ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS OF AMPLE LINE BUNDLES ON ...

12 Pages·2002·0.39 MB·English

PROCEEDINGSOFTHE
AMERICANMATHEMATICALSOCIETY
Volume131,Number1,Pages291–302
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. WedefineaGaussianmeasureonthespaceH0(M,LN)ofalmost
J
holomorphic sections of powers of an ample line bundle L over a symplectic
manifold(M,ω),andcalculatethejointprobabilitydensitiesofsectionstaking
prescribed values and covariant derivatives at a finite number of points. We
prove that they have a universalscalinglimitas N →∞. 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,ω,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 different class H0(M,LN)
J
of ‘almost holomorphic’ sections defined 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¨o projectors
Π :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 finite 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 ν and consider the joint probability
J N
distribution (JPD)
DN =DN(x1,...,xn,ξ1,... ,ξn;z1,...,zn)dxdξ
(z1,...,zn)
ofarandomsections ∈SH0(M,LN)havingtheprescribedvaluesx1,...,xn and
N
derivatives ξ1,... ,ξn at n points z1,...,zn ∈ M. To be more precise, we choose
ReceivedbytheeditorsAugust3,2001.
2000 Mathematics Subject Classification. Primary53D50,53D35,60D05.
Research partially supported by NSF grants #DMS-9800479, #DMS-0100474 (first author)
and#DMS-0071358(second author).
(cid:13)c2002 American Mathematical Society
291
292 BERNARD SHIFFMAN AND STEVE ZELDITCH
localcomplex coordinates{z ,...,z }anda nonvanishinglocalsectione ofL on
1 m L
an open set containing the points {z1,...,zn}; 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∗LN,sN (cid:12)zp, ξqp =(cid:11)(cid:12) e∗LN,N−12∇∂/∂zqsN (cid:12)zp,
ξmp+q = e∗LN,N−12∇∂/∂z¯qsN (cid:12)zp, 1≤p≤n,1≤q ≤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,therebyfinishingthe
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,ω) with curvature ω. Let P ∈ M, and choose a
0
local frame e for L and local complex coordinates centered at P so that ω| and
0 P0
g| are the usual Euclidean Ka¨hler form and metric respectively, (cid:107)e (cid:107) =1, and
P0 L P0
∇e | =0. Then
L P0
(cid:0) (cid:1)
DN √ √ −→D∞ (z1,...,zn)∈Cmn ,
(z1/ N,...,zn/ N) (z1,...,zn)
where the measures D∞(z1,...,zn) =γ∆∞(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 γ∆∞(z). We recall that a (complex) Gaussian
measure on Ck is a measure of the form
e−(cid:104)∆−1z,z¯(cid:105)
(2) γ = dz,
∆ πkdet∆
where dz denotes Lebesgue mea(cid:0)sure o(cid:1)n Ck, and ∆ is a positive definite Hermitian
k×k matrix. The matrix ∆= ∆ is the covariance matrix of γ :
αβ ∆
(3) (cid:104)z z¯ (cid:105) =∆ , 1≤α,β ≤k .
α β γ∆ αβ
Sincetheuniversallimitmeasuresγ∆∞(z) vanishalongnon-holomorphicdirections,
theyaresingularmeasuresonthespaceofall1-jets. Todealwithsingularmeasures,
weintroducein§3generalizedGaussianmeasureswhosecovariancematrices(3)are
only semi-positive definite; a generalized Gaussian measure is simply a Gaussian
measuresupportedonthesubspacecorrespondingtothepositiveeigenvaluesofthe
covariance matrix.
Thecovariancematrix∆∞(z)isgivenalongholomorphicdirectionsbythesame
formula as in the holomorphic case [BSZ1, (97)], namely
(cid:18) (cid:19)
∞ m! A∞(z) B∞(z)
(4) ∆ (z)= ,
c (L)m B∞(z)∗ C∞(z)
1
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 293
where z =(z1,...,zn) and
A∞(z)pp(cid:48) = (cid:40)ΠH1 (zp,0;zp(cid:48),0), ΠH1 (u,0;v,0)= π1meu·v¯−12(|u|2+|v|2) ,
B∞(z)p = (zqp(cid:48) −zqp(cid:48)(cid:48))ΠH1 (zp,0;zp(cid:48),0) for 1≤q ≤m,
p(cid:48)q(cid:48) 0 for m+1≤q ≤2m,
 (cid:16) (cid:17)
 δqq(cid:48) +(z¯qp(cid:48) −z¯qp)(zqp(cid:48) −zqp(cid:48)(cid:48)) ΠH1 (zp,0;zp(cid:48),0)
C∞(z)pp(cid:48)qq(cid:48) =  for 1≤q,q(cid:48) ≤m,
0 for max(q,q(cid:48))≥m+1.
In other words,the coefficients of ∆∞(z) correspondingto the anti-holomorphic
directionsvanish,whilethecoefficientscorrespondingtotheholomorphicdirections
are given by the Szeg¨o kernel ΠH for the reduced Heisenberg group (see [BSZ1,
1
§1.3.2]) and its covariant derivatives.
A technically interesting novelty in the proof is the role of the ∂¯ operator. In
the holomorphic case, DN is supported on the subspace of jets of sections
(z1,...,zn)
satisfying∂¯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 →∞.
An alternate statement of Theorem 1.1 involves equipping H0(M,LN) with a
J
(cid:101)
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
(cid:101)
of sections. We show (Theorem 4.1) that these Gaussian measures DN also have
the same scaling limit D∞, so that asymptotically the probabilities are the same
as in the holomorphic case, where universality was established in [BSZ2]. It is
(cid:101)
then easy to see that DN =γ , where ∆N is the covariance matrix of the
(z1,...,zn) ∆N
randomvariables in (1). The main step in the proofis to show that the covariance
(cid:101)
matrices ∆N underlying DN tend in the scaling limit to a semi-positive matrix
∆∞. It follows that the scaled distributions D(cid:101)N tend to the generalized Gaussian
γ∆∞ ‘vanishing in the ∂¯-directions.’
In a subsequent article [ShZe3], we obtain further probabilistic results on holo-
morphicandalmostholomorphicsections. Regardingalmostholomorphicsections,
we prove that a sequence {s } of almost holomorphic sections satifies the bounds
N
(cid:112) (cid:112)
(cid:107)sN(cid:107)∞/(cid:107)sN(cid:107)L2 =O( logN), (cid:107)∂¯sN(cid:107)∞/(cid:107)sN(cid:107)L2 =O( logN)
almost surely. Hence almost holomorphic sections satisfy bounds similar to those
satisfied 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 =⇒(cid:107)∂¯s(z)|<|∂s(z)| ∀z ∈M
294 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 difficult. On
small balls, our methods rather easily give lower bounds for quantitative transver-
sality of the type
C
µ{s: |∂¯s(z)|<|∂s(z)| ∀z ∈B (z )such thats(z)=0} > 1− ε .
√D 0 N1−ε
N √
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¨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,ω,J) denote a compact almost-complex symplectic manifold such that
[1ω] is an integral cohomology class, and such that ω(Jv,Jw) = ω(v,w) and
π
ω(v,Jv) > 0. We further let (L,h,∇) → M denote a quantizing Hermitian line
bundle andmetric connectionwith curvature iΘ =ω. We denoteby LN the Nth
2 L
tensor power of L.
We give M the Riemannian metric g(v,w) = ω(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 =−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 ∈M ispreferred
1 m 0
at P if any two of the following conditions (and hence all three) are satisfied:
0
i) ∂/∂z | ∈T1,0(M), for 1≤j ≤m,
j P0
ii) ω(P )=ω ,
0 0
iii) g(P )=g ,
0 0
where ω is the standard symplectic form and g is the Euclidean metric:
0 0
(cid:88)m (cid:88)m
i
ω = dz ∧dz¯ = (dx ⊗dy −dy ⊗dx ),
0 j j j j j j
2
j=1 j=1
(cid:88)m
g = (dx ⊗dx +dy ⊗dy ).
0 j j j j
j=1
Here we write z = x + iy , and we let { ∂ , ∂ } denote the dual frame to
j j j ∂zj ∂z¯j
{dz ,dz¯ }.
j j
Asin[BSZ1,BSZ2,ShZe2],itisadvantageoustoworkontheassociatedprincipal
S1 bundle X → M, and our Szeg¨o kernels will be defined there. Let π : L∗ → M
denote the dual line bundle to L with dual metric h∗, and put X = {v ∈ L∗ :
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 295
(cid:107)v(cid:107)h∗ = 1}. We let rθx = eiθx (x ∈ X) denote the S1 action on X. We then
identify sections s of LN with equivariant function sˆon L∗ by the rule
N (cid:0) (cid:1)
(5) sˆ (λ)= λ⊗N,s (z) , λ∈X ,
N N z
where λ⊗N = λ⊗···⊗λ; then sˆ (r x) = eiNθsˆ (x). We denote by L2 (X) the
N θ 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 ∈ M and a local frame e for L such that (cid:107)e (cid:107) = 1
0 L L P0
and ∇e | = 0. This gives us coordinates (z ,...,z ,θ) on X corresponding to
x=eiθ(cid:107)LeP0(z)(cid:107)e∗(z)∈X. We showed in [ShZe12, §1.2]mthat
L L (cid:20) (cid:21) (cid:20) (cid:21)
∂h ∂ i ∂ ∂h ∂ i ∂
(6) = + z¯ +O(|z|2) , = − z +O(|z|2) ,
j j
∂z ∂z 2 ∂θ ∂z¯ ∂z¯ 2 ∂θ
j j j j
where ∂h (resp. ∂h ) denotes the horizontal lift of ∂ (resp. ∂ ).
∂zj ∂z¯j ∂zj ∂z¯j
The almost-complex Cauchy-Riemannoperator∂¯ on X does not satisfy ∂¯2 =0
b b
in general, and usually it has no kernel. Following a method of Boutet de Monvel
andGuillemin, wedefined in[ShZe2]the spaceHN ofequivariantalmost-CRfunc-
J
tions onX as the kernelofa certaindeformationD¯ ofthe ∂¯ operatoronL2 (X).
0 b N
The space H0(M,LN) is then the corresponding space of sections. The Szeg¨o ker-
J
nel Π (x,y) is the kernel of the orthogonal projection Π : L2 (X) → HN. The
N N N J
dimension d =dimH0(M,LN) satisfies the Riemann-Roch formula (see [BoGu])
N J
c (L)m
(7) d = 1 Nm+O(Nm−1).
N
m!
Since H0(M,LN)is finite dimensional,the Szego¨kernelΠ is smoothandisgiven
J N
by
(cid:88)dN
Π (x,y)= SN(x)SN(y),
N j j
j=1
where {SN} is an orthonormalbasis for H0(M,LN).
j J
It would take us too far afield to discuss the definition or significance of the
spaces H0(M,LN) here; we refer the reader to [ShZe2] for background.
J
3. A generalized Poincar´e-Borel lemma
In this section,we give a generalizationof the Poincar´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
γ = e−12(cid:104)∆√−1x,x(cid:105) dx ···dx ,
∆ 1 n
(2π)n/2 det∆
where ∆ is a positive definite symmetric n×n matrix. The matrix ∆ gives the
second moments of γ :
∆
(8) (cid:104)x x (cid:105) =∆ .
j k γ∆ jk
This Gaussian measure is also characterized by its Fourier transform
(cid:80)
(9) γ(cid:99)∆(t1,...,tn)=e−12 ∆jktjtk.
296 BERNARD SHIFFMAN AND STEVE ZELDITCH
Ifwe let ∆ be the n×n identity matrix,we obtainthe standardGaussianmeasure
on Rn,
1
γn := (2π)n/2e−12|x|2dx1···dxn,
withthe propertythatthex areindependentGaussianvariableswithmean0 and
j
variance 1.
ByageneralizedcomplexGaussianmeasureonCn,wemeanageneralizedGauss-
ian measure γc on Cn ≡R2n with moments
∆
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
z =0, z z =0, z z¯ =∆ , 1≤j,k ≤n,
j γc j k γc j k γc jk
∆ ∆ ∆
where ∆ = (∆ ) is an n×n positive semi-definite Hermitian matrix; i.e. γc =
jk ∆
γ , where ∆c is the 2n×2n real symmetric matrix of the inner product on R2n
1∆c
2
inducedby∆. As weareinterestedhereincomplexGaussians,wedropthe‘c’and
write γc =γ . In particular, if ∆ is a strictly positive Hermitian matrix, then γ
∆ ∆ ∆
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
∆ is positive semi-definite. In this case, we use (9) to define a measure γ , which
∆
we call a generalized Gaussian. If ∆ has null eigenvalues, then the generalized
Gaussianγ is a Gaussianmeasure onthe subspace Λ ⊂Rn spannedby the pos-
∆ +
itive eigenvectors. (Precisely, γ∆ = ι∗γ∆|Λ+, where ι : Λ+ (cid:44)→ Rn is the inclusion.
For the completely degenerate case ∆ = 0, we have γ = δ .) Of course, (8) also
∆ 0
holds for semi-positive ∆. 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 γ on Rn is a generalized Gaussian on Rm:
∆
(10) T∗γ∆ =γT∆T∗
Another useful property of generalized Gaussians is the following fact:
Lemma 3.1. Themap∆(cid:55)→γ isacontinuousmapfromthepositivesemi-definite
∆
matrices to the space of positive measures on Rn (with the weak topology).
Proof. Suppose that ∆N → ∆0. We must show that (∆N,ϕ) → (∆0,ϕ) for a
compactlysupportedtestfunction ϕ. We canassume thatϕ is C∞. Itthenfollows
from (9) that
(γ ,ϕ)=(γ(cid:100),ϕ(cid:98))→(γ(cid:100),ϕ(cid:98))=(γ ,ϕ).
∆N ∆N ∆0 ∆0
We shall use the following ‘generalizedPoincar´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 →∞. Suppose that d1NTNTN∗ →∆. Then TN∗νdN →γ∆.
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 297
Proof. LetVN beak-dimensionalsubspaceofRdN suchthatVN⊥ ⊂kerTN,andlet
pN :RdN →VN denote the orthogonal projection. We decompose TN =BN ◦AN,
where AN =d1N/2pN :RdN →VN, and BN =d−N1/2TN|VN :VN →≈ Rk. Write
AN∗νdN =αN, TN∗νdN =BN∗αN =βN.
We easily see that (abbreviating d=d )
N
(11)
(cid:40) √
σd−k[1− 1|x|2](d−k−2)/2 for |x|< d,
αN =AN∗νd =ψddx, ψd = σddk d
0 otherwise,
where dx denotes Lebesgue measure on V , and σ = vol(Sn−1) = 2πn/2 . (The
N n Γ(n/2)
casek =1, d=3of(11)isArchimedes’formula[Arc].) Since[1−|x|2/d](d−k−2)/2 →
e−|x|2/2 uniformly on compacta and σd−k → 1 , we conclude that α → γ .
σddk (2π)k/2 N k
(This is the Poincar´e-Boreltheorem; see Corollary 3.3 below.) Furthermore,
(cid:18) (cid:19) (cid:18) (cid:19)
1 (d−k−2)/2 d−k−2
1− |x|2 ≤exp − |x|2 ≤ek+22e−12|x|2
d 2d
√
for d≥k+2, |x|≤ d, and hence
(12) ψ (x)≤C e−|x|2/2.
dN k
Now let ϕ be a compactly supported continuous test function on Rk. We must
show that
(cid:90) (cid:90)
(13) ϕdβ → ϕdγ .
N ∆
Suppose on the contra(cid:82)ry that (13) does(cid:82)not hold. After passing to a subsequence,
we may assume that ϕdβ → c (cid:54)= ϕdγ . Since the eigenvalues of B are
N ∆ N
bounded, we can assume (again taking a subsequence) that B →B , where
N 0
1
∗ ∗ ∗
B B = lim B B = lim T T =∆.
0 0 N→∞ N N N→∞dN N N
Hence,
(cid:90) (cid:90)
ϕdβ = ϕ(B x)ψ (x)dx
N N dN
Rk VN (cid:90) (cid:90)
e−|x|2/2
→ ϕ(B x) dx= ϕ(B x)dγ (x),
0 (2π)k/2 0 k
VN VN
where the limit holds by dominated convergence, using (12). By (10), we have
B0∗γk =γB0B0∗ =γ∆, and hence
(cid:90) (cid:90)
ϕ(B x)dγ (x)= ϕdγ .
0 k ∆
VN Rk
Thus (13) holds for the subsequence, giving a contradiction.
298 BERNARD SHIFFMAN AND STEVE ZELDITCH
We note that the aboveproofbegan by establishing the Poincar´e-Borel theorem
(which is a special case of Lemma 3.2):
Corollary 3.3 (Poincar´e-Borel). Let P :Rd →Rk be given by
d
√
P (x)= d(x ,...,x ).
d 1 k
Then Pd∗νd →γk.
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 ω = iΘ ,
2 L
where Θ is the curvature of L with respect to a connection ∇. We now describe
L
then-pointjointdistributionarisingfromourprobabilityspace(SH0(M,LN),ν ).
J N
We introduce the Hermitian inner product on H0(M,LN):
(cid:90) J
1
(14) (cid:104)s ,s (cid:105)= hN(s ,s ) ωm (s ,s ∈H0(M,LN)),
1 2 1 2 m! 1 2 J
M
and we write (cid:107)s(cid:107) = (cid:104)s,s(cid:105)1/2. Recall that SH0(M,LN) denotes the unit sphere
2 J
{(cid:107)s(cid:107)=1} in H0(M,LN) and ν 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∗ ⊗LN →ι J1(M,LN)→ρ 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 ∈M.
z J z z
The covariant derivative ∇ : J1(M,LN) → T∗ ⊗LN provides a splitting of (15)
M
and an isomorphism
(16) (ρ,∇):J1(M,LN)−≈→LN ⊕(T∗ ⊗LN).
M
Definition. The n-point joint probability distribution at points P1,...,Pn of M
is the probability measure
(17) DN(P1,...,Pn) :=(JP11 ⊕···⊕JP1n)∗νN
on the space J1(M,LN) ⊕···⊕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 ∈ 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 ≤ p ≤ 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, ξp (1 ≤ p ≤ n, 1 ≤ q ≤ 2m) on SH2 (X) ≡
q N
SH0(M,LN) given by
J
(18) xp(s)=s(zp,0),
1 ∂hs 1 ∂hs
(19) ξp(s)= √ (zp), ξp (s)= √ (zp) (1≤q ≤m),
q N ∂zq m+q N ∂z¯q
for s∈SH0(M,LN).
J
ASYMPTOTICS OF ALMOST HOLOMORPHIC SECTIONS: ADDENDUM 299
We now write
x=(x1,...,xp), ξ =(ξqp)1≤p≤n,1≤q≤2m, z =(z1,...,zn).
Using (16) and the variables xp, ξp to make the identification
q
(20) J1(M,LN) ⊕···⊕J1(M,LN) ≡Cn(2m+1),
P1 Pn
we can write
DN =DN(x,ξ;z)dxdξ,
z
where dxdξ 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),ν ) with the essentially equivalent
J N
Gaussian space H0(M,LN) with the normalized standard Gaussian measure
J
(cid:18) (cid:19)
d dN (cid:88)dN
(21) µN := πN e−dN|c|2dc, s= cjSjN,
j=1
where{SN}isanorthonormalbasisforH0(M,LN). Thismeasureischaracterized
j J
bythepropertythatthe2d realvariables(cid:60)c ,(cid:61)c (j =1,...,d )areindependent
N j j N
Gaussian random variables with mean 0 and variance 1/2d ; i.e.,
N
1
(22) (cid:104)c (cid:105) =0, (cid:104)c c (cid:105) =0, (cid:104)c c¯ (cid:105) = δ .
j µN j k µN j k µN d jk
N
Our normalization guarantees that the variance of (cid:107)s(cid:107) is unity:
2
(cid:104)(cid:107)s(cid:107)2(cid:105) =1.
2 µN
We then consider the Gaussian joint probability distribution
(23) D(cid:101)N(P1,...,Pn) =D(cid:101)N(x,ξ;z)dxdξ =(JP11 ⊕···⊕JP1n)∗µN.
Sinceµ isGaussianandthemapJ1 ⊕···⊕J1 islinear,itfollowsthatthejoint
N P1 Pn
probability distribution is a generalized Gaussian measure of the form
(24) DN(x,ξ;z)dxdξ =γ .
∆N(z)
Weshallseebelowthatthe covariancematrix∆N(z)isgivenintermsoftheSzeg¨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
D(cid:101)N √ √ −→D∞ .
(z1/ N,...,zn/ N) (z1,...,nn)
Proof. We use the coordinates (z ,...,z ,θ) on X given by preferred coordinates
1 m
atP ∈M andalocalframee forLwith(cid:107)e (cid:107) =1and∇e | =0asin§2. The
0 L L P0 L P0
covariance matrix ∆N(z) in (24) is a positive semi-definite n(2m+1)×n(2m+1)
Hermitian matrix. If the map J1 ⊕···⊕J1 is surjective, then ∆N(z) is strictly
z1 zn
(cid:101)
positivedefiniteandDN(x,ξ;z)isasmoothfunction. Ontheotherhand,ifthemap
(cid:101)
is not surjective, then DN(x,ξ;z) is a distribution supported on a linear subspace.
(cid:101)
For example, in the integrable holomorphic case, DN(x,ξ;z) is supported on the
holomorphic jets, as follows from the discussion below.
300 BERNARD SHIFFMAN AND STEVE ZELDITCH
By (8), we have
(cid:18) (cid:19)
A B
∆N(z)= ,
B∗ 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¯p(cid:48) , B= Bp = xpξ¯p(cid:48)(cid:105) , C= Cpq = ξpξ¯p(cid:48)(cid:105) ,
p(cid:48) µN p(cid:48)q(cid:48) q(cid:48) µN p(cid:48)q(cid:48) q q(cid:48) µN
(cid:48) (cid:48)
p,p =1,...,n, q,q =1,...,2m.
(We note that A, B, C are n×n, n×2mn, 2mn×2mn matrices, respectively;
p,q index the rows, and p(cid:48),q(cid:48) index the columns.)
We now describe the the entries of the matrix ∆N in terms of the Szego¨ kernel.
(cid:80)
We have by (22) and (25), writing s= dN c SN,
j=1 j j
(26)
Ap =(cid:10)xpx¯p(cid:48)(cid:11) = (cid:88)dN (cid:10)c c¯ (cid:11) SN(zp,0)SN(zp(cid:48),0)= 1 Π (zp,0;zp(cid:48),0).
p(cid:48) µN j k µN j k dN N
j,k=1
We need some more notation to describe the matrices B and C. Write
1 ∂h 1 ∂h
∇ = √ , ∇ = √ , 1≤q ≤m.
q m+q
N ∂zq N ∂z¯q
We let ∇1, resp. ∇2, denote the differential operator on X ×X given by applying
q q
∇ to the first, resp. second, factor (1 ≤ q ≤ 2m). By differentiating (26), we
q
obtain
(27) Bp = 1 ∇2 Π (zp,0;zp(cid:48),0),
p(cid:48)q(cid:48) d q(cid:48) N
N
(28) Cpq = 1 ∇1∇2 Π (zp,0;zp(cid:48),0).
p(cid:48)q(cid:48) d q q(cid:48) N
N
We now use the scaling asymptotics of the almost holomorphic Szego¨ kernel
Π (x,y) given in [ShZe2]. In addition to the above assumption on the local frame
N
e , we further assume that
L
∇2e | =−(g+iω)⊗e | ∈T∗ ⊗T∗ ⊗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−mΠ (√u , θ;√v , ϕ)
N N N N N
(29) = π1mei(θ−ϕ)+u·v¯−12(|u|2+|v|2)
(cid:104) (cid:105)
(cid:80)
× 1+ Kr=1N−r2br(P0,u,v)+N−K2+1RK(P0,u,v,N) ,
where (cid:107)RK(P0,u,v,N)(cid:107)Cj({|u|+|v|≤ρ}) ≤CK,j,ρ for j =1,2,3,....
It follows from (26)–(28) and (29), recalling (6)–(7), that
(cid:18) (cid:19)
(30) ∆N(√z )→∆∞(z)= m! A∞(z) B∞(z)
N c1(L)m B∞(z)∗ C∞(z)
in the notation of (4).