Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

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ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

ROBERTBERMAN, JOHANNESSJÖSTRAND

Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

Tome XVI, n^o4 (2007), p. 719-771.

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Annales de la Facult´e des Sciences de Toulouse Vol. XIV, n^◦4, 2007 pp. 719–771

Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

⁽^∗⁾

Robert Berman⁽¹⁾, Johannes Sj¨ostrand⁽²⁾

ABSTRACT.— In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle withnon-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds.

R^´ESUM ´E.— Dans ce travail nous obtenons un développement asympto- tique complet du noyau de Bergman-Hodge d’une puissance élevée d’un fibré en droites holomorphe à courbure non-dégénerée. Nous explorons aussi quelques relations avec des sections asymptotiquement holomorphes sur une variété symplectique.

LetLbe a Hermitian holomorphic line bundle over a compact complex Hermitian manifoldX.Denote byΘ the curvature two-form of the canonical connection∇onL.Bythe Hodge theorem, the Dolbeault cohomologygroup H^0,q(X, L) is isomorphic to the space H^0,q(X, L) of harmonic (0, q)−forms with values in L, i.e the null space of the Hodge Laplacian ∆q. Denote byΠq the corresponding Hodge projection, i.e. the orthogonal projection from L²(X, L) onto H^0,q(X, L). We will assume that Θ is non-degenerate of constant signature (n₋, n+), i.e. the number of negative eigenvalues of Θ isn₋ (the index of Θ).Then it is well-known, bythe theorems of Kodaira and H¨ormander, thatH^0,q(X, L^k) is trivial whenq=n₋,for a suﬃciently high tensor power L^k. (See also [18].) We will studythe asymptotics with respect tok of the corresponding Hodge projections Πq,kin the non-trivial case when q = n₋. The case when n₋ = 0, i.e. when L is a positive line bundle and Πq,k is the Bergman projection on the space of holomorphic sections with values in L^k, has been studied extensivelybefore (compare the historical remarks below).

Letπ1 and π2 be the projections on the ﬁrst and the second factor of X×X.Denote byKk the Schwartz kernel of Πq,k(the subscripts kwill be omitted in the sequel) with respect to the volume form ωn on X induced bythe Hermitian metric onX,so thatKis a section ofL(π₂^∗(Λ^0,q(T^∗X)⊗ L^k), π^∗₁(Λ^0,q(T^∗X)⊗L^k)).

Let t, s be local unitarysections of L over X, Y respectively, where X, Y ⊆X.Then onX×Y we can write

K(x, y) =Kt,s(x, y;1

k)t(x)^ks(y)^∗^k,

where Kt,s is a local section of L(π2^∗(Λ^0,q(T^∗X)), π^∗1(Λ^0,q(T^∗X))) so that forx∈X, u∈C₀^∞(Y; Λ^0,q(T^∗X⊗L^k)),

u(x) =t(x)^k

X

Kt,s(x, y;1 k)

u(y), s(y)^∗^k

ωn(dy), We saythat a kernel

R(x, y) =Rt,s(x, y;1

k)t(x)^ks(y)^∗^k, isnegligible if

∂_x^α∂_y^βRt,s(x, y;1

k) =Oα,β,N(k⁻^N),

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locallyuniformlyon everycompact set inX ×Y , for all multiindices α, β and all N in N. Notice that this statement does not depend on the choice oft, sand on the local coordinatesx, y.

Our main result tells us that K is negligible near everypoint (x0, y0) withx0=y0and that for (x, y) near a diagonal point (x0, x0)

Kt,s(x, y) =b(x, y;1

k)e^kψ(x,y)+Rt,s(x, y), Rt,s negligible, (1.1) whereψ is smooth function withψ(x, x) = 0,Reψ(x, y)− |x−y|² and

b(x, y;1

k)∼kⁿ(b0(x, y) +b1(x, y)1 k+...)

inC^∞(neigh(x0, x0);L(π^∗₂(Λ^0,q(T^∗X)), π₁^∗(Λ^0,q(T^∗X))). Moreover, letCbe the graph of ¹_idψinT^∗X×T^∗Xover the diagonal. ThenClocallyrepresents the graph of the canonical connection∇ofL⊗L^∗over the diagonal inX×X and the semiclassical wave front of K. See Theorem 5.1 and the preceding explanations in Section 5 for a more precise local statement.

We will also explore some relations to the work [44] of B.Shiﬀman and S.Zelditch, where so called asymptotic holomorphic sections on symplectic manifolds are studied.

1.1. Overview

After locallyfixing a unitaryframe forL, we identifythe Hodge Lapla- cian ∆q acting on (0, q)−forms with values inL^k,with a local semiclassical differential operator (setting h = 1/k). Since the curvature form of L is assumed to be non-degenerate the characteristic varietyΣ of ∆q is symplectic. Modifying the approach in [40] we then construct associated local asymptotic heat kernels in Section 3 and investigate the limit when the time variable tends to infinity. In Section 4 it is shown that the limit operator is an asymptotic local projection operator. The complex canonical relation of the local projection operators is expressed in terms of the stable outgoing and incoming manifolds associated to Σ in Section 3. Assuming, in Section 5, that the number of negative eigenvalues of the curvature ofLis equal to q everywhere on X we get a complete asymptotic expansion of the global projection operator Πq. In Section 6 we investigate some relations to [44], where so called asymptotic holomorphic sections on symplectic manifolds are studied. We introduce a certain almost complex structure, closelyrelated to the stable manifolds introduced in Section 3, making the curvature form ofLpositive. It is shown that forksufficientlylarge the dimension of

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the null space of ∆qcoincides with the dimension of the corresponding space of asymptotically holomorphic sections (after a suitable twisting ofL). In Section 7 the interplaybetween diﬀerent complex structures is illustrated byhomogeneous line bundles over ﬂag manifolds.

1.2. Historical remarks

Most of the earlier results concern the positivelycurved case n₋ = 0.

G. Tian [49], followed byW. Ruan [43] and Z. Lu [32], computed increas- inglymanyterms of the asymptotic expansion on the diagonal, using Tian’s method of peak solutions. T. Bouche [11] also got the leading term using heat kernels.

S. Zelditch [51], D. Catlin [14] established the complete asymptotic expansion atx=ybyusing a result of L. Boutet de Monvel, J. Sj¨ostrand [13]

for the asymptotics of the Szegö kernel on a strictlypseudoconvex bound- ary(after the pioneering work of C. Fefferman [21]), here on the bound- aryof the unit disc bundle, and a reduction idea of L. Boutet de Monvel, V. Guillemin [13]. Scaling asymptotics away from the diagonal (roughly with a second order polynomial instead ofψin (1.1) and corresponding more general amplitudes) was obtained byP. Bleher, B. Shiffman, S. Zelditch [6] and the asymptotics as in (1.1) by L. Charles [15], using again the reduction method. In the recent work [4] B. Berndtsson and the authors have worked out a short and direct proof for the asymptotics as in (1.1).

In more general situations, asymptotic expansions on the diagonal and in the scaling sense awayfrom the diagonal were obtained byB. Shiﬀman, S. Zelditch [44] and X. Dai, K. Liu, X. Ma [17]. See also the works byX. Ma and G. Marinescu [34] for related spectral results and [35] for asymptotics on the diagonal.

Without a positive curvature assumption there have been fewer results.

J.M. Bismut [5] used the heat kernel method in his approach to Demailly’s holomorphic Morse inequalities. Using local holomorphic Morse inequalities [2], the leading asympotics of the Hodge projections were obtained by the ﬁrst author in [3] without assuming that the curvature is non-degenerate.

X. Ma has pointed out to us that the method and results of [17] can be extended to the case of non-positive holomorphic line bundles byusing a spectral gap estimate from [34] and this was recentlycarried out in the preprint [36]. The result of Theorem 5.1 was announced in [47].

In this quick review, we omitted results awayfrom the diagonal, since our work onlyconcerns the asymptotics moduloO(k^−∞).

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1.3. Why the heat kernel method?

Originallywe thought about a direct semiclassical adaptation of the methods in [13] and both L. Boutet de Monvel and more recentlythe referee have suggested such an approach to us. For a long time our attempts in that direction were stalled bysome algebraic problems in the casen₋ >0, and onlyrecently(after ﬁnishing the present paper) did we get an idea about how to circumvent the algebraic diﬃculty.

We believe however that the heat kernel method has its own interest and is not reallylonger than the adaptation of [13]. Undoubtedlyit can also be used to obtain the complete asymptotics of the inverse of ∆q whenq=n₋ and the partial inverse on the orthogonal of the kernel whenq=n₋. Acknowledgements. — The first author has been partiallysupported bya Marie Curie grant. The second author has benefitted from the hospitalityof Chalmers and Gothenburg Universityin 2000–02. We express our gratitude to Bo Berndtsson for manystimulating discussions and for continued interest in this work. We have also benefitted from discussions with L. Boutet de Monvel L. Charles, X. Ma, H. Seppänen and G. Zhang, as well as with M. Shubin who suggested a similar problem to one of us in 1994. Finally, we thank the referee for several interesting remarks.

2.Holomorphic line bundles and the ∂-complex, a review

LetL be a Hermitian holomorphic line bundle overX. Later, we shall use a local holomorphic non-vanishing section s. We write the point-wise norm ofsas

|s|²=|s|²h1 =:e⁻^2φ.

The curvature form ofLcan be identiﬁed with the Levi form∂∂φ.

Add a Hermitian metric onT^1,0X: H(ν, µ) =

Hj,kνkµ_j, ifν= νj

∂

∂zj

, µ= µj

∂

∂zj

. (2.1)

We have a natural dualitybetween T1,0^∗ X andT^1,0X, satisfying dzj, ∂

∂zk=δj,k,

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so ifω =

ωkdzk, then ω, ν=

ωjνj. For each x∈X, we can choose z1, ..., zn centered atxso that

Hj,k(x) =δj,k; Hx( ∂

∂zj

, ∂

∂zk

) =δj,k.

The metricH also determines a metric on Λ^0,q(T^∗X) such that in the special coordinates above, we have that

dzj₁∧...∧dzj_q, 1j1< j2< ... < jq n,

is an orthonormal basis of Λ^0,qT_x^∗X. Then we have a natural metric also on L⊗Λ^0,qT^∗X.

Let us also ﬁx some smooth positive integration densitym(dz) on X. (For instance, we can takem(dz) =ωn(dz); the induced volume form.) Then we get a natural scalar product on

E^0,q(L) =C^∞(X;L⊗Λ^0,qT^∗X), so if

∂: ..→ E^0,q(L)→ E^0,q+1(L)→..

is the∂ complex, then

∂^∗: ..← E^0,q(L)← E^0,q+1(L)←...

is also a well-deﬁned complex.

If ω is a 0,1-form, let ω : Λ^0,q+1T_x^∗X → Λ^0,qT_x^∗X be the adjoint of left exterior multiplication ω^∧ : Λ^0,qT_x^∗X → Λ^0,q+1T_x^∗X. Here we use the Hermitian inner product H^∗ on Λ^0,qT_x^∗X that is naturallyobtained from H. Without that inner product, we can still deﬁne ν : Λ^0,q+1T_x^∗X → Λ^0,qT_x^∗X, whenν =

νj ∂

∂z_j is a vector ﬁeld of type 0,1, as the transpose ofν^∧: Λ^0,qTxX →Λ^0,q+1TxX. We have the standard identity,

ω^∧ν+νω^∧=ω, νid.

In the present case we have the analogous identity,

ω₁^∧ω₂+ω₂ω₁^∧=H^∗(ω1, ω2)id, (2.2) when ω1, ω2 are (0,1)-forms. Notice also that ω₂ depends anti-linearlyon ω2.

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Let e1(z), ..., en(z) be an orthonormal frame for Λ^0,1T^∗X. Let Z1(z), ..., Zn(z) be the dual basis of Λ^0,1T X, so that on scalar functions,

∂= n

1

ej(z)^∧⊗Zj(z, ∂

∂z).

Iff(z)ej₁∧...∧ejq is a typical term in a general (0, q)-form, we get

∂(f(z)ej₁∧...∧ej_q)

= n j=1

Zj(f)e^∧_jej₁∧...∧ej_q+ q k=1

(−1)^k⁻¹f(z)ej₁∧..∧(∂ej_k)∧..∧ej_q

= ( n j=1

Zj(f)e^∧_j)ej1∧...∧ejk+ ( n j=1

(∂ej)^∧e_j)(f(z)ej1∧...∧ejq).

So for the given orthonormal frame we have the identiﬁcation

∂≡ n j=1

(e^∧_j ⊗Zj+ (∂ej)^∧e_j) (2.3) and correspondingly

∂^∗≡ n j=1

(e_j⊗Z_j^∗+e^∧_j(∂ej)), whereZ_j^∗ is the formal complex adjoint ofZj inL²(m).

Ifsis a trivializing local holomorphic section ofL, thens^kis a trivializing local section ofL^k, and the corresponding metrichk onL^k satisﬁes

|s^k|²h_k =|s|^2kh₁ =e⁻^2kφ(z). Hence if

ω = s^kω∈ E^0,q(X;L^k),

w = s^kw∈ E^0,q+1(X;L^k),

we get for∂,∂^∗, acting on (0, q)-forms with coeﬃcients inL^k:

∂(s^kω) = s^k n j=1

(e^∧_j ⊗Zj+ (∂ej)^∧e_j)ω,

∂^∗(s^kw) = s^k n j=1

(e_j⊗(Z_j^∗+ 2kZj(φ)) +e^∧_j(∂ej))w.

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We next derive more symmetric representations for∂,∂^∗in spaces without exponential weights, byusing the following local representation,

ω= (se^φ)^kω∈ E^0,q(X;L^k), (2.4) so that

E^0,q(X)ω→(se^φ)^kω∈ E^0,q(X;L^k) is locallyunitaryin view of the fact that|s(x)e^φ(x)|h₁(x)= 1:

|ω(x)|²hk(x)⊗Hm(dx) =

|ω(x)|²H(x)m(dx). (2.5) Using (2.3), which makes sense directlyon elements ofE^0,q(X, L^k), we get

∂ω= (se^φ)^k∂sω, (2.6) where,

∂sω= n j=1

(e^∧_j ⊗(Zj+kZj(φ)) + (∂ej)^∧e_j). (2.7) Now the formal adjoint of∂sfor the scalar product given bythe right hand side of (2.5) is

∂^∗sw= n j=1

(e_j⊗(Zj^∗+kZj(φ)) +e^∧j(∂ej)), (2.8) where in view of the unitarityof the relation (2.4),

∂^∗w= (se^φ)^k∂^∗_sw, (2.9) where

w= (se^φ)^kw. (2.10)

Now rewrite things semiclassically. Put h= 1

k, (2.11)

h∂s= n j=1

(e^∧_j ⊗(hZj+Zj(φ)) +h(∂ej)^∧e_j), (2.12)

h∂^∗_s= n j=1

(e_j⊗(hZ_j^∗+Zj(φ)) +he^∧_j(∂ej)). (2.13) HerehZj is a semiclassical diﬀerential operator.

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Proposition 2.1. — Using the representation (2.4), we can identify the Hodge Laplacian with

∆ = (2.14)

(h∂s)(h∂^∗_s) + (h∂^∗_s)(h∂s) = n

j=1

1⊗(hZ_j^∗+Zj(φ))(hZj+Zj(φ))

+

j,k

e^∧_je_k⊗[hZj+Zj(φ), hZ_k^∗+Zk(φ)]

+O(h)(hZ+Z(φ)) +O(h)(hZ^∗+Z(φ)) +O(h²), whereO(h)(hZ+Z(φ))indicates a remainder term of the formh

kak(z) (hZk+Zk(φ))withak smooth, matrix-valued, and similarly for the two other remainder terms in (2.14).

Proof. — We make a straightforward calculation.

(h∂s)(h∂s)^∗+ (h∂s)^∗(h∂s) =

1j,kn

(e^∧_j ⊗(hZj+Zj(φ)))(e_k⊗(hZ_k^∗+Zk(φ))) +(e_k⊗(hZk^∗+Zk(φ)))(e^∧j ⊗(hZj+Zj(φ)))

+(e^∧_j ⊗(hZj+Zj(φ)))(he^∧_k(∂ek)) + (he^∧_k(∂ek))(e^∧_j ⊗(hZj+Zj(φ))) +h((∂ej)^∧e_j)(e_k⊗(hZk^∗+Zk(φ))) + (e_k⊗(hZk^∗+Zk(φ)))h((∂ej)^∧e_j) +h((∂ej)^∧e_j)he^∧_k(∂ek)+he^∧_k(∂ek)h((∂ej)^∧e_j)

.

Using (2.2), we see that the sum of the ﬁrst two terms inside the general term of the sum is equal to

(e^∧_je_k+e_ke^∧_j)⊗((hZ_k^∗+Zk(φ))(hZj+Zj(φ)))

+e^∧je_k[hZj+Zj(φ), hZk^∗+Zk(φ)]

=δj,k(hZ_k^∗+Zk(φ))(hZk+Zk(φ)) +e^∧_je_k[hZj+Zj(φ), hZ_k^∗+Zk(φ)].

The proposition follows.

Let qj be the semiclassical principal symbol of hZj +Zj(φ), that we shall write down more explicitlylater, viewed as a function on the “real”

cotangent spaceT^∗X. (We refer to [42, 19] for standard terminologyabout semiclassical pseudodiﬀerential operators, and to [26, 48] for the fact that

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the Weyl quantization permits to deﬁne the symbol of such an operator modulo O(h²) even on a manifold.) The semiclassical principal symbol of

∆ is

p0= 1⊗ n j=1

q_jqj. (2.15)

The semiclassical subprincipal symbol of ∆ is a well-deﬁned endomorphism of Λ^0,qT^∗X at everypoint (x, ξ)∈Σ on the doublycharacteristic manifold Σ⊂T^∗X, given byq1=...=qn = 0. For an operator of the form (hZ_k^∗+ Zk(φ))(hZj+Zj(φ)) this subprincipal symbol is given by _2i^h{q_k, qj}and the contribution from the double sum in (2.14) to the subprincipal symbol of ∆ is

h i

j,k

e^∧_je_k⊗ {qj, qk}.

Thus on Σ, we get the subprinicipal symbol of ∆:

hp1=h(1⊗

j

−1

2i{qj, qj}+

j,k

e^∧je_k1

i{qj, qk}). (2.16) Sincep1is invariantlydefined on Σ as well as the first sum, the double sum is also invariantlydefined.

To compute further, we choose holomorphic coordinatesz1, ..., zn, zj= xj+iyj. We make the following ﬁberwise bijections between Λ^1,0T^∗X,T^∗X, Λ^0,1T^∗X:

n 1

ζjdzj ↔Re ( n

1

ζjdzj)↔ n

1

ζ_jdzj. (2.17) Writing

ζj=ξj−iηj, we get

Re (

ζjdzj) =

(ξjdxj+ηjdyj), so in local coordinates, we have bijections between

(z, ζ)∈Λ^1,0T^∗X, (x, y;ξ, η)∈T^∗X, (z, ζ)∈Λ^0,1T^∗X.

The semiclassical symbol ofh_∂z^∂

j = ¹₂(h_∂x^∂

j +ih_∂y^∂

j) is ₂ⁱ(ξj+iηj) = ₂ⁱζj. Hence the symbol of

h ∂

∂zj

+ ∂φ

∂zj

is i

2ζ_j+ ∂φ

∂zj

,

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so in the coordinates (z, ζ), the equation for Σ becomes:

ζ_j=−2 i

∂φ

∂zj

, or equivalently,

ζj =2 i

∂φ

∂zj

, j= 1,2, .., n. (2.18) For later use we here compute the principal symbolqj ofhZj+Zj(φ):

Let the orthonormal framee1, ..., en be given by ej(z) =

k

aj,k(z)dzk,

and the corresponding dual basisZ1, ..., Zn of Λ^0,1T_z^∗X by Zj=

k

bj,k

∂

∂zk

,

where the invertible matrices (aj,k) and (bj,k) are related by

t(bj,k)(aj,k) = 1.

Then it follows from the calculations above that qj=

k

bj,k(i

2ζ_k+ ∂φ

∂zk

). (2.19)

Proposition 2.2. — In the(z, ζ)-coordinates, the Poisson bracket{f, g} of two C¹-functionsf, g is given by

1

2{f, g}= 1

2Hfg= (∂f

∂ζ · ∂g

∂z+∂f

∂ζ ·∂g

∂z)−(∂f

∂z ·∂g

∂ζ +∂f

∂z ·∂g

∂ζ) (2.20) Proof. — Consider the real canonical 1-form on T^∗X:

Re (

ζjdzj) =

(ξjdxj+ηjdyj).

Hence the real symplectic form becomes

d(

(ξjdxj+ηjdyj)) = Re (

dζj∧dzj) = Reσ=:ω, where σ =

dζj∧dzj. If f is a smooth real function on the real phase space, the corresponding Hamilton ﬁeldHf is given by

ω, t∧Hf=t, df. (2.21)

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Witht= 2Re (aj ∂

∂zj +bj ∂

dζj), the right hand side becomes

2Re

(aj

∂f

∂zj

+bj

∂f

∂ζj

), while the left hand side is equal to

Reσ, t∧Hf= Re

(bjdzj, Hf −ajdζj, Hf).

Varyingt, we conclude that dzj, Hf= 2∂f

∂ζj

, dζj, Hf=−2∂f dzj

, so

1

2Hf = (∂f

∂ζ · ∂

∂z−∂f

∂z · ∂

∂ζ) + (∂f

∂ζ · ∂

∂z−∂f

∂z · ∂

∂ζ).

In particular, we get (2.20) This expression now extends to the case when f, gare complex-valued functions which completes the proof.

Of course (2.20) can also be obtained bystraightforward calculation from {f, g}=∂f

∂ξ

∂g

∂x+∂f

∂η

∂g

∂y−∂f

∂x

∂g

∂ξ−∂f

∂y

∂g

∂η, ∂

∂x= ∂

∂z+∂

∂z, ∂

∂y=1 i(∂

∂z−∂

∂z), ...

(2.22) Now return to the expressions (2.14), (2.15). If z0 is a ﬁxed point, we choose holomorphic coordinates z1, ..., zn as above in such a waythat Zj = _∂z^∂

j, ej =dzj at z0. Thenbj,k(z0) =δj,k in (2.19) and at the corresponding pointρ0= (z0, ζ0)∈Σ, we have

{qj, qk}(ρ0) ={i

2ζj+ ∂φ

∂zj

,−i

2ζk+ ∂φ

∂zk}.

Applying (2.20), we now get 1

2{qj, q_k}= i 2

∂²φ

∂zj∂zk

+ ∂²φ

∂zk∂zj

i

2 =i ∂²φ

∂zj∂zk

. We rewrite this as

1

2i{qj, q_k}= ∂²φ

∂zj∂zk

, (2.23)

and recognize here the coeﬃcients of the Levi-matrix appearing also in∂∂φ.

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Proposition 2.3. —Σis symplectic at a point(z0;ξ0, η0)iﬀ(_∂z^∂²^φ

j∂zk)(z0) is non-degenerate. Indeed, if we identifyΛ^1,0T^∗X andT^∗X, by means of the ﬁrst bijection in (2.17), then the real symplectic formωbecomesRe (

dζj∧ dzj)and its restriction to Σcan be identiﬁed with ²_i∂∂φ.

Proof. — With the above mentioned identiﬁcation, Σ takes the form (2.18) which can be written more invariantlyas

ζ·dz= 2

i∂φ. (2.24)

Hence, σ_|Σ=d

n 1

2 i

∂φ

∂zj

∧dzj = n j=1

n k=1

2 i

∂²φ

∂zk∂zj

dzk∧dzj =2 i∂∂φ.

This is a real form, so it is also the restriction to Σ of Reσand it is non- degenerate preciselywhen (_∂z^∂²^φ

j∂z_k) is (cf [45]).

Back to the general case, we recall the condition for having the apriori estimate

hu+

(hZj+Zj(φ))u+

(hZj^∗+Zj(φ))uC∆qu, (2.25) foru∈C₀^∞(neigh (z0); Λ^0,qT^∗X).

Proposition 2.4. — (2.25) does not hold precisely when n₋ q n−n+, where(n+, n₋)is the signature of (_∂z^∂²^φ

j∂zk(z0)).

This is essentiallywell-known since the ∂-estimates of L. H¨ormander (see [27]), and in the context of more general hypoelliptic operators it was obtained in [46] in the non-degenerate symplectic case. The constant C in formula (2.25) is also closelyrelated to the curvature term appearing in the Bochner-Kodaira-Nakano formula [24, 18]. The result will not be used explicitlysince the heat equation method below will give enough control (and would allow to recover it easily, compare Proposition 3.1).

3.The associated heat equations

We work locallynear a pointz0∈X, where ( ∂²φ

∂zj∂zk

) is non-degenerate of signature (n+, n₋), (3.1)

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so that the characteristic manifold Σ of ∆q is symplectic. We review some results of A. Menikoﬀ, J. Sj¨ostrand [40], [41] that applyto the present situation with minor changes:

In those works, we considered a scalar classical pseudodiﬀerential operator with principal symbolp0 vanishing to preciselythe second order on a conic symplectic submanifold ofT^∗X. In the present work, we have a semiclassical diﬀerential operator with a leading symbolp0 in (2.3) that we can view as scalar;p0=n

1qjqj andp0 is no longer homogeneous, and Σ is no longer conic in the ﬁber variables.

In this section we consider the problem:

(h∂t+ ∆q)u(t, x) = 0, u(0, x) =v(x). (3.2) We shall applythe standard WKB construction of an approximative solution operator and applyarguments from [40] together with a “Witten trick” to get additional properties to be used later. See Proposition 3.3 for the precise statement about the solution to (3.2). Following a standard idea, we will see how to reduce ourselves to the homogeneous situation (in the proof of Proposition 3.3). Since the non-scalar nature of the operator appears only in the subprincipal terms, it will onlyaﬀect the transport equations which can be treated verymuch as in the scalar case. The reallynew feature is the exponential convergence of the heat parametrix whent→ ∞in the case of (0, n₋)-forms.

We forget about most of the complex structure ofX and work in some smooth local coordinates x = (x1, ..., x2n) deﬁned on X ⊂⊂ X. At least for small t 0, we look for an approximate solution of (3.2) of the form u(t, x) =U(t)v(x),

u(t, x) = 1 (2πh)²ⁿ

e^hⁱ^(ψ(t,x,η)⁻^y^·^η)a(t, x, η;h)u(y)dydη, (3.3) whereais a matrix-valued classical symbol of order 0:

a(t, x, η;h)∼ ∞

0

ak(t, x, η)h^k, a_|t=0= 1, (3.4) andψwith Imψ0 should solve the eikonal equation,

i∂tψ(t, x, η) +p0(x, ψ_x(t, x, η)) = 0 +O((Imψ)^∞), ψ_|t=0=x·η. (3.5) The amplitude a is determined bya sequence of transport equations that will be reviewed later. (Here we follow the convention thatu=O((Imψ)^∞)

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means thatu=O((Imψ)^N) for everyN 0, uniformlyor locallyuniformly depending on the context.)

According to the general theoryin [37, 38], this equation can be solved locally, provided that we also denote by p0 an almost holomorphic extension. The general theoryalso tells us thatU(t) is associated to a canonical transformation,

κt= exp(−itHp₀). (3.6)

(Hereκtdepends slightlyon the choice of almost holomorphic extension of p0, so κt(ρ) is well-deﬁned onlyup to |Imρ|^∞. In [38] we also made the assumption that p0(x, ξ) is positivelyhomogeneous of degree 1 inξ, but as noticed for instance in [40] and will be reviewed in the proof of Proposition 3.3, one can easilyreduce the general case to the homogeneous one, by adding a variable x0 and consider the homogeneous symbol ξ0p(x, ξ/ξ0), then restrict the results toξ0= 1.)

So far, we onlyused the non-negativityof (the real part of)p0. Now we use thatp0∼dist (·,Σ)². It follows that

ψ(t, x, η) =x·η+O(tdist (x, η; Σ)²), (3.7) Imψ(t, x, η)∼tdist (x, η; Σ)², (3.8) for 0tt0, andt0>0 ﬁxed. Correspondingly, we have

κt|Σ= id, (3.9)

Whent >0, κtis a strictlypositive canonical (3.10) transformation with graph (κt)∩(T^∗X)²= diag (Σ×Σ).

Recall that a positive canonical transformation is strictlypositive if the graphκintersectsT^∗X×T^∗Xcleanlyalong a smooth submanifold. Thanks to these simplifying features, all essential properties ofψandκtare captured bytheir Taylor expansions att= 0 and at Σ.

In [40] it was shown that (3.5) can be solved for allt0, and that we have,

Imψ(t, x, η)∼dist (x, η; Σ)², (3.11) uniformlyfor t 1, that (3.9), (3.10) remain valid for all t > 0, and ﬁnallythat there exists a smooth function ψ(∞, x, η), well-deﬁned mod O(dist (x, η; Σ)^∞) such that for allk, α:

∂_t^k∂^α_x,η(ψ(t, x, η)−ψ(∞, x, η)) =O(e⁻^t/C), (3.12)

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uniformlyon [0,+∞[×Σ. (In [41] we also established asymptotic expansions whent→ ∞in terms of exponentials int. We do not need those improved results here.) Here we have locallyuniformlyonX×R²ⁿ:

ψ(∞, x, η) =x·η+O(dist (x, η; Σ)²), Imψ(∞, x, η)∼dist (x, η; Σ)². (3.13) Further, the canonical relationC_∞generated bythe phaseψ(∞, x, η)−y·η is strictlypositive with

C_∞∩(T^∗X×T^∗X) = diag (Σ×Σ), (3.14) andC_∞can be described in the following way:

There are two almost holomorphic manifoldsJ+, J₋ ⊂T^∗X^C(where the latter set is the almost complexiﬁcation ofT^∗X) intersecting T^∗X cleanly along Σ, with the following properties:

codimCJ_±=n, J_± ⊂p⁻₀¹(0), (3.15) J_± are involutive andJ₋ =J+,

1

iσ(t, t)>0,∀t∈Tρ(J+)\Tρ(Σ^C), ρ∈Σ.

Here the involutivityof J+ (and similarlyfor J₋) means that J+ is given bythe equations q1=... =qn = 0, wheredq1, ..., dqn areC-linearly independent and {qj,qk} = 0 on J+. Further the complexiﬁcation Σ^C is contained inJ+ andH^q1, ..., H^qn spanTρJ+/TρΣ^C. The positivityproperty above is equivalent to the fact that the Hermitian matrix (¹_i{qj,q_k}) is positive deﬁnite. In terms of J_±, we can describe the limiting canonical relationC_∞ as{(ρ, µ)∈J+×J₋; then-dimensional bicharacteristic leaves throughρ,µofJ+ andJ₋ respectively, intersect Σ^C₊ at the same point}.

Finallywe can also view C_∞ as the limit of Ct = graph (κt), when t→+∞, where the convergence is exponentiallyfast (in the sense of Taylor expansions at diag (Σ×Σ)). We can also viewJ+,J₋ as the stable outgoing and incoming manifolds respectively, for theH₋ip-ﬂow, near the ﬁxed point set Σ^C. Let us also add thatJ_± are uniquelydetermined and that in the casen+=n, we can takeqj =qj.

Next we consider the behaviour ofain (3.3), (3.4), where we recall that a0,a1, ... are successivelydetermined bya sequence of transport equations.

Following [40] this can be done in the following way, where we take some advantage of the fact that we work in the Weyl quantization. (See also appendix b of [26].): Formally, with ψ = ψ(t,·, η), P = ∆q and with the

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exponent windicating that we take theh-Weyl quantization, we get e⁻^iψ◦P◦e^iψ/h=P(x, ψ_x(x) +ξ;h)^w+O(h²) =

p(x, ψx) +hp1(x, ψx) +1

2(hDx◦pξ(x, ψx) +p_ξ(x, ψ_x)◦hDx) +O(h²) = p(x, ψ_x) +hp1(x, ψ_x) +h

ip_ξ(x, ψ_x)· ∂

∂x +h

2idiv (pξ(x, ψx)· ∂

∂x) +O(h²),

where the ”O(h²)” refers to the action on symbols andp1is the subprincipal symbol. This gives the ﬁrst transport equation fora0:

(ν+1

2div (ν) +p1)a0= 0, where

ν= ∂

∂t−ip_ξ(x, ψ_x)· ∂

∂x.

The higher transport equations for aj,j1, are of the form ν(aj) =Fj(t, x, a0, ..., aj−1).

Then ifa(t, x, η;h)∼_∞

0 aj(t, x, η)h^j inC^∞([0,+∞[×X×R²ⁿ), we have (h∂t+ ∆q)(e^hⁱ^ψ(t,x,η)a(t, x, η;h)) =O(h^∞)

locallyuniformlyon [0,+∞[×X×R²ⁿ and similarlyfor the derivatives.

The discussion on page 69 in [40] shows that div (ν) → ¹₂trF exponentiallyfast on Σ, where trF =

fj, andF is the fundamental matrix of p ie the linearization of Hp at the point of Σ and has the spectrum σ(F) ={±ifj},fj0. In the further discussion of the transport equations the onlynew feature is thatp1is now a square matrix rather than a scalar, and whenever we needed a lower bound on Rep1, we now need a lower bound on the set of real parts of the eigenvalues of p1. Proposition 2.2 in [40] becomes

Proposition 3.1. — L etλ∈C(Σ;R)satisfy λ(x, η)< 1

2trF(x, η) + inf Reσ(p1(x, η)), (x, η)∈Σ.

Then for every compact set K ⊂Σ,j ∈ N and (γ, α, β)∈N^1+2n+2n, we have

|∂^γt∂_x^α∂_η^βaj(t, x, η)|Cj,α,β,γe⁻^tλ(x,η), (x, η)∈K, t0.

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We are therefore interested in whether 1

2trF+ inf Reσ(p1)>0 on Σ (3.16) or not. Now

p= n

1

q_jqj, Hp=

(q_jHq_j +qjHq_j).

At a given point ρ0 ∈Σ, we choose the basis Hq₁, ..., Hq_n, Hq₁, ..., Hq_n for Tρ₀(T^∗X)^C/Σ^C, and compute the linearization ofHp:

Hp(ρ0+

tkHqk+

skHq_k) = O((t, s)²) +

j,k

tk{qk, q_j}Hqj+

j,k

sk{q_k, qj}Hq_j.

So the matrixFp of the linearization is expressed in the basis above by 1

iFp=

(¹_i{qk, q_j}) 0 0 (¹_i{qk, qj})

,

where we recall (2.23). Let µ1, ..., µn be the eigenvalues of (∂zj∂zkφ), with µj >0 for 1jn+ andµj<0 forn++ 1jn. Then

(i⁻¹{qk, q_j}) =^t(i⁻¹{qj, q_k}) has the eigenvalues 2µ1, ...,2µn, and

(i⁻¹{q_k, qj}) =−^t(i⁻¹{qj, q_k}) has the eigenvalues −2µ1, ...,−2µn. Hence the non-vanishing eigenvalues ofFp are±2iµ1, ...,±2iµn, and

1

2trFp=µ1+...+µn₊−µn₊+1−...−µn. (3.17) For the ﬁrst term in (2.16), we get

j

−1

2i{qj, q_j}=−1

2itr ({qj, q_k}) =− n

1

µj. (3.18) We can also compute the eigenvalues of the matrix part of the subprincipal symbol appearing in (2.16) and in the subsequent remark about invariance.

We choose holomorphic coordinates such that at the given pointz0: Zj =

∂z_j, ej=dzj and moreover (i⁻¹{qj, qk}) is diagonalized, equal to





2µ1 0 .. 0

0 2µ2 .. 0

.. .. ..

0 0 .. 2µn



.

Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

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Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles

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