Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds
aa r X i v : . [ m a t h . C V ] D ec BERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINEBUNDLES ON KÄHLER MANIFOLDS
DAN COMAN, WEN LU, XIAONAN MA, AND GEORGE MARINESCUA
BSTRACT . Given a sequence of positive Hermitian holomorphic line bundles p L p , h p q on a Kähler manifold X , we establish the asymptotic expansion of the Bergman kernel ofthe space of global holomorphic sections of L p , under a natural convergence assumptionon the sequence of curvatures c p L p , h p q . We then apply this to study the asymptoticdistribution of common zeros of random sequences of m -tuples of sections of L p as p Ñ 8 . C ONTENTS
0. Introduction 11. Localization of the problem 51.1. Lichnerowicz formula 51.2. Spectral gap of the Dirac operator 61.3. Localization of the problem 82. Asymptotic expansion of Bergman kernel 102.1. Rescaling 102.2. Asymptotics of the scaled operators 122.3. Bergman kernel 152.4. Proof of Theorem 0.1 153. Equidistribution of zeros of random sections 17References 220. I
NTRODUCTION
The familiar setting of geometric quantization is a compact Kähler manifold p X, ω q with Kähler form ω endowed with a Hermitian holomorphic line bundle p L, h L q , calledprequantum line bundle, satisfying the prequantization condition(0.1) ω “ ?´ π R L “ c p L, h L q , where R L denotes the curvature of the Chern connection on p L, h L q and c p L, h L q de-notes the Chern curvature form of p L, h L q . The existence of the prequantum line bun-dle p L, h L q allows to consider the Hilbert space of holomorphic sections H p X, L q and Date : December 23, 2020.D. C. is partially supported by the NSF Grant DMS-1700011.W. L. supported by National Natural Science Foundation of China (Grant Nos. 11401232, 11871233).X. M. partially supported by NNSFC No.11829102 and funded through the Institutional Strategy ofthe University of Cologne within the German Excellence Initiative.G. M. partially supported by the DFG funded projects SFB TRR 191 ‘Symplectic Structures in Geometry,Algebra and Dynamics’ (Project-ID 281071066 – TRR 191) and SPP 2265 ‘Random Geometric Systems’(Project-ID 422743078). construct a correspondence between smooth objects on X (classical observables) andoperators on H p X, L q (quantum observables) [1, 16], stated in terms of the semi-classical limit in which Planck’s constant tends to zero. Changing Planck’s constant isequivalent to rescaling the Kähler form, and this is achieved by taking tensor powers L p “ L b p of the line bundle, since the curvature of L p is pω . A pivotal role in this cor-respondence is played by the orthogonal projection on H p X, L p q . Its integral kernel,called Bergman kernel, admits a full asymptotic expansion as p Ñ 8 to any order (cf.[2, 9, 20, 21, 26, 27]).Condition (0.1) in an integrality condition; a prequantum bundle exists if and onlyif the de Rham cohomology class r ω s is integral, r ω s P H p X, Z q . What can one doin general if ω is a not necessarily integral Kähler form? We can then associate to ω a more general sequence positive line bundle p L p , h p q such that their curvatures onlyapproximate multiples of ω . Such a sequence can be thought as a “prequantization” ofthe nonintegral Kähler metric ω .In this paper we establish the asymptotic expansion of the Bergman kernel of theholomorphic space H p X, L p q on Kähler manifold X under a natural approximationassumption of ω by the curvatures of the positive line bundles L p .Let p X, ϑ, J q be a compact Kähler manifold of dim C X “ n with Kähler form ϑ andcomplex structure J . Let p L p , h p q , p ě , be a sequence of holomorphic Hermitian linebundles on X with smooth Hermitian metrics h p . Let ∇ L p be the Chern connection on p L p , h p q with curvature R L p “ p ∇ L p q . Denote by c p L p , h p q the Chern curvature formof p L p , h p q . Let g T X p¨ , ¨q “ ϑ p¨ , J ¨q be the Riemannian metric on T X induced by ϑ and J . The Riemannian volume form dv X has the form dv X “ ϑ n { n ! . We endow the space C p X, L p q of smooth sections of L p with the inner product(0.2) x s , s y p : “ ż X x s , s y h p ϑ n n ! , s , s P C p X, L p q , and we set } s } p “ x s, s y p . We denote by L p X, L p q the completion of C p X, L p q withrespect to this norm. Let H p X, L p q be the space of holomorphic sections of L p and let P p : L p X, L p q Ñ H p X, L p q be the orthogonal projection. The integral kernel P p p x, x qp x, x P X q of P p with respect to dv X p x q is smooth and is called the Bergman kernel. Therestriction of the Bergman kernel to the diagonal of X is the Bergman kernel functionof H p X, L p q , which we still denote by P p , i.e., P p p x q “ P p p x, x q .The main result of this paper is as follows. Theorem 0.1.
Let p X, ϑ q be a compact Kähler manifold of dim C X “ n . Let p L p , h p q , p ě ,be a sequence of holomorphic Hermitian line bundles on X with smooth Hermitian metrics h p . Let ω be a Kähler form on X such that (0.3) A ´ p c p L p , h p q “ ω ` O p A ´ ap q , as p Ñ 8 , in the C - topology,where a ą , A p ą and lim p Ñ8 A p “ `8 . Then as p Ñ 8 , in the C - topology, P p p x q “ A np b p x q ` A n ´ p b p x q ` . . . ` A n ` t ´ a u ` p b ´ t ´ a u ´ p x q ` O p A n ´ ap q , (0.4) where b p x q “ ω n { ϑ n and b “ π p ω n { ϑ n q r Xω , where r Xω is the scalar curvature of ω , and t ´ a u the integer part of ´ a . Note that the following general result was obtained in [3, Theorem 1.2]. Let p X, ϑ q bea compact Kähler manifold of dimension n and p L p , h p q , p ě , be a sequence of holo-morphic Hermitian line bundles on X with singular Hermitian metrics h p that satisfy c p L p , h p q ě a p ϑ , where a p ą and lim p Ñ8 a p “ 8 . If A p “ ş X c p L p , h p q ^ ϑ n ´ denotes ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 3 the mass of the current c p L p , h p q , then A p log P p Ñ in L p X, ϑ n q as p Ñ 8 . Theorem0.1 refines this result under the stronger assumptions that the metrics are smooth and(0.3) holds.Our assumption (0.3) means that for any k P N , there exists C k ą such that(0.5) ˇˇ A ´ p c p L p , h p q ´ ω ˇˇ C k ď C k A ´ ap , where the C k -norm is induced by the Levi-Civita connection ∇ T X . We will give severalnatural examples of sequences p L p , h p q as above. The most straightforward is p L p , h p q “p L b p , h b p q for some fixed prequantum line bundle p L, h q . Then it follows from (0.1) that(0.3) holds for A p “ p and all a ą . In this case we recover from (0.4) the known resulton asymptotic expansion of Bergman kernel of H p X, L p q (cf. [2, 9, 20, 26, 27]). Otherexamples include p L p , h p q “ p L b p , h p q where h p is not necessarily the product h p , e. g. h p “ h p e ´ ϕ p , with suitable weights ϕ p , or tensor powers of several bundles, see Theorem0.3.Our approximation assumption (0.3) (or (0.5)) is natural in the following sense.Given a Kähler form ω one can first approximate its cohomology class r ω s P H p X, R q byintegral classes in H p X, Z q by using diophantine approximation (Kronecker’s lemma)and then one constructs smooth forms representing these approximating classes. By[18, Théorème 1.3, p. 57] condition (0.5) holds true for any k , A p “ p and a “ ` { β p X q , where β p X q denotes the second Betti number of X , but in general with anon-necessarily holomorphic Hermitian line bundle p L p , h p q . In this paper we show thatif there is a good diophantine approximation with holomorphic line bundles we obtaincorresponding good asymptotics of the Bergman kernel.If ă a ă , then Theorem 0.1 reduces to the following result. Corollary 0.2.
Let p X, ϑ q be a compact Kähler manifold of dim C X “ n . Let p L p , h p q , p ě , be a sequence of holomorphic Hermitian line bundles on X with smooth Hermitianmetrics h p . Assume that there exists a Kähler form ω on X such that (0.6) A ´ p c p L p , h p q “ ω ` O p A ´ ap q , as p Ñ 8 , in the C - topology,where ă a ă , A p ą and lim p Ñ8 A p “ `8 . Then (0.7) P p p x q “ A np b p x q ` O p A n ´ ap q , as p Ñ 8 , in the C - topology,where b p x q “ ω n { ϑ n . Note that a similar result was obtained in [3, Theorem 1.3] under different approxi-mation assumptions.An interesting situation when the previous results apply is when L p equals a productof tensor powers of several holomorphic line bundles, L p “ F m ,p b . . . b F m k,p k , where t m j,p u p , ď j ď k , are sequences in N such that m j,p “ r j p ` O p p ´ a q as p Ñ 8 , where a ě and r j ą are given. This means that p m ,p , . . . , m k,p q P N k approximate thesemiclassical ray R ą ¨ p r , . . . , r k q P R k ą with a remainder O p p ´ a q , as p Ñ 8 (cf. also[3, Corollary 5.11]).
Theorem 0.3.
Let p X, ϑ q be a compact Kähler manifold of dim C X “ n . Let p F j , h F j q besmooth holomorphic Hermitian line bundles on X with c p F j , h F j q ě for ď j ď k andone of them is strictly positive, say, c p F , h F q ě εϑ for some ε ą . Let r j ą , ď j ď k ,be positive real numbers and set ω “ ř kj “ r j c p F j , h F j q . Assume that there exist sequences ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 4 t m j,p u p , ď j ď k , in N and a ě , C ą such that (0.8) ˇˇˇ m j,p p ´ r j ˇˇˇ ď Cp a , ď j ď k, f or p ą . Let P p be the Bergman kernel function of H p X, F m ,p b . . . b F m k,p k q . Then P p p x q “ p n b p x q ` p n ´ b p x q ` . . . ` p n ´ k b k p x q ` O p p n ´ a q , (0.9) as p Ñ 8 , in the C - topology,where k “ ´ t ´ a u ´ , b p x q “ ω n { ϑ n and b “ π p ω n { ϑ n q r Xω , where r Xω is the scalarcurvature of ω . We apply Theorem 0.1 to study the asymptotic distribution of common zeros of ran-dom sequences of m -tuples of sections of L p as p Ñ 8 , see [3, 4, 5, 6, 7, 14, 15, 24, 25]for previous results and references. Let p X, ω q be a compact Kähler manifold of dimen-sion n and let p L p , h p q , p ě , be a sequence of Hermitian holomorphic line bundles on X . To study the equidistribution problem in a more general frame, we assume that themetrics h p are of class C and verify condition (0.5) for k “ . Namely, there exist aKähler form ω on X and a ą , C ą , such that for every p ě we have(0.10) ˇˇ A ´ p c p L p , h p q ´ ω ˇˇ C ď C A ´ ap , where A p ą and lim p Ñ8 A p “ 8 .As before we endow the space of global holomorphic sections H p X, L p q with theinner product (0.2) and we set } s } p “ x s, s y p , d p “ dim H p X, L p q . Let P p be the Bergmankernel function of H p X, L p q . We assume that there exist a constant M ą and p P N such that(0.11) A np M ď P p p x q ď M A np , holds for every x P X and p ą p . Note that, under the stronger hypothesis (0.3),condition (0.11) follows easily from Theorem 0.1 (see (0.4)).Given m P t , . . . , n u and p ě we consider the multi-projective space(0.12) X p,m : “ ` P H p X, L p q ˘ m , equipped with the probability measure σ p,m which is the m -fold product of the Fubini-Study volume on P H p X, L p q » P d p ´ . If s P H p X, L p q we denote by r s “ s the currentof integration (with multiplicities) over the analytic hypersurface t s “ u Ă X , and welet r s p “ s : “ r s p “ s ^ . . . ^ r s pm “ s , for s p “ p s p , . . . , s pm q P X p,m ,whenever this current is well-defined (see Section 3). We also consider the probabilityspace p X ,m , σ ,m q : “ ź p “ p X p,m , σ p,m q . In the above setting, we have the following theorem:
Theorem 0.4.
Let p X, ϑ q be a compact Kähler manifold of dimension n and let p L p , h p q , p ě , be a sequence of Hermitian holomorphic line bundles on X with metrics h p of class C . Assume that conditions (0.10) and (0.11) hold. Then there exist C ą and p P N such that for every β ą , m P t , . . . , n u and p ą p there exists a subset E βp,m Ă X p,m with the following properties:(i) σ p,m p E βp,m q ď CA ´ βp ; ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 5 (ii) if s p P X p,m z E βp,m then, for any p n ´ m, n ´ m q form φ of class C on X , (0.13) ˇˇˇA A mp r s p “ s ´ ω m , φ Eˇˇˇ ď C ´ p β ` q log A p A p ` A ´ ap ¯ } φ } C . Moreover, if ř p “ A ´ βp ă `8 then estimate (0.13) holds for σ ,m -a.e. sequence t s p u p ě P X ,m provided that p is large enough. The question of characterizing the positive closed currents on X which can be ap-proximated by currents of integration along analytic subsets of X , and its local versionas well, are important problems in pluripotential theory and have many applications.Results in this direction are obtained in [5, 8]. Theorem 0.4 shows in particular that thesmooth positive closed form ω m can be approximated by currents of integration alonganalytic subsets of X of dimension n ´ m , for each m P t , . . . , n u .The paper is organized as follows. In Section 1 we show that the asymptotic expan-sion of the Bergman kernel can be localized. In Section 2 we establish the asymptoticexpansion of the Bergman kernel near the diagonal and then prove Theorem 0.1. Theproof of Theorem 0.4 is given in Section 3, using the technique of meromorphic trans-forms of Dinh and Sibony [15], as in the papers [7, 8].1. L OCALIZATION OF THE PROBLEM
In this Section we show that the problem is local by using the Lichnerowicz formula.1.1.
Lichnerowicz formula.
The complex structure J induces a splitting T X b R C “ T p , q X ‘ T p , q X , where T p , q X and T p , q X are the eigenbundles of J correspondingto the eigenvalues ?´ and ´?´ respectively. Let T ˚p , q X and T ˚p , q X be the cor-responding dual bundles. Denote by Ω ,j p X, L p q the space of smooth p , j q -forms over X with values in L p and set Ω , ‚ p X, L p q “ ‘ nj “ Ω ,j p X, L p q . We still denote by x¨ , ¨y thefibrewise metric on Λ p T ˚p , q X q b L p induced by g T X and h p .The L -scalar product on Ω , ‚ p X, L p q is given by (0.2). Let B L p , ˚ be the formal adjointof the Dolbeault operator B L p with respect to the scalar product (0.2). The Dolbeault-Dirac operator is given by(1.1) D p : “ ? ´ B L p ` B L p , ˚ ¯ : Ω , ‚ p X, L p q Ñ Ω , ‚ p X, L p q . The Kodaira Laplacian(1.2) l L p : “ B L p B L p , ˚ ` B L p , ˚ B L p : Ω , ‚ p X, L p q Ñ Ω , ‚ p X, L p q , preserves the Z -grading on Ω , ‚ p X, L p q . It is an essentially self-adjoint operator on thespace L , ‚ p X, L p q , the L -completion of Ω , ‚ p X, L p q . We have(1.3) D p “ l L p . For any v P T X with decomposition v “ v , ` v , P T p , q X ‘ T p , q X , let v ˚ , P T ˚p , q X be the metric dual of v , . Then c p v q “ ? p v ˚ , ^ ´ i v , q defines the Clifford action of v on Λ p T ˚p , q X q , where ^ and i denote the exterior and interior product respectively.Let ∇ T X denote the Levi-Civita connection on p T X, g
T X q , then its induced connectionon T p , q X is the Chern connection ∇ T p , q X on p T p , q X, h T p , q X q , where h T p , q X is theHermitian metric on T p , q X induced by g T X . The Chern connection ∇ T p , q X on T p , q X ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 6 induces naturally a connection ∇ Λ p T ˚p , q X q on Λ p T ˚p , q X q . Then by [20, p. 31] we havefor an orthonormal frame t e j u nj “ of p X, g
T X q ,(1.4) D p “ n ÿ j “ c p e j q ∇ Λ p T ˚p , q X qb L p e j , with(1.5) ∇ Λ p T ˚p , q X qb L p “ ∇ Λ p T ˚p , q X q b Id ` Id b ∇ L p . Denote by ∆ Λ p T ˚p , q X qb L p the Bochner Laplacian on Λ p T ˚p , q X q b L p . Then(1.6) ∆ Λ p T ˚p , q X qb L p “ ´ n ÿ j “ ” ∇ Λ p T ˚p , q X qb L p e j ∇ Λ p T ˚p , q X qb L p e j ´ ∇ Λ p T ˚p , q X qb L p ∇ TXej e j ı . Let K X “ det p T ˚p , q X q be the canonical line bundle on X . The Chern connection ∇ T p , q X on T p , q X induces the Chern connection ∇ K ˚ X on K ˚ X “ det p T p , q X q . Denoteby R K ˚ X the curvature of ∇ K ˚ and by r X the scalar curvature of p X, g
T X q . The Lich-nerowicz formula (cf. [20, (1.4.29)]) reads(1.7) D p “ ∆ Λ p T ˚p , q X qb L p ` r X ` ´ R L p ` R K ˚ X ¯ p e i , e j q c p e i q c p e j q . Spectral gap of the Dirac operator.
As in the case of powers L p of a single linebundle we have a spectral gap for the square D p of the Dirac operator acting on L p . Theresult and the proof are analogous to [19, Theorem 1.1], [20, Theorem 1.5.5]. For aHermitian holomorphic line bundle p L, h q on X set(1.8) a L – inf R Lx p u, u q| u | g TX : x P X, u P T p , q x X zt u + . Note that a L p x q “ inf R Lx p u, u q{| u | g TX : u P T p , q x X zt u ( is the smallest eigenvalue ofthe curvature form R Lx with respect to g T Xx for x P X and a L “ inf x P X a L p x q .We denote by Spec p A q the spectrum of a self-adjoint operator A on a Hilbert space. Theorem 1.1.
Let p X, ϑ q be a compact Kähler manifold. There exists C ą such thatfor all Hermitian holomorphic line bundles p L, h q on X the Dirac operator D “ D L on L satisfies the estimate (1.9) ›› Ds ›› L ě p a L ´ C q ›› s ›› L , s P Ω ą p X, L q : “ n à j “ Ω ,j p X, L q . Moreover,
Spec p D q Ă t u Y r a L ´ C, .Proof. We will use the Bochner-Kodaira-Nakano formula [20, (1.4.63)]. The Chern con-nection on K ˚ X b L is given by(1.10) ∇ K ˚ X b L “ ∇ K ˚ X b Id ` Id b ∇ L , and its curvature is(1.11) R K ˚ X b L “ R K ˚ X b Id ` Id b R L . Let t w j u nj “ be an orthonormal frame of T p , q X . Then [20, (1.4.63)] reads(1.12) l L s “ ∆ , ‚ s ` R L b K ˚ X p w j , w k q w k ^ i w j s, for s P Ω , ‚ p X, L q . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 7 where ∆ , ‚ is a holomorphic Kodaira type Laplacian. Since x ∆ , ‚ s, s y ě , (1.12) yields(1.13) ›› B L s ›› ` ›› B L, ˚ s ›› ě @ R K ˚ X b L p w j , w k q w k ^ i w j s, s D , s P Ω , ‚ p X, L q . Since X is compact there exists C ą such that(1.14) @ R K ˚ X p w j , w k q w k ^ i w j s, s D ě ´ C ›› s ›› , s P Ω , ‚ p X, L q . By (1.8) and (1.14) we have(1.15) @ R L p w j , w k q w k ^ i w j s, s D ě a L ›› s ›› , s P Ω ą p X, L q . Then (1.9) follows immediately from (1.3) and (1.13)–(1.15).Since X is compact, D has a discrete spectrum consisting of eigenvalues of finitemultiplicity. Let s P C p X, L q is an eigensection of D with D s “ λs and λ ‰ , then Ds ‰ and(1.16) D p Ds q “ λDs. Now Ds P Ω , p X, L q , so by (1.9) we have λ ě a L ´ C . (cid:3) For a sequence p L p , h p q , p ě , let us denote(1.17) a p – a L p , with a L p in (1.8). We have thus(1.18) ›› D p s ›› L ě p a p ´ C q ›› s ›› L , s P Ω ą p X, L p q “ n à j “ Ω ,j p X, L p q . Note that under hypothesis (0.3) we have lim p Ñ8 a p “ 8 . As a consequence of Theorem1.1 we obtain a Kodaira-Serre vanishing theorem for the sequence L p . Corollary 1.2.
Let p X, ϑ q be a compact Kähler manifold and let p L p , h p q , p ě , be asequence of holomorphic Hermitian line bundles on X such that lim p Ñ8 a p “ . Then for p large enough the Dolbeault cohomology groups of L p satisfy (1.19) H ,j p X, L p q “ , for j ‰ . Hence the kernel of D p is concentrated in degree for p large enough, i.e., (1.20) Ker p D p q “ H p X, L p q , p " . Proof.
By Hodge theory we know that(1.21)
Ker D p ˇˇ Ω ,j p X,L p q » H ,j p X, L p q , where H , ‚ p X, L p q denotes the Dolbeault cohomology groups. Thus (1.19) follows from(1.9). Moreover, (1.21) and (1.19) yield (1.20). (cid:3) We also need a generalization for non-compact manifolds of Theorem 1.1. Let p X, ϑ q be a Hermitian manifold and let p L, h q be a Hermitian holomorphic line bundle on X .If p X, ϑ q is complete, then the square D of the Dirac operator on L is essentially self-adjoint and we denote by Dom p D q the domain of its self-adjoint extension. Theorem 1.3.
Let p X, ϑ q be a complete Kähler manifold such that its Ricci curvature c p K ˚ X , h K ˚ X q is bounded from above. Then there exists C ą such that for all Hermit-ian holomorphic line bundles p L, h q on X with a L ą ´8 we have (1.22) ›› Ds ›› L ě p a L ´ C q ›› s ›› L , s P Dom p D q X n à j “ L ,j p X, L q . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 8
Moreover,
Spec p D q Ă t u Y r a L ´ C, .Proof. The proof follows from the proof of [20, Theorem 6.1.1]. (cid:3)
Localization of the problem.
Let a X be the injectivity radius of p X, g
T X q , and ε P p , a X { q . We denote by B X p x, ε q and B T x X p , ε q the open ball in X and T x X withthe center x and radius ε , respectively. Then we identify B T x X p , ε q with B X p x, ε q bythe exponential map Z ÞÑ exp Xx p Z q for Z P T x X . Let f : R Ñ r , s be a smooth evenfunction such that f p v q “ for | v | ď ε { and f p v q “ for | v | ě ε . Set(1.23) F p a q “ ´ ż f p v q dv ¯ ´ ż e iva f p v q dv. Then F p a q lies in the Schwartz space S p R q and F p q “ . Proposition 1.4.
For any l, m P N , ε ą , there exists C l,m,ε ą such that for p ě and x, x P X , ˇˇ F p D p qp x, x q ´ P p p x, x q ˇˇ C m p X ˆ X q ď C l,m,ε A ´ lp , ˇˇ P p p x, x q ˇˇ C m p X ˆ X q ď C l,m,ε A ´ lp , if d p x, x q ě ε . (1.24) Here the C m norm is induced by ∇ L p and ∇ T X .Proof.
We adapt here the proof of [9, Proposition 4.1], [20, Proposition 4.1.5]. For a P R , set(1.25) φ p p a q “ r? a p , p| a |q F p a q . For a p ą C we have by Theorem 1.1,(1.26) F p D p q ´ P p “ φ p p D p q . By (1.23), for any m P N there exists C m ą such that(1.27) sup a P R | a | m | F p a q| ď C m . Since X is compact there exist t x i u ri “ such that t U i “ B X p x i , ε qu ri “ is a covering of X . We identify B T xi X p , ε q with B X p x i , ε q by the exponential map as above. For Z P B T xi X p , ε q we identify p L p q Z – p L p q x i , Λ p T ˚p , q X q Z – Λ p T ˚p , q x i X q , by parallel transport along the curve r , s Q u ÞÑ uZ with respect to the connection ∇ L p and ∇ Λ p T ˚p , q X q , respectively.Let t e j u nj “ be an orthonormal basis of T x i X . Let ˜ e j p Z q be the parallel transport of e j with respect to ∇ T X along the above curve. Let Γ L p , Γ Λ p T ˚p , q X q be the correspondingconnection forms of ∇ L p and ∇ Λ p T ˚p , q X q with respect to any fixed frame for L p and Λ p T ˚p , q X q which is parallel along the above curve under the trivialization on U i .Denote by ∇ U the ordinary differentiation operator on T x i X in the direction U . By(1.4),(1.28) D p “ n ÿ j “ c p ˜ e j q ` ∇ ˜ e j ` Γ L p p ˜ e j q ` Γ Λ p T ˚p , q X q p ˜ e j q ˘ . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 9
Let t ρ i u be a partition of unity subordinate to t U i u . For ℓ P N , we define a Sobolev normon the ℓ -th Sobolev space H ℓ p X, L p q by(1.29) ›› s ›› H ℓp “ r ÿ i “ ℓ ÿ k “ n ÿ i ,...,i k “ ›› ∇ e i . . . ∇ e ik p ρ i s q ›› L . Denote by R “ ř j Z j e j the radial vector field. By [20, (1.2.32)], L R Γ L p “ i R R L p . Set(1.30) Γ L p “ n ÿ j “ a j p Z q dZ j , a j P C p U i q . Then(1.31) p L R Γ L p q Z “ n ÿ j,k “ ´ Z k B a j B Z k p Z q ¯ dZ j ` n ÿ j “ a j p Z q dZ j . Evaluating at the point tZ yields(1.32) BB t ` ta j p tZ q ˘ dZ j “ p L R Γ L p q tZ “ p i R R L p q tZ . From (1.32) we obtain immediately Γ L P “ and (cf. also [12, (2-16)])(1.33) Γ L p Z “ ż p L R Γ L p q tZ dt “ ż p i R R L p q tZ dt, which allows us to estimate the term Γ L p in (1.28). From (1.28), (1.29) and (1.33),(1.34) } s } H p ď C ` } D p s ›› L ` A p } s } L ˘ . Let Q be a differential operator of order m with scalar principal symbol and with com-pact support in U i , then(1.35) “ D p , Q ‰ “ n ÿ j “ “ c p ˜ e j q Γ L p p ˜ e j q , Q ‰ ` n ÿ j “ “ c p ˜ e j qp ∇ ˜ e j ` Γ Λ p T ˚p , q X q p ˜ e j qq , Q ‰ , where the sums are differential operators of orders m ´ and m , respectively. By (1.34)and (1.35), ›› Qs ›› H p ď C ` } D p Qs } L ` A p } Qs } L ˘ (1.36) ď C ` } QD p s } L ` A p } s } H mp ˘ . Due to (1.36), for every m P N there exists C m ą such that for p ě ,(1.37) ›› s ›› H m ` p ď C m ` } D p s } H mp ` A p } s } H mp ˘ . This means that(1.38) } s } H m ` p ď C m A m ` p m ` ÿ j “ A ´ jp } D jp s } L . Using the Sobolev estimate (1.38) we can now repeat the proof of [9, Proposition 4.1]and conclude the proof of Proposition 1.4. (cid:3)
From Proposition 1.4 and the finite propagation speed of solutions of hyperbolic equa-tions [20, Theorem D.2.1, (D.2.17)], F p D p qp x, x q only depends on the restriction of D p to B X p x, ε q and the asymptotics P p p x, x q as p Ñ 8 are localized on a neighborhood of x . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 10
2. A
SYMPTOTIC EXPANSION OF B ERGMAN KERNEL
In this Section we establish the asymptotic expansion of the Bergman kernel near thediagonal and then prove Theorem 0.1.2.1.
Rescaling.
To get uniform estimates of the Bergman kernel in terms of p , we adaptthe approach of [12, §2] by using holomorphic coordinates instead of normal coordi-nates as in [9, §4.2] and [20, §4.1.3].Let ψ : X Ą U Ñ V Ă C n be a holomorphic local chart such that P V and V isconvex (by abuse of notation, we sometimes identify U with V and x with ψ p x q ). Thenfor x P V : “ t y P C n , y P V u , we will use the holomorphic coordinates induced by ψ and let ă ε ă be such that B p x, ε q Ă V for any x P V . We choose ε ď a X { inorder to use the estimates given in the proof of Proposition 1.4.For x P V consider the holomorphic family of holomorphic local coordinates ψ x : ψ ´ p B p x, ε qq Ñ B p , ε q , ψ x p y q : “ ψ p y q ´ x . The L -norm on B p x, ε q is given by(2.1) } s } L ,x “ ż B p x, ε q | s p y q| dv X p y q . Let S p be a unitary section of p L p , h p q which is parallel with respect to ∇ L p along thecurve r , s Q u ÞÑ uZ for | Z | ď ε . Lemma 2.1.
There exists a holomorphic frame σ p : “ e f p S p of L p on B p x, ε q such that (2.2) } f p } C k p B p x, ε qq ď C k } R L p } k ` n ` , for some constant C k independent of x P V and p .Proof. Denote by Γ L p the connection form of ∇ L p with respect to the frame S p of L p andby p Γ L p q , the p , q -part of Γ L p . As Bp Γ L p q , “ , by [11, Chapter VIII, Theorem 6.1and (6.4)], there exists f p P C p B p x, ε qq satisfying(2.3) B f p “ ´p Γ L p q , , and(2.4) } f p } L ,x ď c } Γ L p } L ,x , where c is a constant independent of x P V and p . Using elliptic estimate, we have(2.5) } f p } k ` ,x ď c ,k ` }B f p } k,x ` } f p } L ,x ˘ , where } ¨ } k,x denotes the Sobolev norm on the Sobolev space H k p B p x, ε qq and c ,k is aconstant independent of x P V and p . Denote by ϕ p be the real part of f p . From (2.3)we know that σ p : “ e f p S p forms a holomorphic frame of L p on B p x, ε q with(2.6) | σ p | h p p Z q “ e ϕ p p Z q . The estimate (2.2) follows from (1.33), (2.3), (2.4), (2.5) and Sobolev embeddingtheorem. (cid:3)
Remark 2.2.
Note that on a Stein manifold M we have H p M, O ˚ q – H p M, Z q due toCartan’s theorem B (see e. g. [17, p. 201]), thus any holomorphic line bundle L overa Stein contractible manifold, for example a coordinate ball, is holomorphically trivial(this is due to Oka [22]). Lemma 2.1 gives a a proof with estimates of this result over acoordinate ball. ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 11
Consider the holomorphic family of holomorphic trivializations of L p associated withthe coordinate ψ x and the frame σ p . These trivializations are given by Ψ p,x : L p | ψ ´ p B p x, ε qq Ñ B p , ε q ˆ C with Ψ p,x p y, v p q : “ p ψ x p y q , v p { σ p p y qq for v p a vector in the fiber of L p over the point y .Consider a point x P V . Denote by ϕ p,x “ ϕ p ˝ ψ ´ x the function ϕ p in (2.6) in localcoordinate ψ x . Denote by ϕ r s p,x and ϕ r s p,x the first and second order Taylor expansion of ϕ p,x , i.e., ϕ r s p,x p Z q : “ n ÿ j “ ´ B ϕ p B z j p x q z j ` B ϕ p B z j p x q z j ¯ , (2.7) ϕ r s p,x p Z q : “ Re n ÿ j,k “ ´ B ϕ p B z j B z k p x q z j z k ` B ϕ p B z j B z k p x q z j z k ¯ , where we write z “ p z , . . . , z n q the complex coordinate of Z .Let ρ : R Ñ r , s be a smooth even function such that(2.8) ρ p t q “ | t | ă ρ p t q “ | t | ą . We denote in the sequel X “ R n » T x X and equip X with the metric g T X p Z q : “ g T X p ρ p| Z |{ ε q Z q . Now let ă ε ă ε to be determined and define(2.9) φ ε,p p Z q : “ ρ p| Z |{ ε q ϕ p,x p Z q ` ` ´ ρ p| Z |{ ε q ˘` ϕ p p x q ` ϕ r s p,x p Z q ` ϕ r s p,x p Z q ˘ . Let h L p, ε be the metric on L p, “ X ˆ C defined by(2.10) | | h Lp, ε p Z q : “ e ´ φ ε,p p Z q . Let ∇ L p, ε be the Chern connection on p L p, , h L p, ε q and R L p, ε be the curvature of ∇ L p, ε . By(0.5) and (2.2), there exists C ą independent of p such that for | Z | ď ε , ď j ď ,we have(2.11) ˇˇˇ f p,x p Z q ´ ` f p p x q ` f r s p,x p Z q ` f r s p,x p Z q ˘ˇˇˇ C j ď CA p | Z | ´ j . From (0.5) and (1.17), we may assume that a p { A p ě µ holds for all p P N ˚ , here µ isa constant depending only on ω . By (2.9)–(2.11), there exists ă ε ă ε small enoughsuch that the following estimate holds for every x P U :(2.12) inf ! ?´ R L p, ε,Z p u, J u q L | u | g TX : u P T Z X zt u and Z P X ) ě a p . In the sequel we fix ε ą small such that (2.12) holds. Let(2.13) D X p “ ? ´ B L p, ` pB L p, q ˚ ¯ be the Dirac-Dolbeault operator on X associated to the above data, where pB L p, q ˚ isthe adjoint of B L p, with respect to the metrics g T X and h L p, ε . Note that over the ball B p x , ε q , D p is just the restriction of D X p .Let ∇ T p , q X be the holomorphic Hermitian connection on p T p , q X , h T p , q X q withcurvature R T p , q X . It induces naturally a connection ∇ T p , q X on T p , q X . Set r ∇ T X “ ∇ T p , q X ‘ ∇ T p , q X . Then r ∇ T X is a connection on T X b R C .Let T be the torsion of the connection r ∇ T X and T ,as be the anti-symmetrization ofthe tensor V, W, Y
Ñ x T p V, W q , Y y . Let ∇ Cl be the Clifford connection on Λ p T ˚p , q X q ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 12 (cf. [20, (1.3.5)]). Define the operator c p¨q on Λ p T ˚ X b R C q by c p e i ^ . . . ^ e i j q “ c p e i q . . . c p e i j q for ď i ă . . . ă i j ď n . Set(2.14) ∇ A U “ ∇ Cl U ´ c p i U T ,as q . Then as explained in [20, (1.4.27)-(1.4.28)], ∇ A preserves the Z -grading on Λ p T ˚p , q X q .Let ∇ A b L p, be the connection on Λ p T ˚p , q X q b L p, induced by ∇ A and ∇ L p, as in(1.5). Denote by ∆ A b L p, the Bochner Laplacian on Λ p T ˚p , q X q b L p, associated to ∇ A b L p, . By [20, (1.2.51), (1.4.29)], we have p D X p q “ ∆ A b L p, ` r X ` c ´ R L p, `
12 Tr r R T p , q X s ¯ ´ c p dT ,as q ´ ˇˇ T ,as ˇˇ , (2.15)where the norm | A | for A P Λ p T ˚ X q is given by | A | “ ř i ă j ă k | A p e i , e j , e k q| . ByTheorem 1.3 we get from (2.12) the existence of C ą such that for any p P N ˚ ,(2.16) Spec ` p D X p q ˘ Ă t u Y r a p ´ C, . Note that from (2.13), p D X p q preserves the Z -grading on Ω , ‚ p X , L p q .Let S p,x be the unitary section of p L p, , h p, q which is parallel with respect to ∇ L p, ε along the curve r , s Q u Ñ uZ for any Z P X . The unitary frame S p,x provides anisometry L p, » C . Let P p be the orthogonal projection from C p X , L p, q » C p X , C q on Ker D X p , and let P p p x, x q be the smooth kernel of P p with respect to the volumeform dv X p x q . Proposition 2.3.
For any l, m P N , there exists C l,m ą such that for x, x P B T x X p , ε q , (2.17) ˇˇˇ P p p x, x q ´ P p p x, x q ˇˇˇ C m ď C l,m A ´ lp . Proof.
Using (1.23) and (2.16), we know that P p ´ F p D p q verifies also (1.24) for x, x P B T x X p , ε q , thus we get (2.17). (cid:3) Now under the natural identification
End p L p q » C (which does not depend on S p,x ),we will consider p D X p q acting on C p X , C q . Let dv T X be the Riemannian volume formof p T x X, g T x X q . Let κ p Z q be the smooth positive function defined by the equation(2.18) dv X p Z q “ κ p Z q dv T X p Z q , with κ p q “ . For s P C p R n , C q , Z P R n and t “ ? A p , set p δ t s qp Z q “ s p Z { t q , ∇ t, ‚ “ δ ´ t tκ { ∇ L p, κ ´ { δ t , L t “ δ ´ t t κ { p D X p q κ ´ { δ t . (2.19)2.2. Asymptotics of the scaled operators.
Let t w j u nj “ be an orthonormal basis of T p , q x X . Then(2.20) e j ´ “ ? p w j ` w j q and e j “ ?´ ? p w j ´ w j q , j “ , . . . , n, ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 13 form an orthonormal basis of T x X . Set ∇ , ‚ “ ∇ ‚ ` γ x p Z, ¨q , with γ “ ´ π ?´ ω, L “ ´ n ÿ j “ p ∇ ,e j q ´ γ x p w j , w j q . (2.21) Lemma 2.4.
The following holds as t Ñ : (2.22) ∇ t, ‚ “ ∇ , ‚ ` O p t min p , a q q , L t “ L ` O p t min p , a q q . Proof.
By (2.15), when we restrict on C p R n , C q , we get p D X p q “ ∆ A b L p, ´ R L p, p ˜ w j , ˜ w j q` r X ´ Tr “ R T p , q X ‰ p ˜ w j , ˜ w j q ´ c p dT ,as q ´ ˇˇ T ,as ˇˇ , (2.23)where ˜ w j denotes the parallel transport of w j along the curve r , s Q u Ñ uZ . Let Γ A bethe connection form of ∇ A with respect to the orthonormal frame of Λ p T ˚p , q X q whichis parallel along the above curve. Denote by Γ L p, the connection form of ∇ L p, withrespect to the frame S p,x . Set g ij p Z q “ g T X p e i , e j qp Z q “ x e i , e j y| Z and let p g ij p Z qq bethe inverse of the matrix p g ij p Z qq and ∇ T X e i e j “ Γ kij p Z q e k for the Levi-Civita connection ∇ T X on p T X , g T X q . Then ∇ t, ‚ “ κ p tZ q ` ∇ ‚ ` t Γ A tZ ` t Γ L p, tZ ˘ κ ´ p tZ q , L t “ g ij p tZ q ` ∇ t,e i ∇ t,e j ´ t Γ kij p tZ q ∇ t,e k ˘ ´ t R L p, tZ p ˜ w j , ˜ w j q` r X tZ ´ Tr “ R T p , q X tZ ‰ p ˜ w j , ˜ w j q ´ t c ` p dT ,as q tZ ˘ ´ t ˇˇ p T ,as q tZ ˇˇ . (2.24)If | tZ | ă ε , then ρ p| tZ |{ ε q “ and D X p “ D p , in particular, Γ A tZ “ on Λ p T ˚p , q X q “ C , T “ , T ,as “ and r X ´ Tr “ R T p , q X ‰ p ˜ w j , ˜ w j q “ . Thus for | tZ | ă ε , theoperators ∇ t, ‚ , L t are given by ∇ t, ‚ “ κ p tZ q ` ∇ ‚ ` t Γ L p tZ ˘ κ ´ p tZ q , L t “ g ij p tZ q ` ∇ t,e i ∇ t,e j ´ t Γ kij p tZ q ∇ t,e k ˘ ´ t R L p tZ p ˜ w j , ˜ w j q . (2.25)By [20, (1.2.31)],(2.26) Γ L p Z p e j q “ R L p x p Z, e j q ` O ` | Z | | Γ L p | C ˘ , which implies(2.27) t Γ L p tZ p e j q “ t R L p x p Z, e j q ` O ` t | Γ L p | C ˘ . By (2.21), (0.5) turns to(2.28) ˇˇˇ R L p A p ´ γ ˇˇˇ C k ď πC k A ap , By (2.28),(2.29) ˇˇˇ t R L p ´ γ ˇˇˇ C k ď πC k t a . Combining (1.33), (2.27) and (2.29) yields(2.30) t Γ L p tZ p e j q “ γ x p Z, e j q ` O p t min p , a q q . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 14
By (2.29),(2.31) t R L p tZ p ˜ w j , ˜ w j q “ γ x p w j , w j q ` O p t min p , a q q . Moreover,(2.32) κ p Z q “ det p g ij p Z qq { . By [20, (1.2.19)],(2.33) g ij p Z q “ δ ij ` O p| Z | q . Then (2.22) follows from (2.25) and (2.30)–(2.33). The proof of Lemma 2.4 is com-pleted. (cid:3)
To prove Theorem 0.1, we need a refinement of Lemma 2.4. Note that a “ a ` a with a “ ´ t ´ a u ´ P N and ă a ď . Theorem 2.5.
The following holds as t Ñ : If ă a ď , then (2.34) L t “ L ` a ` ÿ r “ t r O r ` O p t a q . If ă a ď , then (2.35) L t “ L ` a ÿ r “ t r O r ` O p t a q . Proof.
Consider the Taylor expansion(2.36) Γ L p Z p e j q “ k ÿ r “ ÿ | α |“ r ` B α Γ L p ˘ x p e j q Z α α ! ` O ` | Z | k ` | Γ L p | C k ` ˘ . From [20, (1.2.30)] and (2.36), we obtain(2.37) Γ L p Z p e j q “ k ÿ r “ r ` ÿ | α |“ r ´ ` B α R L p ˘ x p Z, e j q Z α α ! ` O ` | Z | k ` | Γ L p | C k ` ˘ . Combining (1.33), (2.28) and (2.37) yields(2.38) t Γ L p tZ p e j q “ k ÿ r “ t r ´ r ` ÿ | α |“ r ´ ` B α γ ˘ x p Z, e j q Z α α ! ` O p t min p a,k q q . If ă a ď , then we take k “ a ` in (2.38) and we obtain(2.39) t Γ L p tZ p e j q “ a ` ÿ r “ t r ´ r ` ÿ | α |“ r ´ ` B α γ ˘ x p Z, e j q Z α α ! ` O p t a q . If ă a ď , then we take k “ a ` in (2.38) and obtain(2.40) t Γ L p tZ p e j q “ a ` ÿ r “ t r ´ r ` ÿ | α |“ r ´ ` B α γ ˘ x p Z, e j q Z α α ! ` O p t a q . Then (2.34) and (2.35)) follow from (2.25), (2.31), (2.39) and (2.40). (cid:3)
ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 15
Bergman kernel.
Now we discuss the eigenvalues and eigenfunctions of L indetail. We choose t w j u nj “ an orthonormal basis of T p , q x X such that(2.41) γ x p w j , w j q “ a j , a j ą . Let t w j u nj “ be its dual basis. Then t e j u nj “ given by (2.20) forms an orthonomal basis of T x X . We use the coordinates on R n » T x X induced by e j as(2.42) R n Q p Z , . . . , Z n q ÞÝÑ n ÿ j “ Z j e j P T x X. In what follows we also introduce the complex coordinates z “ p z , . . . , z n q on C n » R n .Thus Z “ z ` z , and w j “ ? BB z j , w j “ ? BB z j . We will also identify z to ř j z j BB z j and z to ř j z j BB z j when we consider z and z as vector fields. Remark that(2.43) ˇˇˇ BB z j ˇˇˇ “ ˇˇˇ BB z j ˇˇˇ “ , so that | z | “ | z | “ | Z | . It is very useful to rewrite L by using the creation and annihilation operators. Set(2.44) b j “ ´ ∇ , BB zj , b ` j “ ∇ , BB zj , b “ p b , . . . , b n q . Then by (2.21) and (2.41), we have(2.45) b j “ ´ BB z j ` a j z j , b ` j “ BB z j ` a j z j . Then(2.46) L “ n ÿ j “ b j b ` j . Let P : ` L p R n q , } ¨ } L ˘ Ñ Ker p L q be the orthogonal projection. Denote by P p x, y q the Schwartz kernel of P . By [20, Theorem 4.1.20],(2.47) P p Z, Z q “ n ź j “ a j π exp ” ´ ÿ j a j ` | z j | ` | z j | ´ z j z j ˘ı In particular,(2.48) P p , q “ n ź j “ a j π “ ω n ϑ n ¨ Proof of Theorem 0.1.
Denote by x¨ , ¨y and }¨} the inner product and the L -normon C p X , C q induced by g T X . For s P C p X , C q , set } s } t, : “ } s } “ ż R n | s p Z q| dv T X p Z q , } s } t,m : “ m ÿ l “ n ÿ j ,...,j l “ } ∇ t,e j . . . ∇ t,e jl s } t, . (2.49)By (2.29), we have the following analogue of [20, Theorem 4.1.9]. Theorem 2.6.
There exist C , C , C ą such that for t P p , q and any s, s P C p R n , C q , @ L t s, s D t ě C ›› s ›› t, ´ C ›› s ›› t, , ˇˇˇ@ L t s, s D t, ˇˇˇ ď C ›› s ›› t, ›› s ›› t, . (2.50) ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 16
Proof.
The analogue of [20, (4.1.39)] holds: @ L t s, s D t, “ ›› ∇ t s ›› t, ´ t A δ ´ t ` R L p, p ˜ w j , ˜ w j q ´ r X ` Tr “ R T p , q X ‰ p ˜ w j , ˜ w j q` c p dT ,as q ` | T ,as | ˘ s, s E . (2.51)By (2.29) and (2.51), we obtain the first inequality of (2.50). From (2.24), we get thesecond inequality of (2.50). (cid:3) Proof of Theorem 0.1.
From Theorem 2.6, we can obtain the analogue of [20, Theorems4.1.10 – 4.1.12] exactly the same way as [20, Theorems 4.1.10 – 4.1.12] follow from[20, Theorems 4.1.9]. Recall that a p { A p ě µ ą for all p P N ˚ large enough. By (2.16)and (2.19), there exists t P p , q such that for t P p , t q ,(2.52) Spec p L t q Ă t u Y ” µ , `8 ¯ . Let δ be the counterclockwise oriented circle in C of center and radius µ { . Then p λ ´ L t q ´ exists for λ P δ .For m P N , let Q m be the set of operators t ∇ t,e i . . . ∇ t,e ij u j ď m . For k, r P N , set(2.53) I k,r : “ ! p k , r q “ p k i , r i q ( ji “ , j ÿ i “ k i “ k ` j, j ÿ i “ r i “ r, k i , r i P N ˚ ) . Then there exist a kr P R such that A kr p λ, t q “ p λ ´ L t q ´ k B r L t B t r p λ ´ L t q ´ k . . . B r j L t B t r j p λ ´ L t q ´ k j , (2.54) B r B t r p λ ´ L t q ´ k “ ÿ p k , r qP I k,r a kr A kr p λ, t q . The analogue of [20, Theorems 4.1.13 – 4.1.14] is as follows. For any m P N , k ą p m ` r ` q , p k , r q P I k,r , there exist C ą , N P N such that for any λ P δ , t P p , t s , Q, Q P Q m ,(2.55) ›› QA kr p λ, t q Q s ›› t, ď C ` ` | λ | ˘ N ÿ | β |ď r } Z β s } t, . Moreover, if ă a ď , then for r P t , , . . . , a ` u and any k ą , there exist C ą , N P N such that for t P r , t s , λ P δ , we have ››› B r L t B t r ´ B r L t B t r ˇˇˇ t “ ››› t, ´ ď Ct ÿ | α |ď r ` ›› Z α s ›› , . ››› B r B t r p λ ´ L t q ´ k ´ ÿ p k , r qP I k,r a kr A kr p λ, q ››› , (2.56) ď Ct p ` | λ | q N ÿ | α |ď r ` } Z α s } , . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 17 If ă a ď , then for r P t , , . . . , a ´ u we have similar estimates to (2.56) and for r “ a and k ą , there exist C ą , N P N , such that for t P r , t s , λ P δ , we have ››› B r L t B t r ´ B r L t B t r ˇˇˇ t “ ››› t, ´ ď Ct a ÿ | α |ď r ` ›› Z α s ›› , , ››› B r B t r p λ ´ L t q ´ k ´ ÿ p k , r qP I k,r a kr A kr p λ, q ››› , (2.57) ď Ct a p ` | λ | q N ÿ | α |ď r ` } Z α s } , . By [20, Theorems 4.1.16 – 4.1.18 and 4.1.21], we obtain the following analogue of[20, Theorem 4.1.24]: for any m P N , q ą , there exists C ą such that for Z, Z P T x X , | Z | , | Z | ď q { a A p , ˇˇˇ A np P p p Z, Z q ´ a ` ÿ r “ F r p a A p Z, a A p Z q κ ´ p Z q κ ´ p Z q A ´ r p ˇˇˇ C m p X q (2.58) ď CA ´ a ´ p , for 12 ă a ď ˇˇˇ A np P p p Z, Z q ´ a ÿ r “ F r p a A p Z, a A p Z q κ ´ p Z q κ ´ p Z q A ´ r p ˇˇˇ C m p X q ď CA ´ a ´ p , for 0 ă a ď
12 ; where F r p Z, Z q “ J r p Z, Z q P p Z, Z q and J r p Z, Z q are polynomials in Z , Z with thesame parity as r and J p Z, Z q “ . Set now Z “ Z “ in (2.58). Then (0.4) followsfrom (2.48) and (2.58). The proof of Theorem 0.1 is completed. (cid:3)
3. E
QUIDISTRIBUTION OF ZEROS OF RANDOM SECTIONS
In this section we prove Theorem 0.4. Assume throughout this section the setting ofTheorem 0.4 and let m P t , . . . , n u . We will denote by ω FS the Fubini-Study form on aprojective space P d , normalized so that ω d FS is a probability measure.Let us start by introducing notation and recalling some facts needed for the proof. If t S pj u d p j “ is an orthonormal basis of H p X, L p q then the Bergman kernel function P p of H p X, L p q is given by(3.1) P p p x q “ d p ÿ j “ | S pj p x q| h p , x P X. Let U be a contractible Stein open set in X and write S pj “ f pj e p , where e p is a localholomorphic frame of L p and f pj is a holomorphic function on U . The Fubini-Studycurrent γ p of H p X, L p q is defined by(3.2) γ p | U “ dd c log d p ÿ j “ | f pj | , where d “ B ` B , d c “ πi pB ´ Bq . These are positive closed currents of bidegree p , q ,smooth away from the base locus Bs H p X, L p q of H p X, L p q . We have(3.3) γ p “ c p L p , h p q ` dd c log P p . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 18
Let Φ p : X P d p ´ be the Kodaira map defined by the basis t S pj u d p j “ , so(3.4) Φ p p x q “ r f p p x q : . . . : f pd p p x qs for x P U. Then γ p “ Φ ˚ p p ω FS q .If s P H p X, L p q we denote by r s “ s the current of integration (with multiplicities)along the analytic hypersurface t s “ u . One has the Lelong-Poincaré formula (see [20,Theorem 2.3.3])(3.5) r s “ s “ c p L p , h p q ` dd c log | s | h p . Recall that X p,m “ ` P H p X, L p q ˘ m , d p “ dim H p X, L p q . Set(3.6) d p,m : “ dim X p,m “ m p d p ´ q . Let π k : X p,m Ñ P H p X, L p q be the canonical projection onto the k -th factor. We endow X p,m with the Kähler form ω p,m : “ c p,m ` π ˚ ω FS ` . . . ` π ˚ m ω FS ˘ , where the constant c p,m is chosen so that ω d p,m p,m “ σ p,m is a probability measure on X p,m .It follows that(3.7) c p,m “ ˜ ` p d p ´ q ! ˘ m d p,m ! ¸ { d p,m . Lemma 3.1.
In the hypotheses of Theorem 0.4, the following hold for p ą p :(i) γ p are smooth p , q forms on X .(ii) For σ p,m -a.e. s p “ p s p , . . . , s pm q P X p,m we have that the analytic set t s pi “ u X . . . X t s pi k “ u has pure dimension n ´ k for each ď k ď m and ď i ă . . . ă i k ď m .In particular the current r s p “ s : “ r s p “ s ^ . . . ^ r s pm “ s is well defined and isequal to the current of integration with multiplicities over the common zero set t s p “ u : “t s p “ u X . . . X t s pm “ u .Proof. By (0.11) we have P p p x q ą for all x P X and p ą p , hence Bs H p X, L p q “ H and p i q follows from (3.3). Since Bs H p X, L p q “ H for p ą p , [6, Proposition 4.1]implies that, for σ p,m -a.e. s p “ p s p , . . . , s pm q P X p,m , the analytic hypersurfaces t s p “ u , . . . , t s pm “ u are in general position, i.e. t s pi “ u X . . . X t s pi k “ u has dimensionat most n ´ k for each ď k ď m and ď i ă . . . ă i k ď m . Hence R : “ r s pi “ s ^ . . . ^ r s pi k “ s (3.8)is a well defined positive closed current of bidegree p k, k q by [10, Corollary 2.11], sup-ported in the set t s pi “ u X . . . X t s pi k “ u . Moreover, by the Lelong-Poincaré formula(3.5), ż X R ^ ϑ n ´ k “ ż X c p L p , h p q k ^ ϑ n ´ k ą . So t s pi “ uX . . . Xt s pi k “ u ‰ H , hence it has pure dimension n ´ k . The last assertionof p ii q now follows from [10, Corollary 2.11, Proposition 2.12]. (cid:3) The proof of Theorem 0.4 uses results of Dinh and Sibony [15, Section 3.1] onmeromorphic transforms. As in [15, Example 3.6 (c)], [7, Section 4.2], [13] weconsider the meromorphic transform Φ p, from X to P H p X, L p q defined by its graph Γ p, “ p x, s q P X ˆ P H p X, L p q : s p x q “ ( . This is related to the Kodaira map Φ p from ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 19 (3.4). Its m -fold product Φ p,m (see [15, Section 3.3]) is the meromorphic transformfrom X to X p,m with graph Γ p,m “ p x, s p , . . . , s pm q P X ˆ X p,m : s p p x q “ . . . “ s pm p x q “ ( . Using Lemma 3.1 p ii q and arguing as in [7, Section 4.2], it follows that Φ p,m is a mero-morphic transform of codimension n ´ m , with fibers Φ ´ p,m p s p q “ t x P X : s p p x q “ . . . “ s pm p x q “ u , where s p “ p s p , . . . , s pm q P X p,m . Moreover, for s p P X p,m generic, the current Φ ˚ p,m p δ s p q “ r s p “ s “ r s p “ s ^ . . . ^ r s pm “ s “ Φ ˚ p, p δ s p q ^ . . . ^ Φ ˚ p, p δ s pm q is a well defined positive closed current of bidegree p m, m q on X . Here δ x denotes theDirac mass at a point x , and F ˚ p T q denotes the pull-back of a current T by a meromor-phic transform F as defined in [15, Section 3.1]. Following the proof of [6, Theorem1.2] (see also [7, Lemma 4.5]), we can show that Φ ˚ p,m p σ p,m q “ γ mp , for all p ą p .We consider the intermediate degrees of Φ p,m of order d p,m , resp. d p,m ´ [15, Section3.1]:(3.9) δ p,m : “ ż X Φ ˚ p,m p ω d p,m p,m q ^ ω n ´ m , δ p,m : “ ż X Φ ˚ p,m p ω d p,m ´ p,m q ^ ϑ n ´ m ` . As in the proof of [7, Lemma 4.4] we obtain that(3.10) δ p,m “ ż X c p L p , h p q m ^ ϑ n ´ m , δ p,m “ c p,m ż X c p L p , h p q m ´ ^ ϑ n ´ m ` . We will need the following estimates:
Lemma 3.2. (i) For every p ě and m P t , . . . , n u , we have em ă c p,m ă em .(ii) There exist constants M ą and p ą p such that, for every p ą p , we have M ´ A np ď d p ď M A np , (3.11) M ´ A mp ď δ p,m ď M A mp , M ´ A p ď δ p,m δ p,m ď M A p , @ m P t , . . . , n u . (3.12) Proof. p i q We have that [23, p. 200] e ă k ! ` ke ˘ k ? k ď e , for every k ě .Since k k ă this implies that ke ă ` k ! ˘ k ă k . Hence by (3.7) and (3.6), em ă c p,m “ ` p d p ´ q ! ˘ {p d p ´ q ` d p,m ! ˘ { d p,m ă em . p ii q We infer from (0.10) that there exists p P N such that(3.13) ω ď A p c p L p , h p q ď ω , for all p ą p .By (3.10) we obtain ´ m A mp ż X ω m ^ ϑ n ´ m ď δ p,m ď m A mp ż X ω m ^ ϑ n ´ m , ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 20 which readily implies the first estimate from (3.12). Using this and part p i q , we obtainthe estimate on δ p,m { δ p,m from (3.12), by increasing the constant M . Finally, using(0.11), we get A np M ż X ϑ n n ! ď d p “ ż X P p ϑ n n ! ď M A np ż X ϑ n n ! , for all p ą p . (cid:3) Our next result deals with the part of the proof of Theorem 0.4 which uses the Dinh-Sibony meromorphic transform technique and equidistribution theorem [15, Theorem4.1, Lemma 4.2 (d)]. For p ą p , m P t , . . . , n u and ε ą , let(3.14) E p,m p ε q : “ ď } φ } C ď s p P X p,m : ˇˇ@ r s p “ s ´ γ mp , φ Dˇˇ ě A mp ε ( , where φ is a p n ´ m, n ´ m q form of class C on X . We also assume that the set of s p P X p,m for which the current r s p “ s is not well defined is contained in E p,m p ε q . Notethat, by Lemma 3.1, the latter is a set of measure since p ą p . Proposition 3.3.
In the hypotheses of Theorem 0.4, there exist constants ν, α, ζ ą and p ą p , such that for every p ą p , m P t , . . . , n u and ε ą we have σ p,m p E p,m p ε qq ď ν A ζp e ´ αA p ε . Proof.
Fix m P t , . . . , n u . We apply [15, Lemma 4.2 (d)] to the sequence of meromor-phic transforms Φ p,m : p X, ϑ q p X p,m , ω p,m q of codimension n ´ m and the probabilitymeasures σ p,m “ ω d p,m p,m on X p,m . Let E p,m p ε q : “ ď } φ } C ď s p P X p,m : ˇˇ@ r s p “ s ´ γ mp , φ Dˇˇ ě δ p,m ε ( , where p ą p and δ p,m is the degree of Φ p,m defined in (3.9). By [15, Lemma 4.2 (d)] itfollows that σ p,m p E p,m p ε qq ď ∆ p p η ε,p q , where η ε,p : “ ε δ p,m δ p,m ´ R p .Here R p : “ R p X p,m , ω p,m , σ p,m q , ∆ p p t q : “ ∆ p X p,m , ω p,m , σ p,m , t q , where t ą ,are quantities defined in [15, Sections 2.1, 2.2] and are related to certain compactclasses of quasiplurisubharmonic functions on X p,m (see also [7, Section 4.1]). By theappendix of [15] (see also [7, Lemma 4.6]) we infer that R p ď ν m ` ` log d p,m ˘ , ∆ p p t q ď ν ` d p,m ˘ ζ e ´ α t , t ą , where ν , ζ , α ą are constants depending only on m .Let M , p be as in Lemma 3.2. Then by (3.12) we have for p ą p , η ε,p ě εA p M ´ R p ě εA p M ´ ν m ` ` log d p,m ˘ . Hence σ p,m p E p,m p ε qq ď ∆ p p η ε,p q ď ν ` d p,m ˘ ζ e ´ α A p ε , where ν , ζ ą are constants depending only on m and α “ α { M . Using again(3.12) we have δ p,m ď M A mp , so E p,m p ε q Ă E p,m p ε { M q . Therefore σ p,m p E p,m p ε qq ď σ p,m p E p,m p ε { M qq ď ν ` d p,m ˘ ζ e ´ α A p ε { M . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 21
Since by (3.11), d p,m ă md p ď mM A np for p ą p , the conclusion follows. (cid:3) Proposition 3.4.
In the hypotheses of Theorem 0.4, there exist C ą and p P N suchthat for every β ą , m P t , . . . , n u and p ą p there exists a subset E βp,m Ă X p,m with thefollowing properties:(i) σ p,m p E βp,m q ď CA ´ βp ;(ii) if s p P X p,m z E βp,m then, for any p n ´ m, n ´ m q form φ of class C on X , ˇˇˇ A mp A r s p “ s ´ γ mp , φ Eˇˇˇ ď C p β ` q log A p A p } φ } C . Moreover, if ř p “ A ´ βp ă `8 then the last estimate holds for σ ,m -a.e. sequence t s p u p ě P X ,m provided that p is large enough.Proof. For every β ą , m P t , . . . , n u and p ą p , let ε p “ p β ` ζ q log A p αA p , E βp,m : “ E p,m p ε p q , where p , α, ζ are as in Proposition 3.3 and the set E p,m p ε q is defined in (3.14). ByProposition 3.3, we have that σ p,m p E βp,m q ď ν A ζp e ´ αA p ε p “ νA ´ βp . If s p “ p s p , . . . , s pm q P X p,m z E βp,m then, by the definition of E βp,m , the current r s p “ s “r s p “ s ^ . . . ^ r s pm “ s is well defined and ˇˇˇ A mp A r s p “ s ´ γ mp , φ Eˇˇˇ ď ε p } φ } C , for any p n ´ m, n ´ m q form φ of class C . So assertions p i q and p ii q hold with the constant C : “ max ν, α , ζα ( . The last assertion follows from these using the Borel-Cantelli lemma(see e.g. the proof of [7, Theorem 4.2]). (cid:3) Proposition 3.5.
In the hypotheses of Theorem 0.4, there exist C ą and p P N suchthat for every m P t , . . . , n u , p ą p and every p n ´ m, n ´ m q form φ of class C on X ,we have ˇˇˇA γ mp A mp ´ ω m , φ Eˇˇˇ ď C ˆ log A p A p ` A ´ ap ˙ } φ } C . Proof.
There exists c ą such that for every real p n ´ m, n ´ m q form φ of class C , m P t , . . . , n u , and every real p , q form θ on X one has(3.15) ´ c } φ } C ϑ n ´ m ` ď dd c φ ď c } φ } C ϑ n ´ m ` , (3.16) ´ c } φ } C } θ } C ϑ n ´ m ` ď φ ^ θ ď c } φ } C } θ } C ϑ n ´ m ` . For p ą p let R p : “ γ mp A mp ´ ω m , ρ p : “ m ´ ÿ j “ γ jp A jp ^ ω m ´ ´ j , α p : “ c p L p , h p q A p ´ ω . By (0.10), respectively by (3.3), we have that } α p } C ď C A ap , γ p A p ´ ω “ α p ` A p dd c log P p . ERGMAN KERNELS AND EQUIDISTRIBUTION FOR SEQUENCES OF LINE BUNDLES 22
Hence if φ is a real p n ´ m, n ´ m q form of class C we obtain that(3.17) x R p , φ y “ A´ γ p A p ´ ω ¯ ^ ρ p , φ E “ ż X ρ p ^ α p ^ φ ` ż X log P p A p ρ p ^ dd c φ . Using (3.16) we infer that ´ c C A ap } φ } C ϑ n ´ m ` ď α p ^ φ ď cC A ap } φ } C ϑ n ´ m ` , hence(3.18) ˇˇˇ ż X ρ p ^ α p ^ φ ˇˇˇ ď cC A ap } φ } C ż X ρ p ^ ϑ n ´ m ` . By (3.15), the total variation of the signed measure ρ p ^ dd c φ verifies | ρ p ^ dd c φ | ď c } φ } C ρ p ^ ϑ n ´ m ` . Therefore ˇˇˇ ż X log P p A p ρ p ^ dd c φ ˇˇˇ ď c } φ } C ż X | log P p | A p ρ p ^ ϑ n ´ m ` . We choose p ą p such that (3.13) holds for p ą p and A p ą M for p ą p . By(0.11) it follows that A n ´ p ď P p ď A n ` p , so | log P p | ď p n ` q log A p , hold on X for p ą p . We infer that(3.19) ˇˇˇ ż X log P p A p ρ p ^ dd c φ ˇˇˇ ď nc } φ } C log A p A p ż X ρ p ^ ϑ n ´ m ` for p ą p .Using (3.13) and (3.3) we have, for p ą p and ď j ď m ´ , that ż X γ jp A jp ^ ω m ´ ´ j ^ ϑ n ´ m ` “ ż X c p L p , h p q j A jp ^ ω m ´ ´ j ^ ϑ n ´ m ` ď j ż X ω m ´ ^ ϑ n ´ m ` . Hence(3.20) ż X ρ p ^ ϑ n ´ m ` “ m ´ ÿ j “ ż X γ jp A jp ^ ω m ´ ´ j ^ ϑ n ´ m ` ă m ż X ω m ´ ^ ϑ n ´ m ` . By (3.17), (3.18), (3.19) and (3.20) we conclude that if p ą p then |x R p , φ y| ď m ´ c C A ap } φ } C ` nc log A p A p } φ } C ¯ ż X ω m ´ ^ ϑ n ´ m ` , for every m P t , . . . , n u and every real p n ´ m, n ´ m q form φ of class C . This impliesthe proposition. (cid:3) Proof of Theorem 0.4.
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Email address : [email protected] S CHOOL OF M ATHEMATICS AND S TATISTICS , H
UAZHONG U NIVERSITY OF S CIENCE AND T ECHNOLOGY ,W UHAN
HINA
Email address : [email protected] I NSTITUT DE M ATHÉMATIQUES DE J USSIEU -P ARIS R IVE G AUCHE , U
NIVERSITÉ DE P ARIS , CNRS, F-75013P
ARIS , F
RANCE
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ATHEMATISCHES I NSTITUT , W
EYERTAL
ÖLN , G
ERMANY I NSTITUTE OF M ATHEMATICS ‘S IMION S TOILOW ’, R
OMANIAN A CADEMY , B
UCHAREST , R
OMANIA
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