ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS OF AMPLE LINE BUNDLES

12 Pages·2002·0.21 MB·English

PROCEEDINGSOFTHE
AMERICANMATHEMATICALSOCIETY
S0002-9939(02)06557-7
ArticleelectronicallypublishedonMay8,2002
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS
OF AMPLE LINE BUNDLES ON SYMPLECTIC MANIFOLDS:
AN ADDENDUM
BERNARD SHIFFMAN AND STEVE ZELDITCH
(Communicated byChristopherD.Sogge)
Abstract. Wede(cid:12)neaGaussianmeasureonthespaceH0(M;LN)ofalmost
J
holomorphic sections of powers of an ample line bundle L over a symplectic
manifold(M;!),andcalculatethejointprobabilitydensitiesofsectionstaking
prescribed values and covariant derivatives at a (cid:12)nite number of points. We
prove that they have a universalscalinglimitas N !1. This resultwillbe
used in another paper to extend our previous work on universality of scaling
limitsofcorrelationsbetween zerostothealmost-holomorphicsetting.
1. Introduction
Thisnoteisanaddendumtoourstudyin[ShZe2]ofalmostholomorphicsections
of powers of ample line bundles LN ! M over almost complex symplectic man-
ifolds (M;!;J). Motivated by the important role now played by ‘asymptotically
holomorphic’sectionsinsymplecticgeometry(see[Aur,Don]andmanyrelatedar-
ticles), we studied in [ShZe2] a conceptually related but di(cid:11)erent class H0(M;LN)
J
of ‘almost holomorphic’ sections de(cid:12)ned by a method of Boutet de Monvel and
Guillemin [BoGu]. The main results of [ShZe2] were the scaling limit law of the
almost-complex Szeg(cid:127)o projectors
(cid:5) :L2(M;LN)!H0(M;LN)
N J
and various geometric consequences of it.
Our purpose in this addendum is to develop[ShZe2]in a probabilistic direction,
inthespiritofourearlierworkwithP.Bleherontheholomorphiccase[BSZ1],and
tocompletethediscussionin[BSZ2]ofcorrelationsbetweenzerosinthesymplectic
case. The space H0(M;LN) is (cid:12)nite dimensional and carries a natural Hermit-
J
ian inner product (see (7) and (14)). We endow the unit sphere SH0(M;LN) in
J
H0(M;LN) with Haar probability measure (cid:23) and consider the joint probability
J N
distribution (JPD)
DN =DN(x1;:::;xn;(cid:24)1;::: ;(cid:24)n;z1;:::;zn)dxd(cid:24)
(z1;:::;zn)
ofarandomsections 2SH0(M;LN)havingtheprescribedvaluesx1;:::;xn and
N
derivatives (cid:24)1;::: ;(cid:24)n at n points z1;:::;zn 2 M. To be more precise, we choose
ReceivedbytheeditorsAugust3,2001.
2000 Mathematics Subject Classi(cid:12)cation. Primary53D50,53D35,60D05.
Research partially supported by NSF grants #DMS-9800479, #DMS-0100474 ((cid:12)rst author)
and#DMS-0071358(second author).
(cid:13)c2002 American Mathematical Society
1
2 BERNARD SHIFFMAN AND STEVE ZELDITCH
localcomplex coordinatesfz ;:::;z ganda nonvanishinglocalsectione ofL on
1 m L
an open set containing the points fz1;:::;zng; then DN is the JPD of the
(z1;:::;zn)
n(2m+1) complex random variables
(cid:10) (cid:11)(cid:12) (cid:10) (cid:11)(cid:12)
(1) xp =(cid:10) e(cid:3)LN;sN (cid:12)zp; (cid:24)qp =(cid:11)(cid:12) e(cid:3)LN;N(cid:0)12r@=@zqsN (cid:12)zp;
(cid:24)mp+q = e(cid:3)LN;N(cid:0)12r@=@z(cid:22)qsN (cid:12)zp; 1(cid:20)p(cid:20)n;1(cid:20)q (cid:20)m:
As the name implies, almost holomorphic sections behave in the large N limit
much as do holomorphic sections on complex manifolds. The main result of this
notebearsthisoutbyshowingthattheJPDinthealmostcomplexsymplecticcase
hasthesameuniversalscalinglawasinthe holomorphiccase,thereby(cid:12)nishingthe
proof of Theorem 1.2 of [BSZ2] on universal scaling limits of correlations between
zeros in the symplectic case.
Theorem 1.1. Let L be the complex line bundle over a 2m-dimensional compact
integral symplectic manifold (M;!) with curvature !: Let P 2 M, and choose a
0
local frame e for L and local complex coordinates centered at P so that !j and
0 P0
gj are the usual Euclidean Ka(cid:127)hler form and metric respectively, ke k =1, and
P0 L P0
re j =0. Then
L P0
(cid:0) (cid:1)
DN p p (cid:0)!D1 (z1;:::;zn)2Cmn ;
(z1= N;:::;zn= N) (z1;:::;zn)
where the measures D1(z1;:::;zn) =(cid:13)(cid:1)1(z) are the same universal Gaussian measures
as in the holomorphic case; in particular, they are supported on the holomorphic
1-jets.
Let us review the formula for (cid:13)(cid:1)1(z). We recall that a (complex) Gaussian
measure on Ck is a measure of the form
e(cid:0)h(cid:1)(cid:0)1z;z(cid:22)i
(2) (cid:13) = dz;
(cid:1) (cid:25)kdet(cid:1)
where dz denotes Lebesgue mea(cid:0)sure o(cid:1)n Ck, and (cid:1) is a positive de(cid:12)nite Hermitian
k(cid:2)k matrix. The matrix (cid:1)= (cid:1) is the covariance matrix of (cid:13) :
(cid:11)(cid:12) (cid:1)
(3) hz z(cid:22) i =(cid:1) ; 1(cid:20)(cid:11);(cid:12) (cid:20)k :
(cid:11) (cid:12) (cid:13)(cid:1) (cid:11)(cid:12)
Sincetheuniversallimitmeasures(cid:13)(cid:1)1(z) vanishalongnon-holomorphicdirections,
theyaresingularmeasuresonthespaceofall1-jets. Todealwithsingularmeasures,
weintroduceinx3generalizedGaussianmeasureswhosecovariancematrices(3)are
only semi-positive de(cid:12)nite; a generalized Gaussian measure is simply a Gaussian
measuresupportedonthesubspacecorrespondingtothepositiveeigenvaluesofthe
covariance matrix.
Thecovariancematrix(cid:1)1(z)isgivenalongholomorphicdirectionsbythesame
formula as in the holomorphic case [BSZ1, (97)], namely
(cid:18) (cid:19)
1 m! A1(z) B1(z)
(4) (cid:1) (z)= ;
c (L)m B1(z)(cid:3) C1(z)
1
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 3
where z =(z1;:::;zn) and
A1(z)pp0 = ((cid:5)H1 (zp;0;zp0;0); (cid:5)H1 (u;0;v;0)= (cid:25)1meu(cid:1)v(cid:22)(cid:0)12(juj2+jvj2) ;
B1(z)p = (zqp0 (cid:0)zqp00)(cid:5)H1 (zp;0;zp0;0) for 1(cid:20)q (cid:20)m;
p0q0 0 for m+1(cid:20)q (cid:20)2m;
8 (cid:16) (cid:17)
>< (cid:14)qq0 +(z(cid:22)qp0 (cid:0)z(cid:22)qp)(zqp0 (cid:0)zqp00) (cid:5)H1 (zp;0;zp0;0)
C1(z)pp0qq0 = >: for 1(cid:20)q;q0 (cid:20)m;
0 for max(q;q0)(cid:21)m+1:
In other words,the coe(cid:14)cients of (cid:1)1(z) correspondingto the anti-holomorphic
directionsvanish,whilethecoe(cid:14)cientscorrespondingtotheholomorphicdirections
are given by the Szeg(cid:127)o kernel (cid:5)H for the reduced Heisenberg group (see [BSZ1,
1
x1.3.2]) and its covariant derivatives.
A technically interesting novelty in the proof is the role of the @(cid:22) operator. In
the holomorphic case, DN is supported on the subspace of jets of sections
(z1;:::;zn)
satisfying@(cid:22)s=0. Inthealmostcomplexcase,sectionsdonotsatisfythisequation,
so DN is a measure on the higher-dimensional space of all 1-jets. However,
(z1;:::;zn)
Theorem 1.1 says that the mass in the non-holomorphic directions shrinks to zero
as N !1.
An alternate statement of Theorem 1.1 involves equipping H0(M;LN) with a
J
e
Gaussian measure, and letting DN be the corresponding joint probability
(z1;:::;zn)
distribution, which is a Gaussian measure on the complex vector space of 1-jets
e
of sections. We show (Theorem 4.1) that these Gaussian measures DN also have
the same scaling limit D1, so that asymptotically the probabilities are the same
as in the holomorphic case, where universality was established in [BSZ2]. It is
e
then easy to see that DN =(cid:13) , where (cid:1)N is the covariance matrix of the
(z1;:::;zn) (cid:1)N
randomvariables in (1). The main step in the proofis to show that the covariance
e
matrices (cid:1)N underlying DN tend in the scaling limit to a semi-positive matrix
(cid:1)1. It follows that the scaled distributions DeN tend to the generalized Gaussian
(cid:13)(cid:1)1 ‘vanishing in the @(cid:22)-directions.’
In a subsequent article [ShZe3], we obtain further probabilistic results on holo-
morphicandalmostholomorphicsections. Regardingalmostholomorphicsections,
we prove that a sequence fs g of almost holomorphic sections sati(cid:12)es the bounds
N
p p
ksNk1=ksNkL2 =O( logN); k@(cid:22)sNk1=ksNkL2 =O( logN)
almost surely. Hence almost holomorphic sections satisfy bounds similar to those
satis(cid:12)ed by asymptotically holomorphic sections in [Don, Aur].
Finally,wementionsomeintriguingquestionsrelatingourprobabalisticapproach
toalmostholomorphicsectionstothe now-standardcomplexity-theoreticapproach
to asymptotically holomorphic sections in symplectic geometry, due to Donaldson
andfurtherdevelopedbyAurouxandothers. Fromananalyticalviewpoint(which
ofcourseisjustonetechnicalsideoftheirwork),thekeyresultsareexistencetheo-
rems for one or severalasymptoticallyholomorphic sections satisfying quantitative
transversality conditions, such as
s(z)=0 =)k@(cid:22)s(z)j<j@s(z)j 8z 2M
4 BERNARD SHIFFMAN AND STEVE ZELDITCH
in the case of one section. Can one use the probabalistic method to prove such ex-
istence results? It is the global nature of the problem which makes it di(cid:14)cult. On
small balls, our methods rather easily give lower bounds for quantitative transver-
sality of the type
C
(cid:22)fs: j@(cid:22)s(z)j<j@s(z)j 8z 2B (z )such thats(z)=0g > 1(cid:0) " :
pD 0 N1(cid:0)"
N p
However,thereareroughlyC Nm ballsofradius1= N,soonecannotsimplysum
m
this small-ball estimate. To globalize, one would need to partition M into small
cubesasin[Don]andthenanalyzethedependenceoftransversalityconditionsfrom
one cube to another.
Remark. Thispaperand[ShZe2]wereoriginallycontainedinthearchivedpreprint
[ShZe1]. Inrevisingthatpaper forpublicationas[ShZe2],we expandedthe section
ontheSzego(cid:127)kernel,addedadiscussionontransversality,andrelocatedthematerial
on the JPD to this article. In particular, Theorem 1.1 is Theorem 0.2 of [ShZe1],
and Theorem 4.1 is Theorem 5.4 of [ShZe1].
2. Background
To avoid duplication, we only set up some basic notation and background, and
refer to [BSZ1, BSZ2, ShZe2] for further discussion and details.
Let (M;!;J) denote a compact almost-complex symplectic manifold such that
[1!] is an integral cohomology class, and such that !(Jv;Jw) = !(v;w) and
(cid:25)
!(v;Jv) > 0. We further let (L;h;r) ! M denote a quantizing Hermitian line
bundle andmetric connectionwith curvature i(cid:2) =!. We denoteby LN the Nth
2 L
tensor power of L.
We give M the Riemannian metric g(v;w) = !(v;Jw). We denote by T1;0M;
resp. T0;1M, the holomorphic, resp. anti-holomorphic, sub-bundle of the complex
tangent bundle TM; i.e., J =i on T1;0M and J =(cid:0)i on T0;1M.
We now recall our notion of ‘preferred coordinates’ from [ShZe2]. They are
important because the universal scaling laws are only valid in such coordinates. A
coordinatesystem(z ;:::;z )onaneighborhoodU ofapointP 2M ispreferred
1 m 0
at P if any two of the following conditions (and hence all three) are satis(cid:12)ed:
0
i) @=@z j 2T1;0(M), for 1(cid:20)j (cid:20)m,
j P0
ii) !(P )=! ,
0 0
iii) g(P )=g ,
0 0
where ! is the standard symplectic form and g is the Euclidean metric:
0 0
Xm Xm
i
! = dz ^dz(cid:22) = (dx (cid:10)dy (cid:0)dy (cid:10)dx );
0 j j j j j j
2
j=1 j=1
Xm
g = (dx (cid:10)dx +dy (cid:10)dy ):
0 j j j j
j=1
Here we write z = x + iy , and we let f @ ; @ g denote the dual frame to
j j j @zj @z(cid:22)j
fdz ;dz(cid:22) g.
j j
Asin[BSZ1,BSZ2,ShZe2],itisadvantageoustoworkontheassociatedprincipal
S1 bundle X ! M, and our Szeg(cid:127)o kernels will be de(cid:12)ned there. Let (cid:25) : L(cid:3) ! M
denote the dual line bundle to L with dual metric h(cid:3), and put X = fv 2 L(cid:3) :
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 5
kvkh(cid:3) = 1g. We let r(cid:18)x = ei(cid:18)x (x 2 X) denote the S1 action on X. We then
identify sections s of LN with equivariant function s^on L(cid:3) by the rule
N (cid:0) (cid:1)
(5) s^ ((cid:21))= (cid:21)(cid:10)N;s (z) ; (cid:21)2X ;
N N z
where (cid:21)(cid:10)N = (cid:21)(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)(cid:21); then s^ (r x) = eiN(cid:18)s^ (x). We denote by L2 (X) the
N (cid:18) N N
space of such equivariant functions transforming by the Nth character.
When working on X, covariant derivatives on sections of L become horizontal
derivativesofequivariantfunctions. Weconsiderpreferredcoordinates(z ;:::;z )
1 m
centered at a point P 2 M and a local frame e for L such that ke k = 1
0 L L P0
and re j = 0. This gives us coordinates (z ;:::;z ;(cid:18)) on X corresponding to
x=ei(cid:18)kLeP0(z)ke(cid:3)(z)2X. We showed in [ShZe12, x1.2]mthat
L L (cid:20) (cid:21) (cid:20) (cid:21)
@h @ i @ @h @ i @
(6) = + z(cid:22) +O(jzj2) ; = (cid:0) z +O(jzj2) ;
j j
@z @z 2 @(cid:18) @z(cid:22) @z(cid:22) 2 @(cid:18)
j j j j
where @h (resp. @h ) denotes the horizontal lift of @ (resp. @ ).
@zj @z(cid:22)j @zj @z(cid:22)j
The almost-complex Cauchy-Riemannoperator@(cid:22) on X does not satisfy @(cid:22)2 =0
b b
in general, and usually it has no kernel. Following a method of Boutet de Monvel
andGuillemin, wede(cid:12)ned in[ShZe2]the spaceHN ofequivariantalmost-CRfunc-
J
tions onX as the kernelofa certaindeformationD(cid:22) ofthe @(cid:22) operatoronL2 (X).
0 b N
The space H0(M;LN) is then the corresponding space of sections. The Szeg(cid:127)o ker-
J
nel (cid:5) (x;y) is the kernel of the orthogonal projection (cid:5) : L2 (X) ! HN: The
N N N J
dimension d =dimH0(M;LN) satis(cid:12)es the Riemann-Roch formula (see [BoGu])
N J
c (L)m
(7) d = 1 Nm+O(Nm(cid:0)1):
N
m!
Since H0(M;LN)is (cid:12)nite dimensional,the Szego(cid:127)kernel(cid:5) is smoothandisgiven
J N
by
XdN
(cid:5) (x;y)= SN(x)SN(y);
N j j
j=1
where fSNg is an orthonormalbasis for H0(M;LN).
j J
It would take us too far a(cid:12)eld to discuss the de(cid:12)nition or signi(cid:12)cance of the
spaces H0(M;LN) here; we refer the reader to [ShZe2] for background.
J
3. A generalized Poincar(cid:19)e-Borel lemma
In this section,we give a generalizationof the Poincar(cid:19)e-Borellemma concerning
the asymptotics of linear push-forwards of measures on spheres of growing dimen-
sions, which we shall use in the proof of Theorem 1.1.
Recall that a Gaussian measure on Rn is a measure of the form
(cid:13) = e(cid:0)12h(cid:1)p(cid:0)1x;xi dx (cid:1)(cid:1)(cid:1)dx ;
(cid:1) 1 n
(2(cid:25))n=2 det(cid:1)
where (cid:1) is a positive de(cid:12)nite symmetric n(cid:2)n matrix. The matrix (cid:1) gives the
second moments of (cid:13) :
(cid:1)
(8) hx x i =(cid:1) :
j k (cid:13)(cid:1) jk
This Gaussian measure is also characterized by its Fourier transform
P
(9) (cid:13)c(cid:1)(t1;:::;tn)=e(cid:0)12 (cid:1)jktjtk:
6 BERNARD SHIFFMAN AND STEVE ZELDITCH
Ifwe let (cid:1) be the n(cid:2)n identity matrix,we obtainthe standardGaussianmeasure
on Rn,
1
(cid:13)n := (2(cid:25))n=2e(cid:0)12jxj2dx1(cid:1)(cid:1)(cid:1)dxn;
withthe propertythatthex areindependentGaussianvariableswithmean0 and
j
variance 1.
ByageneralizedcomplexGaussianmeasureonCn,wemeanageneralizedGauss-
ian measure (cid:13)c on Cn (cid:17)R2n with moments
(cid:1)
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
z =0; z z =0; z z(cid:22) =(cid:1) ; 1(cid:20)j;k (cid:20)n;
j (cid:13)c j k (cid:13)c j k (cid:13)c jk
(cid:1) (cid:1) (cid:1)
where (cid:1) = ((cid:1) ) is an n(cid:2)n positive semi-de(cid:12)nite Hermitian matrix; i.e. (cid:13)c =
jk (cid:1)
(cid:13) , where (cid:1)c is the 2n(cid:2)2n real symmetric matrix of the inner product on R2n
1(cid:1)c
2
inducedby(cid:1). As weareinterestedhereincomplexGaussians,wedropthe‘c’and
write (cid:13)c =(cid:13) . In particular, if (cid:1) is a strictly positive Hermitian matrix, then (cid:13)
(cid:1) (cid:1) (cid:1)
is given by (2).
Thepush-forwardofaGaussianmeasurebyasurjectivelinearmapisalsoGauss-
ian. In the next section, we shall push forward Gaussian measures on the spaces
H0(M;LN) by linear maps that are sometimes not surjective. Since these non-
J
surjectivepush-forwardsaresingularmeasures,we needto consider the case where
(cid:1) is positive semi-de(cid:12)nite. In this case, we use (9) to de(cid:12)ne a measure (cid:13) , which
(cid:1)
we call a generalized Gaussian. If (cid:1) has null eigenvalues, then the generalized
Gaussian(cid:13) is a Gaussianmeasure onthe subspace (cid:3) (cid:26)Rn spannedby the pos-
(cid:1) +
itive eigenvectors. (Precisely, (cid:13)(cid:1) = (cid:19)(cid:3)(cid:13)(cid:1)j(cid:3)+, where (cid:19) : (cid:3)+ ,! Rn is the inclusion.
For the completely degenerate case (cid:1) = 0, we have (cid:13) = (cid:14) .) Of course, (8) also
(cid:1) 0
holds for semi-positive (cid:1). One useful property of generalized Gaussians is that
the push-forward by a (not necessarily surjective) linear map T : Rn ! Rm of a
generalized Gaussian (cid:13) on Rn is a generalized Gaussian on Rm:
(cid:1)
(10) T(cid:3)(cid:13)(cid:1) =(cid:13)T(cid:1)T(cid:3)
Another useful property of generalized Gaussians is the following fact:
Lemma 3.1. Themap(cid:1)7!(cid:13) isacontinuousmapfromthepositivesemi-de(cid:12)nite
(cid:1)
matrices to the space of positive measures on Rn (with the weak topology).
Proof. Suppose that (cid:1)N ! (cid:1)0. We must show that ((cid:1)N;’) ! ((cid:1)0;’) for a
compactlysupportedtestfunction ’. We canassume that’ is C1. Itthenfollows
from (9) that
((cid:13) ;’)=((cid:13)d;’b)!((cid:13)d;’b)=((cid:13) ;’):
(cid:1)N (cid:1)N (cid:1)0 (cid:1)0
We shall use the following ‘generalizedPoincar(cid:19)e-Borellemma’ relating spherical
measures to Gaussian measures in our proof of Theorem 1.1 on asymptotics of the
joint probability distributions for SH0(M;LN).
J
Lemma 3.2. Let TN : RdN ! Rk, N = 1;2;:::, be a sequence of linear maps,
where dN !1. Suppose that d1NTNTN(cid:3) !(cid:1). Then TN(cid:3)(cid:23)dN !(cid:13)(cid:1).
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 7
Proof. LetVN beak-dimensionalsubspaceofRdN suchthatVN? (cid:26)kerTN,andlet
pN :RdN !VN denote the orthogonal projection. We decompose TN =BN (cid:14)AN,
where AN =d1N=2pN :RdN !VN, and BN =d(cid:0)N1=2TNjVN :VN !(cid:25) Rk. Write
AN(cid:3)(cid:23)dN =(cid:11)N; TN(cid:3)(cid:23)dN =BN(cid:3)(cid:11)N =(cid:12)N:
We easily see that (abbreviating d=d )
N
(11)
( p
(cid:27)d(cid:0)k[1(cid:0) 1jxj2](d(cid:0)k(cid:0)2)=2 for jxj< d;
(cid:11)N =AN(cid:3)(cid:23)d = ddx; d = (cid:27)ddk d
0 otherwise;
where dx denotes Lebesgue measure on V , and (cid:27) = vol(Sn(cid:0)1) = 2(cid:25)n=2 . (The
N n (cid:0)(n=2)
casek =1; d=3of(11)isArchimedes’formula[Arc].) Since[1(cid:0)jxj2=d](d(cid:0)k(cid:0)2)=2 !
e(cid:0)jxj2=2 uniformly on compacta and (cid:27)d(cid:0)k ! 1 , we conclude that (cid:11) ! (cid:13) .
(cid:27)ddk (2(cid:25))k=2 N k
(This is the Poincar(cid:19)e-Boreltheorem; see Corollary 3.3 below.) Furthermore,
(cid:18) (cid:19) (cid:18) (cid:19)
1 (d(cid:0)k(cid:0)2)=2 d(cid:0)k(cid:0)2
1(cid:0) jxj2 (cid:20)exp (cid:0) jxj2 (cid:20)ek+22e(cid:0)12jxj2
d 2d
p
for d(cid:21)k+2; jxj(cid:20) d, and hence
(12) (x)(cid:20)C e(cid:0)jxj2=2:
dN k
Now let ’ be a compactly supported continuous test function on Rk. We must
show that
Z Z
(13) ’d(cid:12) ! ’d(cid:13) :
N (cid:1)
Suppose on the contraRry that (13) doesRnot hold. After passing to a subsequence,
we may assume that ’d(cid:12) ! c 6= ’d(cid:13) . Since the eigenvalues of B are
N (cid:1) N
bounded, we can assume (again taking a subsequence) that B !B , where
N 0
1
(cid:3) (cid:3) (cid:3)
B B = lim B B = lim T T =(cid:1):
0 0 N!1 N N N!1dN N N
Hence,
Z Z
’d(cid:12) = ’(B x) (x)dx
N N dN
Rk VN Z Z
e(cid:0)jxj2=2
! ’(B x) dx= ’(B x)d(cid:13) (x);
0 (2(cid:25))k=2 0 k
VN VN
where the limit holds by dominated convergence, using (12). By (10), we have
B0(cid:3)(cid:13)k =(cid:13)B0B0(cid:3) =(cid:13)(cid:1), and hence
Z Z
’(B x)d(cid:13) (x)= ’d(cid:13) :
0 k (cid:1)
VN Rk
Thus (13) holds for the subsequence, giving a contradiction.
8 BERNARD SHIFFMAN AND STEVE ZELDITCH
We note that the aboveproofbegan by establishing the Poincar(cid:19)e-Borel theorem
(which is a special case of Lemma 3.2):
Corollary 3.3 (Poincar(cid:19)e-Borel). Let P :Rd !Rk be given by
d
p
P (x)= d(x ;:::;x ):
d 1 k
Then Pd(cid:3)(cid:23)d !(cid:13)k:
4. Proof of Theorem 1.1
We return to our complex Hermitian line bundle (L;h) on a compact almost
complex 2m-dimensional symplectic manifold M with symplectic form ! = i(cid:2) ,
2 L
where (cid:2) is the curvature of L with respect to a connection r. We now describe
L
then-pointjointdistributionarisingfromourprobabilityspace(SH0(M;LN);(cid:23) ).
J N
We introduce the Hermitian inner product on H0(M;LN):
Z J
1
(14) hs ;s i= hN(s ;s ) !m (s ;s 2H0(M;LN));
1 2 1 2 m! 1 2 J
M
and we write ksk = hs;si1=2. Recall that SH0(M;LN) denotes the unit sphere
2 J
fksk=1g in H0(M;LN) and (cid:23) denotes its Haar probability measure.
J N
We let J1(M;LN) denote the space of 1-jets of sections of LN. Recall that we
have the exact sequence of vector bundles
(15) 0!T(cid:3) (cid:10)LN !(cid:19) J1(M;LN)!(cid:26) LN !0:
M
We consider the jet maps
J1 :H0(M;LN)!J1(M;V) ; J1s= the 1-jet of s at z; for z 2M:
z J z z
The covariant derivative r : J1(M;LN) ! T(cid:3) (cid:10)LN provides a splitting of (15)
M
and an isomorphism
(16) ((cid:26);r):J1(M;LN)(cid:0)(cid:25)!LN (cid:8)(T(cid:3) (cid:10)LN):
M
De(cid:12)nition. The n-point joint probability distribution at points P1;:::;Pn of M
is the probability measure
(17) DN(P1;:::;Pn) :=(JP11 (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)JP1n)(cid:3)(cid:23)N
on the space J1(M;LN) (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)J1(M;LN) .
P1 Pn
Since we are interested in the scaling limit of DN, we need to describe this
measuremoreexplicitly: SupposethatP1;:::;Pn lieinacoordinateneighborhood
of a point P 2 M, and choose preferred coordinates (z ;:::;z ) at P . We let
0 1 m 0
zp;:::;zp denote the coordinates of the point Pp (1 (cid:20) p (cid:20) n), and we write
1 m
zp = (zp;:::;zp). (The coordinates of P are 0.) We consider the n(2m + 1)
1 m 0
complex-valued random variables xp; (cid:24)p (1 (cid:20) p (cid:20) n; 1 (cid:20) q (cid:20) 2m) on SH2 (X) (cid:17)
q N
SH0(M;LN) given by
J
(18) xp(s)=s(zp;0);
1 @hs 1 @hs
(19) (cid:24)p(s)= p (zp); (cid:24)p (s)= p (zp) (1(cid:20)q (cid:20)m);
q N @zq m+q N @z(cid:22)q
for s2SH0(M;LN).
J
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 9
We now write
x=(x1;:::;xp); (cid:24) =((cid:24)qp)1(cid:20)p(cid:20)n;1(cid:20)q(cid:20)2m; z =(z1;:::;zn):
Using (16) and the variables xp; (cid:24)p to make the identi(cid:12)cation
q
(20) J1(M;LN) (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)J1(M;LN) (cid:17)Cn(2m+1);
P1 Pn
we can write
DN =DN(x;(cid:24);z)dxd(cid:24);
z
where dxd(cid:24) denotes Lebesgue measure on Cn(2m+1).
Before proving Theorem 1.1 on the scaling limit of DN, we state and prove a
z
corresponding result replacing (SH0(M;LN);(cid:23) ) with the essentially equivalent
J N
Gaussian space H0(M;LN) with the normalized standard Gaussian measure
J
(cid:18) (cid:19)
d dN XdN
(21) (cid:22)N := (cid:25)N e(cid:0)dNjcj2dc; s= cjSjN;
j=1
wherefSNgisanorthonormalbasisforH0(M;LN). Thismeasureischaracterized
j J
bythepropertythatthe2d realvariables<c ;=c (j =1;:::;d )areindependent
N j j N
Gaussian random variables with mean 0 and variance 1=2d ; i.e.,
N
1
(22) hc i =0; hc c i =0; hc c(cid:22) i = (cid:14) :
j (cid:22)N j k (cid:22)N j k (cid:22)N d jk
N
Our normalization guarantees that the variance of ksk is unity:
2
hksk2i =1:
2 (cid:22)N
We then consider the Gaussian joint probability distribution
(23) DeN(P1;:::;Pn) =DeN(x;(cid:24);z)dxd(cid:24) =(JP11 (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)JP1n)(cid:3)(cid:22)N:
Since(cid:22) isGaussianandthemapJ1 (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)J1 islinear,itfollowsthatthejoint
N P1 Pn
probability distribution is a generalized Gaussian measure of the form
(24) DN(x;(cid:24);z)dxd(cid:24) =(cid:13) :
(cid:1)N(z)
Weshallseebelowthatthe covariancematrix(cid:1)N(z)isgivenintermsoftheSzeg(cid:127)o
kernel.
We have the following alternate form of Theorem 1.1:
Theorem 4.1. Under the hypotheses and notation of Theorem 1.1, we have
DeN p p (cid:0)!D1 :
(z1= N;:::;zn= N) (z1;:::;nn)
Proof. We use the coordinates (z ;:::;z ;(cid:18)) on X given by preferred coordinates
1 m
atP 2M andalocalframee forLwithke k =1andre j =0asinx2. The
0 L L P0 L P0
covariance matrix (cid:1)N(z) in (24) is a positive semi-de(cid:12)nite n(2m+1)(cid:2)n(2m+1)
Hermitian matrix. If the map J1 (cid:8)(cid:1)(cid:1)(cid:1)(cid:8)J1 is surjective, then (cid:1)N(z) is strictly
z1 zn
e
positivede(cid:12)niteandDN(x;(cid:24);z)isasmoothfunction. Ontheotherhand,ifthemap
e
is not surjective, then DN(x;(cid:24);z) is a distribution supported on a linear subspace.
e
For example, in the integrable holomorphic case, DN(x;(cid:24);z) is supported on the
holomorphic jets, as follows from the discussion below.
10 BERNARD SHIFFMAN AND STEVE ZELDITCH
By (8), we have
(cid:18) (cid:19)
A B
(cid:1)N(z)= ;
B(cid:3) C
(cid:0) (cid:1) (cid:10) (cid:11) (cid:0) (cid:1) (cid:10) (cid:0) (cid:1) (cid:10)
(25) A= Ap = xpx(cid:22)p0 ; B= Bp = xp(cid:24)(cid:22)p0i ; C= Cpq = (cid:24)p(cid:24)(cid:22)p0i ;
p0 (cid:22)N p0q0 q0 (cid:22)N p0q0 q q0 (cid:22)N
0 0
p;p =1;:::;n; q;q =1;:::;2m:
(We note that A; B; C are n(cid:2)n; n(cid:2)2mn; 2mn(cid:2)2mn matrices, respectively;
p;q index the rows, and p0;q0 index the columns.)
We now describe the the entries of the matrix (cid:1)N in terms of the Szego(cid:127) kernel.
P
We have by (22) and (25), writing s= dN c SN,
j=1 j j
(26)
Ap =(cid:10)xpx(cid:22)p0(cid:11) = XdN (cid:10)c c(cid:22) (cid:11) SN(zp;0)SN(zp0;0)= 1 (cid:5) (zp;0;zp0;0):
p0 (cid:22)N j k (cid:22)N j k dN N
j;k=1
We need some more notation to describe the matrices B and C. Write
1 @h 1 @h
r = p ; r = p ; 1(cid:20)q (cid:20)m:
q m+q
N @zq N @z(cid:22)q
We let r1, resp. r2, denote the di(cid:11)erential operator on X (cid:2)X given by applying
q q
r to the (cid:12)rst, resp. second, factor (1 (cid:20) q (cid:20) 2m). By di(cid:11)erentiating (26), we
q
obtain
(27) Bp = 1 r2 (cid:5) (zp;0;zp0;0);
p0q0 d q0 N
N
(28) Cpq = 1 r1r2 (cid:5) (zp;0;zp0;0):
p0q0 d q q0 N
N
We now use the scaling asymptotics of the almost holomorphic Szego(cid:127) kernel
(cid:5) (x;y) given in [ShZe2]. In addition to the above assumption on the local frame
N
e , we further assume that
L
r2e j =(cid:0)(g+i!)(cid:10)e j 2T(cid:3) (cid:10)T(cid:3) (cid:10)L:
L P0 L P0 M M
(In [ShZe3], we called such an e a ‘preferred frame’ at P , and the resulting coor-
L 0
dinateswerecalled‘Heisenbergcoordinates.’) Thenwehave(see[ShZe3],Theorem
2.3)
N(cid:0)m(cid:5) (pu ; (cid:18);pv ; ’)
N N N N N
(29) = (cid:25)1mei((cid:18)(cid:0)’)+u(cid:1)v(cid:22)(cid:0)12(juj2+jvj2)
h i
P
(cid:2) 1+ Kr=1N(cid:0)r2br(P0;u;v)+N(cid:0)K2+1RK(P0;u;v;N) ;
where kRK(P0;u;v;N)kCj(fjuj+jvj(cid:20)(cid:26)g) (cid:20)CK;j;(cid:26) for j =1;2;3;:::.
It follows from (26){(28) and (29), recalling (6){(7), that
(cid:18) (cid:19)
(30) (cid:1)N(pz )!(cid:1)1(z)= m! A1(z) B1(z)
N c1(L)m B1(z)(cid:3) C1(z)
in the notation of (4).