x h α,x , 0 α R . The first statement is that Φ is an analytic
′
isomo{rph∈ism.| hIn pairticula∀r, th∈ere i}s an induced isomorphism Φ : B h /W.
The 2nd statement is that Φ extends analytically across the discreimin′an→t and′ the
extension provides an analytic isomorphism B (cid:27) h/W := Spec(S(h )W). Using the
∗
isomorphismΦ and thenatural projection B B we can thinkof thecoordinates
′ ′
→
t as W-invariant holomorphic functions on h . The 2nd statement is equivalent
i e e ′
to saying that each coordinate t extends holomorphically through the reflection
i
mirrors, the extension is in fact a W-invariant polynomial in h, and the ring of all
W-invariantpolynomialsisS(h )W = C[t ,...,t ].Wereferto[18,21]fortheproof
∗ 1 N
ofallthesestatements.
1.5. The spectral curve. Let us fix a set of simple roots α N h and denote
by x = (x ,...,x ) the coordinatesystem in h correspondin{git}oi=t1he⊂ba∗sis of funda-
1 N
mentalweights ω N h, i.e.,
{ i}i=1 ⊂
N
x = xω, x = α,x .
i i i i
X h i
i=1
As explained above t C[x ,...,x ]W are invariant polynomials and since the
i 1 N
∈
periodmappingisweighted-homogeneous,t arehomogeneouspolynomialsofcer-
i
tain degrees d. Let us assume that the degrees are in an increasing order, then the
i
numbers 1 = d 1 d 1 ...d 1 =: h 1 are known as the Coxeter
1 2 N
exponents(see Ta−ble1≤). Giv−en s≤ CN 1 w−edefine th−ealgebraiccurveV PN
− s
∈ ⊂
t(X ,...,X ) = sXdi, 1 i N 1.
i 1 N i 0 ≤ ≤ −
As we will see later on if s B , then V is non-singular. In fact, the points s for
ss s
∈
which V has singularities are precisely the caustic B B . I am not aware if the
s ss
familyofalgebraiccurvesV ,s B hasanofficialnam−eattached,butsinceitwill
s ss
∈
be the spectral curve for the EO recursion, we will refer to it as the spectral curve
ofthesingularityorjustthespectralcurvewhenthesingularityisunderstoodfrom
thecontext.
Thereisanatural projection
λ : V P1, [X ,X ,...,X ] [Xh,t (X ,...,X )], (7)
s → 0 1 N 7→ 0 N 1 N
which is a branched covering of degree W , where A denotes the number of ele-
ments of the set A. The branching point|s a|re λ = u| ,|...,u , , where u are the
1 N i
∞
critical values of F(s,x). By definition, the period integral (I( 1)(s,λ),1) defines
−
6 TODORMILANOV
locallynearanon-branchingpointλ P1 asectionofthebranched covering(7). It
∈
followsthat theset oframificationpoints
λ 1(u), 1 i N
− i
≤ ≤
isprecisely theintersectionsofV and thereflection mirrors
s
N
α,X = α,ω X = 0, α R ,
i i +
h i Xh i ∈
i=1
whereR isthesetofpositiveroots. Theremainingramificationpointsareλ 1( ).
+ −
∞
They correspond to eigenvectors of the Coxeter transformations with eigenvalue
η := e2π√ 1/h:
−
[X ,X ,...,X ] λ 1( )
0 1 N −
∈ ∞
if and only if X = 0 and N Xω h is an eigenvector with eigenvalue η for a
0 i=1 i i ∈
Coxeter transformation. ItPis easy to see that the ramification index of any point in
λ 1(u)is2, whiletheramificationindexofapointinλ 1( )ish.
− i −
∞
1.6. TheEynard–Orantinrecursion. Wemakeuseofthefollowingformalseries
fα(t,λ;z)= I(k)(t,λ)( z)k, φα(t,λ;z) = I(k+1)(t,λ)( z)kdλ.
X α − X α −
k Z k Z
∈ ∈
Note that φα(t,λ;z) = d fα(t,λ;z). Given n cycles α ,...,α and a semi-simple
λ 1 n
point s B wedefinethefollowingn-pointsymmetricforms
ss
∈
ωgα,1n,...,αn(s;λ1,...,λn) = φα+1(s,λ1;ψ1),...,φα+n(s,λn;ψn) (s), (8)
g,n
D E
where the + means truncation of the terms in the series with negativepowers of z.
Thefunctions(8)willbecalledn-pointseriesofgenusgorsimplycorrelatorforms.
The ancestor correlators (3) are known to be tame (see [14]), which by definition
means that they vanish if k + + k > 3g 3 + n. Hence the correlator (8) is
1 n
··· −
a polynomial expression of the components of the period vectors (5). Thanks to
the translation symmetry, we may assume that s = 0, then (8) is a meromorphic
N
function on the spectral curve V V with possible poles at the ramification
s s
×···×
pointsofthecovering(7).
Let us fix s = (s ,...,s ) CN 1 and denote by γ h an arbitrary cycle, s.t.,
1 N 1 − ∗
(γα) , 0 for all α R. We−defi∈ne a set ofsymmetricm∈eromorphicdifferentialson
| ∈
Vn withpolesalongtheramificationpointsofV
s s
ω (s;p ,...,p ) := ωγ,...,γ(s;λ ,...,λ ), (9)
g,n 1 n g,n 1 n
where the RHS is defined by fixing a reference path for each (s,λ) S , s.t.,
i ′
p = (I( 1)(s,λ),1). Ourmain resultcan bestatedas follows. ∈
i − i
THEEYNARD–ORANTINRECURSIONFORADESINGULARITIES 7
Theorem 1.1. If s B , then the forms ω , 2g 2 + n > 0, satisfy the Eynard–
ss g,n
∈ −
Orantinrecursion associatedwith thebranchedcovering (7) and themeromorphic
function f : V P1, f (x) := γ,x .
γ s γ
→ h i
Therecursion willberecalled lateron (seeSection 3.3). Wewouldlikehowever
toemphasizethattheEynard–Orantinrecursioninourcasediffersfromthestandard
onebytheinitialcondition:
ω (x,y) = (γwγ)B(x,wy), (10)
0,2
X |
w W
∈
where B(x,y) is the Bergman kernel of V and for x = wy one has to regularize
s
the RHS by removing an appropriate singular term (see Section 3.3). Let us point
out that while the Bergman kernel depends on the choice of a Torelli marking of
V , i.e., a symplectic basis of H (V ;Z), our initial condition is independent of the
s 1 s
Torellimarking(seeCorollary2.5). Thisfactcouldbeprovedalsodirectlybyusing
the explicit formula (24) for the holomorphic 1-forms and some standard facts for
W-invariant polynomials. Finally, let us point out that the set of correlators (9)
determinestheset(8),because bydefinition
ωw1γ,...,wNγ(s;λ ,...,λ ) = ω (s;w 1p ,...,w 1p ),
g,n 1 N g,n −1 1 −N N
wherew ,...,w W are arbitrary.
1 N
∈
The branched covering (7) and the meromorphic function f determine a bi-
γ
rational model of V in C2. Following [10] we can introduce the tau-function
s
Z(~,s) := Z (V ,λ, f ) of the birational model of the spectral curve. It has the
~ s γ
form
∞
Z(~,s) = exp ~g 1F(g)(s) ,
−
(cid:16)Xg=0 (cid:17)
where F(g)(s) is called the genus-g free energy of V . It is very natural to compare
s
F(g)(s) with Givental’sprimary genus-g potentials F(g)(s). Unfortunately we could
notsolvethisproblemingeneral, butonlyfor g = 0 and g = 1
W W
F(0)(s) = (γγ)| |F(0)(s), F(1)(s) = | |F(1)(s) = 0,
− | N 2
where the first identity is valid up to quadratic terms in the Frobenius flat coordi-
nates of s B, whilethesecond oneup to a constant independent of s. It is known
∈
thatthegenus-1potentialoftheFrobeniusstructureishomogeneousofdegree0,so
itmustvanishinthecaseofasimplesingularity. Accordingto[10],Z(~,s)satisfies
Hirotabilinearequations. Accordingto[15],thetotalancestorpotential (~;q)of
s
A
anADEsingularitysatisfiestheHirotabilinearequationsofthecorrespondinggen-
eralized KdV hierarchy. It willbeinterestingto clarify therelation between F(g)(s)
and F(g)(s) for g 2, as well as to determine whether Givental’s primary ancestor
potential (~;0)≥alsosatisfies Hirotabi-linearequations.
s
A
8 TODORMILANOV
2. Analytic extension ofthekernel ofthelocal Eynard–Orantin recursion
It was proved in [19] (see also [9]) that the correlator forms (8) satisfy a local
Eynard–Orantin (EO) recursion, whose kernel is defined by the symplecticpairing
of certain period series. In this section, we will prove that these symplectic pair-
ings are convergent and can be extended to the entire spectral curve V . Moreover,
s
the corresponding extensions can be expressed in an elegant way via the so called
Bergmankernel ofV .
s
2.1. The kernel ofthelocal recursion. Recall thesymplecticpairing
Ω(f(z),g(z)) = Res (f( z),g(z))dz, f,g H((z 1)).
z=0 −
− ∈
Thelocalrecursion isdefined intermsofthesymplecticpairings
Ω(φα(s,λ;z),fβ(s,µ;z)) = dλ ∞ ( 1)k+1(I(k+1)(s,λ),I( k 1)(s,µ)),
+ − X − α β− −
k=0
where the infinite series is interpreted formally in a neighborhood of a point µ =
u(s), s.t., the cycle β vanishes over µ = u(s). We are going to prove that this
i s,µ i
infiniteseries expansionis convergenttoa meromorphicfunctionon V V .
s s
Tobeginwith,letusrecallthattheperiodssatisfythefollowingsystem×ofdiffer-
entialequations
∂ I(n)(s,λ) = v I(n+1)(s,λ)
a a s
− •
∂ I(n)(s,λ) = I(n+1)(s,λ)
λ
1
(λ E )∂ I(n)(s,λ) = θ n I(n)(s,λ), (11)
s λ
− • − − 2
(cid:16) (cid:17)
where ∂ := ∂/∂s , E is the Euler vector field E = N deg(t)t∂ , and θ is the
a a i=1 i i ti
Hodge-gradingoperator P
θ : H H, v (D/2 deg(v ))v ,
a a a
→ 7→ −
where D = deg(Hess(f)) = 1 2/h is the conformal dimension of the Frobenius
−
structure. The key to proving the convergence is the so called phase 1-form (see
[14, 2])
(s,ξ) = I(0)(s,ξ) I(0)(s,0) T S,
Wα,β α • β ∈ s∗
wheretheperiodvectorsareinterpretedaselementsinT S andthemultiplicationin
s∗
T S isinducedbytheFrobeniusmultiplicationviathenaturalidentificationT S (cid:27)
s∗ s∗
T S. The dependence on the parameter ξ is in the sense of a germ at ξ = 0, i.e.,
s
Taylor’sseries expansionaboutξ = 0. Thephaseformisapowerseriesinξ whose
coefficients are multivalued1-forms on B .
′
THEEYNARD–ORANTINRECURSIONFORADESINGULARITIES 9
Lemma 2.1. Wehave
(αβ) = ι (s,0) = (I(0)(s,0),E I(0)(s,0)).
| − EWα,β − α • β
Thisisawell knownfact dueoriginallytoK. Saito[21].
Lemma 2.2. The phaseformisweighted-homogeneousof weight0, i.e.,
(ξ∂ +L ) (s,ξ) = 0,
ξ E α,β
W
where L istheLiederivativewithrespect to thevector field E.
E
Proof. Notethat
(s,ξ) = (I(0)(s,ξ),dI( 1)(s,0)).
Wα,β α β−
Itiseasytocheckthat isaclosed1-form,sousingtheCartan’smagicformula
α,β
L = d ι +ι d , wherWeι isthecontractionbythevectorfield E, weget
E s E E s E
L = d (I(0)(s,ξ),(θ+1/2)I( 1)(s,0)) = d ((θ 1/2)I(0)(s,ξ),I( 1)(s,0)).
EWα,β s α β− − s − α β−
Weused thatθ isskew-symmetricwithrespect totheresiduepairingand that
ι d I( 1)(s,0) = EI( 1)(s,0)) = (θ+1/2)I( 1)(s,0),
E s β− β− β−
where the last equality comes from the differential equation (11) with n = 1 and
λ = 0. Furthermore, usingtheLeibnitzruleweget −
((θ 1/2)d I(0)(s,ξ),I( 1)(s,0)) ((θ 1/2)I(0)(s,ξ),d I( 1)(s,0)).
− − s α β− − − α s β−
Thefirst residuepairing is
(AI(1)(s,ξ),(θ+1/2)I( 1)(s,0)) = (AI(1)(s,ξ),E I(0)(s,0)), (12)
α β− − α • β
whereweusedthatθisskew-symmetricandthatd I(0) = AI(1)withA = N (∂/∂s )ds .
s α − α a=1 a• a
Similarly,the2ndresiduepairingbecomes P
((ξ∂ +E)I(0)(s,ξ),d I( 1)(s,0)) = ξ∂ (s,ξ)+(E I(1)(s,ξ),AI(0)(s,0)). (13)
ξ α s β− ξWα,β • α β
On the otherhand, since theFrobenius multiplicationis commutative,[A,E ] = 0,
sotheterms(12)and (13)add upto ξ∂ (s,ξ),whichcompletesthepro•of. (cid:3)
ξ α,β
W
Let us define the following meromorphic 1-forms on V V . Given (x,y)
s s
× ∈
V V ,s.t., x,yarenotramificationpoints,thereareuniquepairs(C,λ)and(C ,µ),
s s ′
×
s.t., x = Φ(C,s λ1) and y = Φ(C ,s µ1), where C and C are paths in B
′ ′ ′
− −
connectingrespectively s λ1and s µ1 withthereference point. Put
e − e −
dλ
K (s,x,y) := (I(0)(s,λ),(θ+1/2)I( 1)(s,µ)),
α,β λ µ α β−
−
10 TODORMILANOV
where the branches of I(0) and I( 1) are determined respectively by the pathsC and
α β−
C . This definition extends analytically across the ramification points and the form
′
has apoleonlyalong thedivisor
(x,y) V V : λ(x) = λ(y) = x = wy ,
s s w W
{ ∈ × } ∪ ∈ { }
wheretheactionofW on his inducedfromtheactionon h , i.e.,
∗
α,wx = w 1α,x , α h , x h.
− ∗
h i h i ∈ ∈
Proposition2.3. Thesymplecticpairing
Ω(φα(s,λ;z),fβ(s,µ;z)) = K (s,x,y),
+ α,β
−
where x = Φ(C,s λ1), y = Φ(C,s µ1) andC is thepath that specifies thevalue
− −
ofthesymplecticpairing.
e e
Proof. Usingthedifferentialequationsfortheperiods, itiseasy to verifythat
d Ω(φα(s,λ;z),fβ(s,µ;z)) = dλI(1)(s,λ) I(0)(s,µ) = d (s µ1,λ µ).
s + − α •s β λWα,β − −
Accordingto Lemma2.2 wehave
1
∂ (s ,λ µ) = d ι (s ,λ µ) , (14)
λ α,β ′ s E α,β ′
W − − ′ λ µ W −
(cid:16) (cid:17)
−
whichby definitionis
1
d (I(0)(s ,λ µ),(θ+1/2)I( 1)(s ,0) .
s′ λ µ α ′ − β− ′
(cid:16) (cid:17)
−
Integrating(14)withrespect to s alongashortpath from s := s u(s)1 to s µ1
′ 0 i
− −
and usingthat I( 1)(s ,0)vanishesas s s , weget
β− ′ ′ → 0
dλ
Ω(φα(s,λ;z),fβ(s,µ;z)) = (I(0)(s,λ),(θ+1/2)I( 1)(s,µ)). (cid:3) (15)
+ λ µ α β−
−
−
2.2. ThelocalkernelandtheBergmankernel. Nowweareinapositiontoprove
the key result in this paper. Let us fix a symplectic basis , g of H (V ;Z),
{Ai Bi}i=1 1 s
s.t., = δ . Thereisauniquesymmetricdifferential B(x,y) Ω1 ⊠Ω1 (2∆)
Ai◦B| i,j ∈ Vs Vs
whichisholomorphiconV V exceptforapoleoforder2withnoresiduealong
s s
thediagonal∆ V V , no×rmalizedby
s s
⊂ ×
B(x,y) = 0, 1 i N
I ≤ ≤
y i
∈A
and
dλ(x)dλ(y)
B(x,y) = +
(λ(x) λ(y))2 ···
−
for any local coordinate λ : U C (U V ) and for all x,y U U. The
s
differential B(x,y) is called the B→ergman ke⊂rnel. We refer to [10] f∈or m×ore details
and references.