Bergman kernels on punctured Riemann surfaces

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1 Bergman kernels on unctured Riemann surfaces Hugues AUVRAY and Xiaonan MA and George MARINESCU Aril 1, 016 Abstract In this aer we consider a unctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the unctures and a holomorhic line bundle which olarizes the metric. We show that the Bergman kernels can be localized around the singularities and its local model is the Bergman kernel of the unctured unit disc endowed with the standard Poincaré metric. As a consequence, we obtain an otimal uniform estimate of the suremum norm of the Bergman kernel, involving a fractional growth order of the tensor ower. Contents 1 Introduction 1 Preliminaries 6.1 Exansion of Bergman kernels on comlete manifolds Functional saces, section saces Bergman kernels on the unctured unit disc Exression of the Bergman kernels on the unctured unit disc Asymtotics of the density functions near the uncture Ellitic Estimates for Kodaira Lalacians on D and Σ Estimate on the unctured disc D : degree Estimate on the unctured Riemann surface Σ : degree Bidegree (0, 1) Sectral Ga and Localization 5 6 Proofs of the main results 9 A Proof of Lemma Introduction In this aer we study the Bergman kernels of a singular Hermitian line bundle over a Riemann surface under the assumtion that the curvature has singularities of Poincaré tye at a finite set. Our first result shows that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the unctured disc endowed with the standard Poincaré metric. The roof follows the rincile that the sectral ga 1

2 of the Kodaira Lalacian imlies the localization of the Bergman metric [MM1]. By a detailed analysis of the local model we deduce a shar uniform estimate of the suremum norm of the Bergman kernels. Let us describe our setting. Let Σ be a comact Riemann surface and let D = {a 1,..., a N } Σ be a finite set. We consider the unctured Riemann surface Σ = Σ\D and a Hermitian form ω Σ on Σ. Let L be a holomorhic line bundle on Σ, and let h be a singular Hermitian metric on L such that: (α) h is smooth over Σ, and for all j = 1,..., N, there is a trivialization of L in the comlex neighborhood V j of a j in Σ, with associated coordinate z j such that 1 h (z j) = log( z j ). (β) There exists ε > 0 such that the (smooth) curvature R L of h satisfies ir L εω Σ over Σ and ir L = ω Σ on V j := V j \{a j }; in articular, ω Σ = ω D in the local coordinate z j on V j and (Σ, ω Σ ) is comlete. Here ω D denotes the Poincaré metric on the unctured unit disc, normalized as follows: (1.1) ω D := idz dz z log ( z ) For 1, let h := h be the metric induced by h on L Σ, where L := L. We denote by H() 0 (Σ, L ) the sace of L -holomorhic sections of L relative to the metrics h and ω Σ, { } (1.) H() 0 (Σ, L ) = S H 0 (Σ, L ) : S := S L h ω Σ <, Σ endowed with the obvious inner roduct. The sections from H 0 () (Σ, L ) extend to holomorhic sections of L over Σ, i. e., (see [MM1, (6..17)]) (1.3) H 0 () (Σ, L ) H 0( Σ, L ). In articular, the dimension d of H 0 () (Σ, L ) is finite. We denote by B C (Σ, R) the Bergman kernel function of the sace H 0 () (Σ, L ), defined as follows: if {S l } l 1 is an orthonormal basis of H 0 () (Σ, L ), then (1.4) B (x) = S l (x) h. d Note that B is indeendent of the choice of basis (see [MM1, (6.1.10)] or [CM, Lemma 3.1]). Let B D be the Bergman kernel function of ( D, ω D, C, log( z ) ). The main result of this aer is a weighted estimate in the C m -norm near the unctures for the global Bergman kernel B comared to the Bergman kernel B D of the unctured disc, uniformly in the tensor owers of the given bundle. Theorem 1.1 Assume that (Σ, ω Σ, L, h) fulfill conditions (α) and (β). Then the following estimate holds: for every integer l, m 0, and every δ > 0, there exists a constant C = C(l, m, δ) such that for all N, and z V 1... V N with the local coordinate z j, in the sense of (.13), (1.5) B B D (z j) C l log( z C m j ) δ.

3 Remark 1. Theorem 1.1 admits a generalization to orbifold Riemann surfaces. Assume that Σ is a comact orbifold Riemann surface, and the finite set D = {a 1,..., a N } Σ does not meet the (orbifold) singular set of Σ. Assume moreover that L is a holomorhic orbifold line bundle on Σ. Let ω Σ be an orbifold Hermitian form on Σ and h an orbifold Hermitian metric on L in the sense of [MM1, 5.4]. The roof of Theorem 1.1 can be modified to show: If conditions (α), (β) hold in this context, then (1.5) holds. In fact, the ellitic estimate [DLM1, (4.14)] and the finite roagation seed of wave oerators hold on orbifolds as observed by [M, 6], so the arguments used in this aer go through for orbifolds to get the conclusion. By [MM1, Theorems 6.1.1, 6..3], for any comact set K Σ we have the following exansion on K in any C m -toology (see Theorem.1), (1.6) 1 B (x) = 1 π + b j (x) j as. j=1 Theorem 1.1 gives a recise descrition of B near the unctures, in terms of the Bergman kernel function of the Poincaré metric on the local model of the unctured unit disc in C. Note that in the case of smooth metrics with ositive curvature the Bergman kernel can be localised and its local model is the Euclidean sace endowed with a trivial bundle of ositive curvature, see [MM1, Sections ]. This kind of localization is insired from the analytic localization technique of Bismut-Lebeau [BL] in local index theory. For the roblem at hand here we have to overcome difficulties linked to the resence of singularities. From a study of the model Bergman kernel functions B D on the unctured unit disc, we get the following ratio estimate as a corollary of Theorem 1.1 and Corollary 3.6: Corollary 1.3 Let (Σ, ω Σ, L, h) be as in Theorem 1.1. Then σ(x) ( h ) 3/ (1.7) su B (x) = su x Σ x Σ,σ H() 0 (Σ,L ) σ = + O() as. π L It is, to our knowledge, the first examle of a uniform L asymtotic descrition of the Bergman kernel function of a singular olarization. This is of articular interest in arithmetic situations. Note that the work of Burgos et al. [BrBK, BuKK] develoed the arithmetic intersection theory for log-singular Hermitian metrics, showing in articular that Arakelov heights can be defined, and alied successfully the theory for the Hilbert modular surfaces. Our results rovide some ossible alications in this direction. For examle, the classical arithmetic Hilbert-Samuel theorem [GiS] for ositive Hermitian line bundles is usually used to roduce global integral sections with small su-norm; a combination of the recent work [BF] with the distortion estimate of our Corollary 1.3 should give some interesting arithmetic consequence for cus forms on arithmetic surfaces and Hilbert modular surfaces. Corollary 1.3 is also quite striking from a Kähler geometry oint of view, as the suremum of the Bergman kernel is equivalent to ( ) n π on comact olarized manifolds of comlex dimension n (cf. Corollary.3). Note also that the behavior of the Bergman kernel on singular Riemann surfaces is relevant for the theory of quantum Hall effect [LCCW] and attracted attention recently. We give an imortant examle where Theorem 1.1 alies. Let Σ be a comact Riemann surface of genus g and consider a finite set D = {a 1,..., a N } Σ. We also denote by D the divisor N j=1 a j and let O Σ (D) be the associated line bundle. The following 3

4 conditions are equivalent: (i) Σ = Σ D admits a comlete Kähler-Einstein metric ω Σ with Ric ωσ = ω Σ, (ii) g + N > 0, (iii) the universal cover of Σ is the uer-half lane H, (iv) L = K Σ O Σ (D) is amle. This follows from the Uniformization Theorem [FK, Chater IV] and the fact that the Euler characteristic of Σ equals χ(σ) = g N and the degree of L is g +N = χ(σ). If one of these equivalent conditions is satisfied, the Kähler-Einstein metric ω Σ is induced by the Poincaré metric on H; (Σ, ω Σ ) and the formal square root of (L, h) satisfy conditions (α) and (β), see Lemma 6.. Theorem 1.1 hence alies to this context. Let Γ be the Fuchsian grou associated with the above Riemann surface Σ, that is, Σ = Γ\H. Then Γ is a geometrically finite Fuchsian grou of the first kind, without ellitic elements. Conversely, if Γ is such a grou, then Σ := Γ\H can by comactified by finitely many oints D = {a 1,..., a N } into a comact Riemann surface Σ such that the equivalent conditions (i)-(iv) above are fulfilled. Let S Γ be the sace of cus forms (Sitzenformen) of weight of Γ endowed with the Petersson inner roduct. We can form the Bergman kernel function of S Γ as in (1.4), denoted by B Γ. We deduce from Corollary 1.3: Corollary 1.4 Let Γ PSL(, R) be a geometrically finite Fuchsian grou of the first kind without ellitic elements. Let B Γ be the Bergman kernel function of cus forms of weight. If Γ is cocomact then uniformly on Γ\H, (1.8) B Γ (x) = π + O(1), as. If Γ is not cocomact then ( ) 3/ (1.9) su B Γ (x) = + O(), as. x Γ\H π Uniform estimates for su x Γ\H B Γ (x) are relevant in arithmetic geometry and were roved in various degrees of generality and sharness in [AU, MU, JK, FJK]. In [FJK] it is roved that in the cofinite but non-cocomact case su x Γ\H B Γ (x) = O( 3/ ) and the result is otimal, at least u to an additive term in the exonent of the form ε for any ε > 0. Estimate (1.9) gives the recise coefficient of the leading term 3/ and is shar (by killing the ε from below from [FJK]). Estimate (1.8) is the consequence of the general exansion of the Bergman kernel on comact manifolds [T, Bou, Ca, Z] (cf. also [DLM1, MM1] and Theorem.1). It turns out that Corollary 1.4 can be formulated so as to underline a certain uniformity in Γ, in the same fashion as in [FJK]: Theorem 1.5 Let Γ 0 PSL(, R) be a fixed Fuchsian subgrou of the first kind without ellitic elements and let Γ Γ 0 be any subgrou of finite index. If Γ 0 is cocomact, then (1.10) B Γ (x) = π + O Γ 0 (1), as. If Γ 0 is not cocomact then ( ) 3/ (1.11) su B Γ (x) = + OΓ0 (), as. x Γ\H π Here the imlied constants in O Γ0 (1), O Γ0 () deend solely on Γ 0. 4

5 Note that (1.10) is a secial case of a more general result which is imlied in [MM1, 6.1.] and which we state as Theorem.5 in Section. We consider further extension of Theorem 1.5 to the case when the grou Γ 0 has ellitic elements. Then the quotients Γ\H are in general orbifolds. By using the result of Dai- Liu-Ma [DLM1, (5.5)] on the Bergman kernel asymtotics on orbifolds and the orbifold version of Theorem 1.1 we obtain the following. Theorem 1.6 Let Γ 0 PSL(, R) be a fixed Fuchsian subgrou of the first kind. Let {x j } q j=1 be the orbifold oints of Γ 0\H and U xj be a small neighborhood of x j in Γ 0 \H. Let Γ Γ 0 be any subgrou of finite index and π Γ : Γ\H Γ 0 \H be the natural rojection. If Γ 0 is cocomact, then as (1.1) B Γ (x) = π + O Γ 0 (1), uniformly on (Γ\H) q j=1 π 1 Γ (U x j ). On each π 1 Γ (U x j ) we have as, ( (1.13) B Γ (x) = 1 + ( ex iθ γ (1 e iθγ ) z )) π + O Γ 0 (1), γ Γ x Γ j {1} where x Γ j π 1 Γ (x j) is in the same comonent of π 1 Γ (U x j ) as x, e iθγ is the action of γ on the fiber of K Γ\H at x Γ j, and z = z(x) is the coordinate of x in normal coordinates z centered at x Γ j in H, and Γ y = {γ Γ : γy = y} the stabilizer of y. In articular, if q 0 = lcm{ Γ 0,xj : j = 1,..., q}, n Γ = max{ Γ y : y π 1 Γ (x j), j = 1,..., q}, then (1.14) su B Γ q 0 q 0 (x) = n Γ x Γ\H π + O Γ 0 (1). If Γ 0 is not cocomact then as ( ) 3/ (1.15) su B Γ (x) = + OΓ0 (). x Γ\H π Here again the imlied constants in O Γ0 (1), O Γ0 () deend solely on Γ 0. Theorems 1.5, 1.6 sharen (in an otimal way) the main result of [FJK] which states that { su B Γ O Γ0 () if Γ 0 is cocomact, (1.16) (x) = x Γ\H O Γ0 ( 3/ ) if Γ 0 is not cocomact. We obtain in this way the the recise leading terms in (1.16). This aer is organized as follows. In Section we recall the Bergman kernel exansion of comlete Kähler manifolds and introduce the functional sace we need further. In Section 3, we study our model situation: the Bergman kernel on the unctured unit disc with Poincaré metric. In Section 4, we establish the basic weighted ellitic estimate on the unctured unit disc with Poincaré metric uniformly with resect to the -th ower of the trivial line bundle with Poincaré metric. In Section 5, we develo the sectral ga roerties of the Kodaira Lalacian and give a rough uniform estimate of an aroximation of the Bergman kernel. In Section 6, by combining the finite roagation seed of the wave 5

6 oerator and Section 5, we establish finally the main results stated in the Introduction. In the Aendix A, we rove a technical result, Lemma 3.4. Acknowledgements. H. A. is artially suorted by ANR-14-CE5-0010, and is thankful to the University of Cologne where this aer was artly written; he would also like to thank Michael Singer for insiring conversations. X. M. is artially suorted by ANR-14- CE and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative. G. M. acknowledges suort from Université Paris Diderot Paris 7 where this aer was artly written and warmly thanks the roject Analyse Comlexe et Géométrie for hositality over many years. Preliminaries In Section.1 we recall by following [MM1] the asymtotics of the Bergman kernels on comlete manifolds and rove some results of indeendent interest about this exansion on Riemann surfaces with locally constant curvature and also about its behavior with resect to coverings. In Section. we introduce some functional and section saces that will be used throughout the aer..1 Exansion of Bergman kernels on comlete manifolds For a Hermitian holomorhic line bundle (L, h) on a comlex manifold we denote by R L its Chern curvature and by c 1 (L, h) = i π RL its Chern form. Let (M, ω M ) be a comlete Kähler manifold of dimension n and (L, h) be a Hermitian holomorhic line bundle on M and K M be the canonical line bundle on M. Then the L -norm on C0 (M, L ), the sace of smooth sections of L with comact suort, is defined for any s C0 (M, L ) by s = s(x) ωm n (.1) L h n! M Let L (M, L ) be the L -comletion of (C0 (M, L ), L ). We denote by, the inner roduct on L (M, L ) induced by this L -norm. Then the Bergman kernel function B (x) C (M, R) is still defined by (1.4) with {S l } l 1 an orthonormal basis of H() 0 (M, L ), the sace of L -holomorhic sections of L on M with resect to (.1). The Bergman kernel B (x, y) is the smooth kernel of the orthonormal rojection from (L (M, L ), L ) onto H() 0 (M, L ). We have (.) B (x, y) = l 1 S l (x) (S l (y)) L x (L y), and B (x, x) = B (x). Here (S l (y)) (L y) is the metric dual of S l (y) with resect to h. The Bergman kernel function (1.4) has the following variational characterization (see [CM, Lemma 3.1]): { } (.3) B (x) = max S(x) h : S H0 () (M, L ), S L = 1. We recall the exansion theorem for the Bergman kernel on a comlete manifold [MM1, Theorem 6.1.1]. Theorem.1 Let (M, ω M ) be a comlete Kähler manifold of dimension n and (L, h) be a Hermitian holomorhic line bundle on M. We assume there exist ε > 0, C > 0 such that 6

7 ir L εω M and Ric ωm Cω M, where Ric ωm = ir K M is the Ricci curvature of ω M. Then there exist coefficients b j C (M), j N, such that for any comact set K M, any k, m N, there exists C k,m,k > 0 such that for N, (.4) 1 n B (x) k b j (x) j C C m k,m,k k 1, (K) j=0 where (.5) b 0 = c 1(L, h) n ω n M, b 1 = b 0 8π (r ω ω log b 0 ), and r ω, ω, are the scalar curvature, resectively the (ositive) Lalacian, of the Riemannian metric associated to ω := c 1 (L, h). We write (.4) shortly as (.6) k B (x) = b j (x) n j + O( n k 1 ). j=0 For comact or certain comlete Kähler-Einstein manifolds the exansion was obtain by Tian [T] for k = 0 and m =. For general k, m and comact manifolds the existence of the exansion was first obtained in [Ca, Z]. The roof of [MM1, Theorem 6.1.1] yields immediately the following localization rincile for Bergman kernels used in the roof of Corollary.4. Namely, the asymtotics of B (x) deend only on the geometric data in any neighborhood of x M. Hence, the Bergman kernel function asymtotics are the same on two oen sets (in two ossibly different manifolds) over which the geometric data are isometric. Theorem. Let (M 1, ω M1 ), (M, ω M ) be comlete Kähler manifolds of dimension n and (L 1, h 1 ) M 1, (L, h ) M be Hermitian holomorhic line bundles. We assume there exist ε > 0, C > 0 such that for j = 1, we have ir L j εω Mj and Ric ωmj Cω Mj. Assume moreover that there are oen sets U j M j, j = 1,, and biholomorhic isometries Φ : U 1 U, Ψ : (L 1 U1, h 1 ) Φ ((L U, h )), where Ψ is also a bundle isomorhism. Let us denote by B j, the Bergman kernel functions of H() 0 (M j, L j ), j = 1,. Then for any k, m N and any comact set K U 1, we have B 1, B, Φ = O( k ) in C m (K) as. In articular, if b 1,j and b,j denote the coefficients of the exansion (.6) of B 1, and B,, then b 1,j = b,j Φ on U 1 for all j N. We immediately obtain from Theorem.1 uniform su-norm bounds for the Bergman kernel on comact subsets. Corollary.3 Under the hyotheses of Theorem.1, let K M be a comact subset such that ir L = ω M on K. Then uniformly on K, ( ) n (.7) B (x) = + O( n 1 ), as. π In the case of dimension one and constant curvature we can state the following. 7

8 Corollary.4 Assume that M in Theorem.1 is a Riemann surface and there exists an oen set V such that ω M has scalar curvature 4 and ir L = ω M on V. Then for any k, m N and any comact set K V, (.8) B (x) = 1 π 1 π + O( k ) in C m (K) as. Proof. From (.5) follows that b 0 = 1 π and b 1 = 1 π (note that r ω = 8π), thus the task is to rove that the coefficients b j of the exansions (1.6), (.6) vanish on V for j. We divide the roof in three stes. Firstly, it is easy to observe that b j are constant functions on V for all j N. Indeed, by [MM1, Theorem 4.1.1] we know that b j, j N, are olynomials in the curvatures R L and R T (1,0)M and their derivatives. On V we have ir L = ω M and ir T (1,0)M = ω M. Thus all the derivatives alluded to above vanish on V, hence b j are olynomials just in R L and R T (1,0)M, hence constant functions on V, for all j N. Secondly, we rove the assertion of the Corollary for a comact Riemann surface Σ 1 with genus g, such that Σ 1 Γ 1 \H, with Γ 1 a cocomact Fuchsian grou. We endow Σ 1 with the metric ω Σ1 induced from the Poincaré metric of H with scalar curvature 4. We consider the line bundle L 1 = T (1,0) Σ 1 = K Σ1 endowed with the metric h 1 induced by ω Σ1. Thus ir L 1 = ω Σ1. Let B 1, (x) be the Bergman kernel function of H 0 (Σ 1, L 1 ). By our observation above, the coefficients b 1,j of the exansion (.6) are constant functions on Σ 1 for all j N. Thus (.9) k B 1, (x) = b 1,j 1 j + O( k 1 ). j=0 By the Riemann-Roch theorem, for > 1, (.10) B 1, (x)ω Σ1 = dim H 0 (Σ 1, L 1 ) = Σ 1 Σ 1 ( 1 ) c 1 (K Σ1, h 1 ), and c 1 (K Σ1, h 1 ) = 1 π ω Σ 1. By lugging the exansion (.9) into (.10), identifying the coefficients of the owers of and taking into account that b 1,j are constants we get (.11) b 1,0 = 1 π, b 1,1 = 1 π, b 1,j = 0 for j. Thirdly, we use the localization rincile for Bergman kernels formulated in Theorem.. We now identify holomorhically and isometrically L on a neighborhood of x V to L 1 on an oen set of Σ 1. Indeed, by [W, Theorem.5.17 and Corollary.5.18], near x, the surface is locally isometric to the Poincaré uer half-lane H, and the holomorhic structure of the surface is determinated by the conformal structure fixed by the metric, thus we obtain a holomorhic and isometric identification Ψ of a convex neighborhood U of x V to an oen set of Σ 1. Then the curvature of the Chern connection on the line with the induced metric h is zero on U. If σ is a holomorhic frame of L Ψ K 1 Σ 1 on U, this means that log σ h = 0, so there is a holomorhic function f on U such that log σ h = Imf (which holds in any dimension). Now e f σ is a holomorhic frame of L Ψ K 1 Σ 1 such that e f σ h = 1 on U, and this yields a holomorhic and bundle L Ψ K 1 Σ 1 isometric identification of L to Ψ K Σ1. By Theorem., we know the asymtotics of B (x) is as same as of B 1,, thus from (.9) and (.11), we get that (.8) holds uniformly on K V. 8

9 Observe that if (Σ, ω Σ, L, h) fulfill conditions (α) and (β), the hyotheses of Corollary.4 are satisfied for V = V 1... V N (note that the scalar curvature of the Poincaré metric (1.1) equals 4), thus (.8) holds on any comact set K V 1... V N. The following result is a direct consequence of the roof of [MM1, Theorem 6.1.4], and for comleteness, we include the roof in Section 6. Theorem.5 Let (M, ω M, L, h) be in Theorem.1 and assume moreover that M is comact. Let π 1 (M) be the fundamental grou of Mand M be the universal covering of M. For any subgrou Γ π 1 (M) with finite index, we define the Bergman kernel B Γ (x, y) on Γ\ M with the ull-back objects from π Γ : Γ\ M M. Then for any k, m N, there exists C k,m > 0 such that for any Γ as above we have (.1) 1 n BΓ (x) k (πγb j )(x) j C C m k,m k 1, (Γ\ M) j=0 where b j are the coefficients of the exansion (.4) on M.. Functional saces, section saces We define a few functional saces, that will be much helful in what follows. (i) C 0 (D, ω D ) is merely the sace of bounded continuous functions on D, endowed with the su norm; notice that the reference to the metric, needed when considering bounds on derivatives, is suerfluous here. (ii) Let U Σ be an oen set. The sace C k (U, ω Σ ) is defined as the set of C k functions on U bounded u to order k on U with resect to the metric ω Σ, and endowed with the natural norm: with C k (U, ω Σ ) = { f C k (U) : f C k (U,ω Σ ) < }, (.13) f C k (U,ω Σ ) = su f C k(x), x U f C k(x) = ( f + Σ f ωσ ( Σ ) k f ωσ ) (x), Σ being the Levi-Civita connection attached to ω Σ. In the same vein, C k (U, ω Σ, L, h ) is the sace of C k sections of L on U such that the following norm is bounded for σ C k (U, ω Σ, L, h ): (.14) σ C k (h )(x) = ( σ h +,Σ σ h,ω Σ (,Σ ) k σ h,ω Σ ) (x), σ C k (U,ω Σ ) := su σ C k (h )(x) <, x U with,σ is the connection on (T Σ) l L induced by the Levi-Civita connection associated with ω Σ and the Chern connection relative to h. (iii) For k 1, the sace L,k( Σ, ω Σ, L, h ) is the Sobolev sace of sections of the line bundle L endowed with the Hermitian norm h over Σ, which are L u to order k, with resect to ω Σ and h. This way, elements of L,k( Σ, ω Σ, L, h ) are sections σ of L with L,k loc regularity on Σ, such that: (.15) σ L,k (h) := Σ ( σ h +,Σ σ h,ω Σ (,Σ ) k σ h,ω Σ ) ωσ <. 9

10 Alternatively, L,k( Σ, ω Σ, L, h ) is the L,k closure of the sace of smooth and com- (h) defined in (.15). For k = 0 we actly suorted sections of L over Σ, with L,k (h) simly denote L,0 (h) by L (h) and the corresonding inner roduct by,. When we aly this definition for the trivial line bundle C endowed with the nontrivial Hermitian norm log( z ) h 0 (the trivial Hermitian norm being h 0 ), we get the sace L,k( D, ω D, C, log( z ) ) h 0 and the norm L,k (D ). (iv) We also need in the localization rocedure below some weighted Sobolev saces on (Σ, ω Σ ) (res. on a double coy of (Σ, ω Σ )). We first define the weight function ρ on Σ as a smooth function, equal to 1 far from the unctures, to log( zj ) near the uncture a j, and everywhere 1. Let now k N, and q 1; the weighted Sobolev sace L q,k wtd (Σ, ω Σ) is defined as the sace of L q,k loc (.16) f q := L q,k wtd functions f on Σ, such that: ρ ( f q ( Σ ) k f q ) ω ωσ Σ <. Σ Notice moreover that Σ Σ is the comlement of a simle normal crossing divisor in a comact Kähler manifold, namely Σ Σ = ( Σ ) D with D = (D Σ) + (Σ D). This way, the natural roduct metric ω Σ Σ is a Kähler metric of Poincaré tye on Σ Σ (see e.g. [Auv, Def. 0.1]). Analogously, L q,k wtd (Σ Σ, ω Σ Σ) is the sace of L q,k loc functions f on Σ Σ, endowed with the roduct metric ω Σ Σ (x, y) = ω Σ (x) + ω Σ (y) such that (.17) f q := L q,k wtd is finite. (x,y) Σ Σ ρ(x)ρ(y) ( f(x, y) q ( Σ Σ ) k f(x, y) q ω Σ Σ ) ωσ (x)ω Σ (y) Lemma.6 a) We have L 1,3 wtd (Σ, ω Σ) C 0 (Σ), i. e., there exists c 0 > 0 such that for all f L 1,3 wtd (Σ, ω Σ) we have (.18) f C 0 (Σ,ω Σ ) c 0 f L 1,3. wtd b) There are continuous embeddings (.19) L,k wtd (Σ Σ, ω Σ Σ) C m (Σ Σ, ω Σ Σ ) for all k, m such that k > m +. Proof. a) is from [Biq, 4.A and Lemme 4.5]. For b), after noticing that the roof of [Biq, Lemme 4.5] remains valid close to the divisor D Σ Σ but far from the crossings (a j1, a j ) Σ Σ, we work around one of these, just as in the roof of [Auv, Lemma 4.]. More recisely, we choose two small unctured discs D r 1 and D r around a j1 and a j in each Σ resectively, and cover the roduct D r 1 D r in Σ Σ with hel of (self-overlaing) holomorhic olydiscs: Φ l1,l : D ɛ D ɛ D r 1 D r (u, v) (e l 1 1+u 1 u, e l 1+v ) 1 v, 10

11 with l 1, l 0, and 0 < ɛ < 1 fixed indeendently of l 1 and l. This way, D r 1 D r l 1,l =0 Φ l 1,l (D ɛ D ɛ ), and we can even assume that D r 1 D r l 1,l =0 Φ l 1,l (D ɛ/ D ɛ/ ). Moreover, for any (l 1, l ), (.0) (Φ l1,l ) idu du idv dv ω Σ Σ = (1 u + ) (1 v := ϖ, ) which does not deend on (l 1, l ). On the other hand, (Φ l1,l ) ρ(x) = l u 1 u, which is of size l1+1 for u ɛ, with derivatives (at every order) of the same size for ϖ, and similarly for (Φ l1,l ) ρ(y) with l+1. Set U = D r1 D r Σ Σ, so that D r 1 D r = U D. Take w L q,k wtd (Σ Σ, ω Σ Σ), q 1, k 0, and ick m 0, m < k q, so that w Cm (U D); what recedes thus yields: (.1) w q C m (U D) su (Φ l1,l ) w q C l 1,l 0 m (D ɛ/ D ɛ/,ϖ) (Φ l1,l ) w q C m (D ɛ/ D ɛ/,ϖ) = c l 1,l =0 l 1,l =0 1 l 1+l + l 1+l + (Φ l1,l ) w q C m (D ɛ/ D ɛ/,ϖ) 1 l 1+l + l 1,l =0 l 1,l =0 (Φ l1,l ) ( ρ(x) 1 q ρ(y) 1 q w ) q C m (D ɛ/ D ɛ/,ϖ) 1 l 1+l (Φ l1,l ) ( ρ(x) 1 q ρ(y) 1 q w ) q L q,k (D ɛ D ɛ,ϖ) by the fixed usual Sobolev embedding (or, more exactly, continuous restriction) L q,k (D ɛ D ɛ, ϖ) C m (D ɛ/ D ɛ/, ϖ) alied to all the (Φ l1,l ) ( ρ(x) 1 q ρ(y) 1 q w ). Now, observe that our choices rovide (.) l 1,l =0 1 l 1+l (Φ l1,l ) ( ρ(x) 1 q ρ(y) 1 q w ) q L q,k (D ɛ D ɛ,ϖ) C(q) ( ρ(x) 1 q ρ(y) 1 q w ) q L q,k (Σ Σ,ω Σ Σ ), as the Φ l1,l (D ɛ D ɛ ) self-overlas are of order l 1+l, and as each Φ l1,l (D ɛ D ɛ ) overlas only a finite number of other Φ l 1,l (D ɛ D ɛ ), this number being bounded indeendently of l 1 and l. Hence w q C m (U D) C ( ρ(x) 1 q ρ(y) 1 q w ) q L q,k (Σ Σ,ω Σ Σ ) C w q, L q,k wtd and one concludes by secializing to q =, and gathering such estimates around the crossings (a j1, a j ) with analogous estimates along the divisor D Σ Σ and far from the crossings, and estimates far from the divisor. 3 Bergman kernels on the unctured unit disc In this section we give a detailed descrition of the Bergman kernel on the unctured unit disc. We first obtain an exlicit formula in 3.1 and then in 3. we get recise asymtotics near the uncture by using a natural rescaling. 11

12 3.1 Exression of the Bergman kernels on the unctured unit disc Let N, and (3.1) H () (D ) := H 0 ()( D, ω D, C, log( z ) h 0 ), be the sace of holomorhic functions S on D with finite L -norm defined in Section. (iii) for k = 0. The urose here is to study of the Bergman kernel of H () (D ), as. Lemma 3.1 For, the set {( l ) 1/z (3.) l : l N, l 1} π( 1)! forms an orthonormal basis of H () (D ). Proof. Let H 0 (D, C) be the sace of holomorhic function on D. By [MM1, (6..17)], we know (3.3) H () (D ) H 0 (D, C). Note that for, l 1, log( z ) ω D = log( z ) idz dz D D z (3.4) 1 = 1 dr dθ log r S 1 0 r =, and z l log( z ) ω D = log( z ) z l idz dz D D z 1 (3.5) = 1 dθ r l 1 log r dr S 1 0 = π (l) 1 π( )! Γ( 1) = l 1 <. By (3.4), (3.5) and the circle invariance of ω D and log( z ) h 0, the set (3.) forms an orthonormal basis of H () (D ). Remark 3. Notice that a similar comutation shows that the elements of H 0 () (Σ, L ) are, for, exactly the sections of L over the whole Σ vanishing on the uncture divisor D = {a 1,..., a N }. Back to D and according to Lemma 3.1, the Bergman kernel of H () (D ), for any, is thus log( y ) (3.6) B D (x, y) = l 1 x l y l. π( )! Here the metric dual of the canonical section 1 with resect to h 0 is identified to 1, hence the metric dual of 1 with resect to log( z ) h 0 is 1 (z) = log( z ) 1. Secializing to the diagonal, we get in articular the Bergman kernel function of H () (D ) for all, (3.7) B D (z) = log( z ) π( )! l 1 z l. This readily rovides the behavior of B D far from 0 D. 1

13 Proosition 3.3 For any 0 < a < 1 and any m 0, there exists c = c(a) > 0 such that B D (z) 1 (3.8) π = C m ({a z <1},ω D ) O(e c ) as. More generally, for any 0 < a < 1 and 0 < γ < 1, there exists c = c(a, γ) > 0 such that B D (z) 1 π = O( ) (3.9) e c1 γ as. C m ({ae γ z <1},ω D ) Proof. Let us recall the celebrated formula from comlex analysis: 1 sin w = 1 (w kπ) on C πz. k Z Thus for t > 0, (3.10) k Z 1 (kiπ + t) = (et/ e t/ ) = le lt. Combining (3.7), (3.10) with an easy induction on, one gets the identity ( 1) log( z ) B D (z) = π (ikπ + log( z ) ) k Z (3.11) ( ( 1) = 1 + log( z ) ) π (ikπ + log( z ) ). k Z, k 0 To obtain (3.8) for m = 0, from (3.11), for, we use log( z ) (3.1) k Z, k 0 k Z, k 0 ikπ + log( z ) < ( 1 + (π) log( z ) log( z ) ) k=1 ikπ + log( z ) log( z ) log( z ) (kπ) + log( z ) for 0 < z e 1/, (kπ) for e 1/ z < 1. For m 1, by considering searately a < z e 1/ and e 1/ z < 1 as above, we get again (3.8) from (3.11). For ae γ z e 1/, from (3.13) log ( 1 + (π) k=1 and (3.1), we get also (3.9). k=1 ) C log( z ) C γ, log( z ) log( z ) (kπ) + log( z ) k=1 C γ (kπ) + 1, Observe that the exected behavior for an Einstein metric of scalar curvature 4 such as ω D, at least on comact subsets of D, according to (1.6), Theorem.1 and Corollary.4, is B D (z) ( 1) π = O( ). From our exlicit descrition of B D, we hence benefit an imrovement, namely exonential decay of the remainder, and extension of such an asymtotic result u to the exterior boundary D of D, as well as an estimate on how close to the singularity 0 D such an exonential decay holds. 13

14 3. Asymtotics of the density functions near the uncture We are also interested in a global descrition of B D u to the singularity 0 D, esecially in the geometric context of Theorem 1.1, and such a descrition requires another angle of attack. Let us simlify notations: for N, set log y +1 b (y) = l y l for y (0, 1), π( 1)! (3.14) ϕ(ξ) = e ξ log ξ for ξ (0, 1), ν() = (π) 1/ e (!) 1 1. Note that by Stirling s formula and (3.14), (3.15) ν() = O( 1 ) as. By (3.7) and (3.14), we have (3.16) B D +1(z) = b ( z ) for z D. Motivated by the observation that at fixed y, the index of the largest term of the sum l y l is determined by y 1/, we further roceed to the change of variable x = y 1/, and focus on the function f : (0, 1) R given by: log(x f (x) := b (x ) +1 ) = l x l π( 1)! = + e ( log x ex l log(x l ) ) (3.17) π! ( ) 3/(1 ) ( = + ν() log x ϕ(x l ) ). π The smooth function ϕ mas (0, 1) to (0, 1], with ϕ(ξ) = 1 iff ξ = e 1. Thus, for l fixed, x ( ϕ(x l ) ) heuristically converges to a thinner and thinner Gaussian-shaed bum of height 1 centered at e 1/l, and log x ( ϕ(x l ) ) can thus be thought of as a series of these bums centered at e 1, e 1/, e 1/3, of resective heights 1, 1, 1 3 (because of the factor log x ) and so on; this actually holds for x in low regime (x e δ, δ > 1/, say), the tail (x e δ, 0 < δ < 1/) consisting in an agglomeration of such bums mixing u with one another to follow an almost constant behavior near x = 1: see Figure 1 below. We develo in the following lines some elementary analysis that justifies these heuristic considerations. First, set (3.18) ψ (ζ) = ( ϕ(e ζ ) ) = e (1 ζ+log ζ) for ζ > 0, G 0 (η) = e η /, G 1 (η) = η 3 e η / for η R. We rove in the aendix A the following estimate, linking ψ to the Gaussian-tye functions G 0 and G 1 : Lemma 3.4 There exists a constant C such that for all ζ > 0 and all 1, ( ) 1 ψ (ζ) G 0 (1 ζ) + 3 G ( ) C 1 (1 ζ) (1 + (1 ζ) ). 14

15 Figure 1 The scaled functions ( ) π 3/f on (0, 1) For 1 and x (0, 1), set ( ( (3.19) G (x) = log x G 0 [1 + log(x l )] ) 1 3 ( G 1 [1 + log(x l )] )). Remembering that we are looking for an aroximation of log x ( ϕ(x l ) ) and keeing in mind the relation (3.18) between ϕ, and ψ, we state: Proosition 3.5 There exists a constant C such that for all 1 and x (0, 1), (3.0) log x ( ϕ(x l ) ) G (x) C + log x. Corollary 3.6 There exists a constant C such that for all 1 and z D, (3.1) ( π ) 3/(1 ) 1B + ν() D +1(z) G ( z / ) C + log z. In articular, (3.) su B D z D (z) = ( ) 3/ + O(). π Proof of Proosition 3.5. Setting ( ) 1 (3.3) δ (ζ) = ψ (ζ) G 0 (1 ζ) + 3 G ( ) 1 (1 ζ). For all 1 and x (0, 1), by (3.18), (3.19) and (3.3), (3.4) log x ( ϕ(x l ) ) G (x) = log x log x ( δ log(x l ) ), l log x 1 δ ( log(x l ) ) ; 15

16 this takes into account the vanishing of δ (ζ) at ζ = 1. bounded above by By Lemma 3.4, the latter is (3.5) C log x (l log x + 1) ;,l log x 1 C we can thus conclude if we bound this quantity above by an exression of tye (1+ log x ). If 0 < log x, by bounding the terms associated with l = (log x) 1, (log x) by 1, where we note u the integer art of u R, we get l log x (l log x + 1) + 1 log x 0 Thus by (3.4) and (3.5), for 0 < log x, (3.6) log x + 1 log x π = +, log x dα 1 + (α 1) dα 1 + (α 1) ( ϕ(x l ) ) C log x G (x) + C 3/ C, C and this yields the uer bound (1 + log x ) as log x 1. This way, the estimate 3 (3.0) is roved on the region {0 < log x }. l log x Let us assume now that log x. Then for all l 1, l log x 1, thus and (3.7) l log x 1 (l log x + 1) = (l log x 1) l log x, 4 log x 1 + (l log x + 1) log x ( l log x 4 ) = π 3. In other words, by (3.4), (3.5) and (3.7), on the region { log x }, (3.8) log x ( ϕ(x l ) ) G (x) and this uer bound yields here again an uer bound 3 (1 + log x ) C log x, C (1 + log x ), since 1 log x when log x. By (3.6) and (3.8), we get (3.0). Proof of Corollary 3.6. The first art of the corollary follows at once from Proosition 3.5. The second art is an immediate consequence of the estimate (3.9) su log x G (x) = 1 + O( 1/ ). x (0,1) 16

17 To establish this estimate, let us rove first that (3.30) 1 su log x x (0,1) ( G 0 [1 + log(x l )] ) = 1 + O( 1/ ). As utting x = e 1 in log x G ( 0 [1+log(x l )] ) gives 1+ l= G 0( [1+log(x l )] ) 1, we get already that the su in (3.30) is bounded below by 1. Now we have (3.31) (log x) 1 +1 l= (log x) 1 e (1+l log x) 1 + e (log x) /4 if log x 1, and as a function of s > 0, e (1+s log x) increases when s > (log x) 1 + 1, thus (3.3) ( (log x) = l= (log x) 1 + R R when s < (log x) 1 and decreases ) G 0 ( [ 1 + log(x l ) ]) G 0 ( [ 1 s log(x) ]) ds G 0 ( s log(x) ) ds = C log x, where we just omit the sum (log x) 1 1, if log x 1; the transition from the second to the third line simly comes from the translation s s + log x 1. Now log x e (1+log x) = ( log x 1)e ( log x 1) + e ( log x 1). By using that the function η ηe η / is bounded on R, we get from (3.31), (3.3) that for x (0, 1), (3.33) log x ( G 0 [1 + log(x l )] ) = inf{1, log x } + O( 1/ ),. With similar methods, one roves that (3.34) su log x x (0,1) ( G 1 [1 + log(x l )] ) = O(1),. From (3.30) and (3.34), we get (3.9). 4 Ellitic Estimates for Kodaira Lalacians on D and Σ In this section, we establish a weighted ellitic estimate for Kodaira Lalacians on (D, ω D ) with weight log( z ) such that the estimate is uniform on, and on D. This is the essential analyze inut in comarison with the comact situation. Let L L be the adjoint of the Dolbeault oerator on (L, h ) over (Σ, ω Σ ). Then the Kodaira Lalacian is defined as (4.1) := ( L + L ) = L L + L L : Ω (0, ) (Σ, L ) Ω (0, ) (Σ, L ). We denote by D the above oerator when Σ = D. 17

18 4.1 Estimate on the unctured disc D : degree 0 Note that the Poincaré metric (1.1) on the unctured disc can be written as (4.) ω D = i log ( log( z ) ). Recall that the norm L, was defined in Section. (iii). In what follows, we (D ) adot the notation L for the trivial line bundle C over the oen unit disc D, thought of as endowed with the singular Hermitian metric h D := log( z ) h 0 ; similarly, for 1, L will imlicitly refer to ( C, log( z ) ) ( h 0 = C, h ) D. Notice that with these conventions, (4.) can be interreted as: (4.3) iω D is the curvature of (L, h D ) (and thus i ω D is that of (L, h D )). We rove in this section the following basic ellitic estimate on the Kodaira Lalacians D, associated to the data ( D, ω D, L, h D ). Proosition 4.1 Let s 1. Then there exists C = C(s, h D ) such that for all 1, and all σ L,s (D ), s (4.4) σ L,s (D ) C 4(s j) ( D ) j σ L (D ). j=0 Our strategy is as follows. We will write the detail roof for s = 1, then by induction, we get it for s. Fixing f C0 (D ), for s = 1, we first establish an estimate analogous to (4.4) with 0, the Lalace-Beltrami oerator of ω D, instead of the Kodaira Lalacian D associated to (D, ω D, L, h ) D. Then we deduce (4.4) by Kähler identities. To facilitate the comutation, we introduce first a new coordinate for D and exlain some basic geometric facts. For z D, we will use the coordinates (t, θ) R (R/πZ) with (4.5) t := log ( log( z ) ) ), z = z e iθ. We denote also t by t, and θ by θ. Then we comute (4.6) z log( z ) z = t i et θ, (z log( z )) 1 dz = 1 dt + i e t dθ. Thus we have (4.7) = dz z = (1 dt + i e t dθ)( t i et θ ), t = 1 dt i e t dθ. From (4.7), we obtain the following useful relation ω D = e t (4.8) dt dθ, log( z ) ω D = e ( 1)t dt dθ, and the metric associated with ω D in the coordinates (t, θ) is (4.9) 1 (dt) + e t (dθ), thus ( t, 1 e t θ ) is an orthonormal frame of ω D. 18

19 Let D be the Levi-Civita connection on (D, ω D ). Using (4.9) and the equality we comute that (4.10) >From (4.10), we get D t θ D θ t = [ t, θ ] = 0, D θ θ, t = D t θ, θ = 1 t θ, θ = e t. (4.11) From (4.11), we get D t t = 0, D θ θ = 4e t t, D t θ = D θ t = θ. (4.1) D dθ = dθ dt + dt dθ, 0 = ( t t t ) 1 et θ θ. D dt = 4e t dθ dθ, Let (res. L ) be the adjoint of on the trivial line bundle (C, h0 ) (res. on (C, log( z ) h 0 )) over (D, ω D ). By (4.6) and (4.8), we have the following exressions in the coordinates (t, θ), (4.13) L = ( t ) and ( t ) dz = dz, t = z log( z ). By (4.6) and (4.8), we get for f C (D ), (4.14) L (f t) = z log( z ) f + (1 )f. z Thus the Kodaira Lalacian associated with (C, log( z ) h 0 ) has the form (4.15) D = L + L = + ( ( t ) + ( t ) ) = 1 0 ( ( t ) + ( t ) ), where we used the Kähler identity for the last equality. Proof of Proosition 4.1. Notice that since the Hermitian line bundles L we consider here are owers of the line bundle ( C, log( z ) ), the Chern connections acting on the sections of these bundles, which are functions, are given by (4.16) f = df + f t, f C (D, L ). Therefore, for these f, and > 1, f L,1 (D ) = ( f + D t f + (1/ f ) )e t log( z ) ω D θ (4.17) ( ( + 1) f + t f + 1 D et θ f ) log( z ) ω D, and, similarly (4.18) f L, (D ) ( 4 f + ( t f + e t θ f ) D + t f + e t t θ f + e t θ f ) log( z ) ω D, 19

20 with the constants hidden in indeendent of. We will comute everything by using the coordinate (t, θ), then means R (R/πZ) and sometimes, we identify S 1 to R/πZ. Thus by (4.8) and simle integrations by arts, we get (4.19) e t θ f log( z ) ω D = e t θ f e ( 1)t dtdθ = (e t θ f)fe( 1)t dtdθ, D and (4.0) t f log( z ) ω D = ( t f)fe ( 1)t dtdθ + D ( 1) f e ( 1)t dtdθ. This way, by the Peter-Paul inequality, xy x ε + εy for x, y 0, ε > 0, we obtain for every ε > 0, (4.1) ( t f + e t θ f ) log( z ) ω D D (ε 1 ( ) 1) + f e ( 1)t dtdθ + ε Taking ε = we get (4.) ( t f + e t θ f ) e ( 1)t dtdθ. ( t f + e t θ f ) log( z ) ω D D ( 4 f + ( t f + e t θ f ) ) e ( 1)t dtdθ. Thus, from (4.18), for f L, (D ), (4.3) f L, (D ) ( 4 f + t f + e t t θ f + e t θ f ) e ( 1)t dtdθ, D with the imlied constant indeendent of. By (4.1), (4.4) ( 0 f) log( z ) ω D D (( = 4 t f) + ( t f) + ( et 4 θ f)) e ( 1)t dtdθ ( t f)( et 4 θ f)e( 1)t dtdθ 8 We deal with the mixed terms as follows: ( t f)( t f)e ( 1)t dtdθ ( t f)( et 4 θ f)e( 1)t dtdθ. 8 ( t f)( t f)e ( 1)t dtdθ: an integration by arts yields: (4.5) 8 ( t f)( t f)e ( 1)t dtdθ = 4( 1) ( t f) e ( 1)t dtdθ, and we do not rovide more efforts, as this quantity has the favorable sign already remember we want a bound below on the L (D )-norm of 0 f; 0

21 8 ( t f)( et 4 θ f)e( 1)t dtdθ: exchanging t and θ via integrations by arts, we get: (4.6) 8 ( t f)( et 4 θ f)e( 1)t dtdθ = 8 ( et t θ f) e ( 1)t dtdθ 8( + 1) ( t f)( et 4 θ f)e( 1)t dtdθ, and collect the extra term 8(+1) ( t f)( et 4 θ f)e( 1)t dtdθ together with the left over right-hand-side mixed term in (4.4), i.e. we deal with: 8( + ) ( t f)( et 4 θ f)e( 1)t dtdθ: (4.7) 8( + ) ( t f)( et 4 θ f)e( 1)t dtdθ ( et 4 θ f) e ( 1)t dtdθ 8( + ) ( t f) e ( 1)t dtdθ, By Cauchy-Schwarz inequalities and (4.0), we get 8( + ) ( t f) e ( 1)t dtdθ ( t f) e ( 1)t dtdθ ( 8( + ) 4 + 4( 1) ( + ) ) f e ( 1)t dtdθ. We sum u what recedes as: (4.8) D ( 0 f) log( z ) ω D (( t f) + ( et t θ f) + ( et 4 θ f)) e ( 1)t dtdθ ( 8( + ) 4 + 4( 1) ( + ) ) f e ( 1)t dtdθ, By (4.3) and (4.8), we get ) (4.9) f ( L, (D ) C 0 f L (D ) + 4 f, L (D ) for some C > 0 indeendent of both 1 and f C0 (D ) with real values; by density, this readily generalizes to f L, (D ) with comlex values, as 0 is a real oerator. We now carry out the relacement of 0 by D in (4.9), to get the desired estimate (4.4). By (4.13) and (4.15), acting on function on D, we have (4.30) D = 1 0 with = z log( z ) z = t i et θ Let f L, (D ). Using inequality (4.1) with ε > 0 to be adjusted, (4.3) and (4.9), we have: f e ( 1)t dtdθ ( t f + et θf )e ( 1)t dtdθ (4.31) ( ε εc 4) f L (D ) + εc 0f L (D ). From (4.30) and (4.31), we are led to: (4.3) 0 f L (D ) = ( D + ) f L (D ) 8 D f L (D ) + 8 f L (D ) 8 D f L (D ) + 8( ε εc 4) f L (D ) + 8 εc 0 f L (D ). 1

22 Take ε = 1 to conclude that: 16C (4.33) 0 f L (D ) 16 f L (D ) + A4 f L (D ) with A = 56C + 9. Plugged back into (4.9), this estimate gives exactly (4.4), with a (new) constant C > 0, uniform for 1 and f L, (D ). The roof of Proosition 4.1 for s = 1 is comleted. Continuing by induction we get it for all s. 4. Estimate on the unctured Riemann surface Σ : degree 0 We now consider the geometric situation of a unctured olarized Riemann surface (Σ, ω Σ, L, h) satisfying conditions (α) and (β). Let a D. By assumtion the following holds: there exists a trivialization of L around a such that in the associated local comlex coordinate z D, we have h = log( z ) on the coordinate disc D r centered at a and of radius r (0, e 1 ). This way, the curvature ω Σ of h coincides with ω D on D r := D r {0}. Proosition 4. For every s N there exists C = C(s, h) such that for all 1, and all σ L,s (h) = L,s( Σ, ω Σ, L, h ), s (4.34) σ L,s (h) C 4(s j) ( ) j σ L (h), where is the Kodaira Lalacian on Σ associated to ω Σ and h. j=0 Proof. Again, we do it for s = 1. In the situation of the Proosition, we denote by h a smooth Hermitian metric on L on the whole Σ such that it coincides with h on Σ D r/. It is an easy exercise to construct h so that its curvature, ω say, is Kähler over the whole (comact) Σ, which we take for granted until the end of this roof. Notice that ω Σ and ω coincide on Σ D r/. Now the rincile of the roof is to glue estimate (4.4) to the analogous estimate for ( Σ, L, h ), that states the existence of a constant C such that for all 1, and all σ L, ( ) h := L, ( Σ, ω, L, h ), (4.35) σ L, (h) C( Σ σ L (h) + 4 σ L (h) ). This estimate, as well as its generalization for σ L,s ( ) h, s 1, can be found for instance in [DLM1, (4.14)] or [MM1, 1.6.]. We denote,σ the formal adjoint of,σ action on Λ(T (0,1) Σ) L. By Lichnerowicz formula [MM1, Remark 1.4.8], (4.36) =,Σ,Σ R L (w, w) + (R L + R T (1,0)Σ )(w, w)w i w, and w is an orthonormal frame of T (1,0) Σ. By (4.3), (4.37) R L (w, w) = 1, R T (1,0)Σ (w, w) = on V = V 1... V N. From (4.36) and (4.37), for any σ L, (h),,σ σ (4.38) L (h) σ, σ C σ L (h).

23 Let χ be a cut-off function suorted near a; assume, more recisely, that (4.39) χ C ( Σ ), 0 χ 1, χ Dr/ 1, χ Σ Dr/3 0. Let 1, and σ L, ( ) h = L, ( Σ, ω Σ, L, h ). Then (1 χ)σ L, (h) and on its suort, h coincides with h; likewise, χσ can be interreted as an element of L, (D ) and on its suort, h can be regarded as h D. Therefore, (4.40) σ L, (h) = χσ + (1 χ)σ L, (h) ( χσ L, (D ) + (1 χ)σ L, C ( D (χσ) L (D ) + Σ [(1 χ)σ] L (h) ) + C 4( χσ L (D ) + (1 χ)σ L (h) where C = su ( C(h D ), C(h) ), with C(h D ), res. C(h), the constant from (4.4), res. from its analogue for ( Σ, L, h ). Thus defined, C is indeendent of σ and. Now χσ L (h D ) = χσ L (h) σ L (h), and (1 χ)σ L (h) σ as well. L (h) The treatments of D (χσ) L (h D ) and Σ [(1 χ)σ] are done in the same sirit, L (h) but require a little extra work. For instance, on D r/3, by (4.15) and (4.16), (4.41) D (χσ) = χ D σ ( χ, ( ) 1,0 σ ) ( σ, χ ) ( 1 ) + T D T D 0χ σ, ) (h) ) hence (4.4) D (χσ) ( χ L (h D ) 4 D σ L (h D ) + χ σ L (h D ) + χ ( ) 1,0 σ L (h D ) + ( 1 0χ)σ ) L (h D ) ( = 4 χ σ L (h) + χ σ L (h) + χ (,Σ ) 1,0 σ L (h) + ( 1 0χ)σ L (h) ), with the L (h D )-norms, res. L (h)-norms, for 1-forms, res. 1-forms with value in L, comuted with ω D h D on D, res. with ω Σ h. Consequently, D (χσ) ( L (h D ) 4C σ L (h) +,Σ σ L (h) + σ ) (4.43) L (h) with C = 1 + max { 1 0χ C 0 (D,ω D ), χ C 0 (D,ω D )}, that does not deend on. From (4.38) and (4.4), we get (4.44) D (χσ) L (h D ) C( σ L (h) + σ ) L (h) for some C indeendent of and σ. Similarly, (4.45) Σ ( ) (1 χ)σ L (h) C( σ L (h) + σ ) L (h) with C again indeendent of and σ. In conclusion, it follows from (4.40), (4.44) and (4.45), there exists C > 0 such that for any 1 and σ L, (h), we have (4.46) σ L, (h) C( σ L (h) + 4 σ ) L. (h) The roof of Proosition 4. for s = 1 is comleted. The roof for general s N follows by induction with the hel of Proosition

24 4.3 Bidegree (0, 1) This subsection will not be used in the rest of this aer, we include it here only for comleteness and its indeendent interest. To rove that Proositions 4.1 and 4. still hold in bidegree (0, 1), or, namely, for σ a section of T (0,1) D L or T (0,1) Σ L, an easy rocedure is to observe that the following diagram: (4.47) C 0 (D ) dz z log( z ) C 0 (D, T (0,1) D ) ie t θ C 0 (D ) dz z log( z ) C 0 (D, T (0,1) D ) commutes, where the horizontal arrows are isometries under h D and (h D ) ω D. Indeed, dz by (4.6) and (4.7), z log( z ) = t, and by (4.7) and (4.14), for g C (D ), we have [ L (g t) = L g + z log( z ) z g z log( z ) ] t (4.48) z g = ( g ie t θ g) t. Proosition 4.3 Let s 1. Then there exists C = C(s, h D ) such that for all 1, and all σ L,s (D ) = L,s( D, ω D, T (0,1) D L, ω D h D ), s (4.49) σ L,s (D ) C 4(s j) ( D ) j σ L (D ). dz j=0 Proof. Indeed, take σ = f z log( z ) = f t C (D, T (0,1)D ). Then for instance, ( ) σ = ( ) f t + f D t + f ( D ) t where D is the Levi-Civita connection of ω D. By (4.7), (4.9) and (4.11), t is uniformly bounded at any order with resect to ω D on D, we get that ( ) σ D ω D ( ( ) f D + f + f ) ω D = f L, (D ), indeendently of. By Proosition 4.1, we thus have (4.50) ( ) σ D ω D C ( f D + 4 f ) ω D for 1, with C indeendent of. By (4.47), we have σ D ω D = f ie t θ f D ω D (4.51) 1 f ω D e t θ f e ( 1)t dtdθ. D By (4.1) with ε = 1 9, (4.8) and (4.33), as in (4.31), we get (4.5) e t θ f e ( 1)t dtdθ C f L (D ) f L (D ). 4

25 As D σ ω D = D f ω D, from (4.51) and (4.5), we get f D ω D 4 σ ω D + C σ ω D. D D This yields, coming back to (4.50), (4.53) ( ) σ D ω D C ( σ D + 4 σ ) ω D. Consequently, for 1, we get (4.54) σ L, (D ) C( σ L (D ) + 4 σ L (D ) ). Now by induction on s, we get Proosition 4.3 for s 1. Using moreover the same gluing rocedure as in roving Proosition 4., we get the analogue of (4.49) on Σ: Proosition 4.4 Let s N. Then there exists C = C(s, h) such that for all 1, and all σ L,s (Σ) = L,s( Σ, ω Σ, T (0,1) Σ L, ω Σ h ), s (4.55) σ L,s (Σ) C 4(s j) ( ) j σ L (Σ). j=0 5 Sectral Ga and Localization We follow in this Section the localization scheme based on the sectral ga and finite roagation seed [MM1] and show that the Bergman kernel localizes near the singularities. As a consequence we obtain a first rough estimate, which will be imroved in the next Section. Let (M, ω M ) be a comlete Kähler manifold. We will denote by R det the curvature of the anticanonical line bundle (KM, hk M ), where h K M is induced by ω M. Let (E, h E ) be a Hermitian holomorhic line bundle on M. Let be the formal adjoint of with resect to, (cf. (.1)). Let E = be the Kodaira Lalace oerator. By [MM1, Corollary 3.3.4] the oerator E : C0 (M, E) C 0 (M, E) is essentially selfadjoint and we will denote its unique self-adjoint extension with the same symbol E. Note that the domain of this extension is Dom( E ) = {σ L (M, E) : E σ L (M, E)}. Consider now a Hermitian holomorhic line bundle (L, h) and denote by := L the Kodaira Lalace oerator corresonding to (L, h ). By [MM1, Theorem 6.1.1] and its roof we have the following. Proosition 5.1 (Sectral ga) Let (M, ω M ) be a comlete Kähler manifold and (L, h) be a Hermitian holomorhic line bundle on M. We assume there exist ε > 0, C > 0 such that ir L εω M and ir det Cω M. Then there exists c = c(c, ε) > 0 such that for all 1 we have (5.1) Sec( ) {0} [c, + ). Corollary 5. The sectral ga (5.1) holds for the Lalacian in the following situations: (1) (M, ω M ) = (D, ω D ), (L, h) = (C, log( z ) h 0 ), () (M, ω M ) = (Σ, ω Σ ), (L, h) as in Theorem

26 Indeed, by (4.3) and ir det = ω D holds on D, combining the condition (β), we know ir L εω Σ, ir det Cω Σ on Σ, for some C > 0. Thus we can aly Proosition 5.1 to Corollary 5.. We assume here, without loss of generality, that the uncture divisor D in Σ is reduced to one oint a. Let e be the holomorhic frame of L near a corresonding to the trivialization in the condition (α). By the assumtion (α), (β), under our trivialization e of L on the coordinate z on D r for some 0 < r < e 1, we have the identification of the geometric data (5.) (Σ, ω Σ, L, h) D r = (D, ω D, C, h D ) D r. We set: F is the normalized Fourier transform of a smooth cut-off function as in [MM1, 4.1], namely ( ) 1 (5.3) F (u) = f(v) dv e ivu f(v) dv R R with f : R [0, 1] a smooth even function such that f(v) = 1 if v ɛ/ and f(v) = 0 if v ɛ for ɛ > 0. Thus F is an even function in the Schwartz sace S (R) with F (0) = 1. Let F be the function satisfying F (u ) = F (u) for all u R. We consider the function (5.4) φ : R R, u 1 [c,+ ) ( u ) F (u) where c > 0 is defined in (5.1); let K := φ ( ) and let K (, ) be the associated kernel; we denote by f (, ) the function associated to K (, ) via the doubled trivialization around a used above; for x D r, we set f,x for the one-variable function y f (x, y); then (5.5) K (x, y) = f (x, y)e (x) (e (y)), and (e (y)) is the metric dual of e (y) with resect to h, that is, e (y) e (y) = e (y) h. χ a cut-off function as in (4.39); ρ : Σ [1, + ) is a smooth function such that ρ(z) = log( z ) on D r. Proosition 5.3 For any l, m 0, γ > 1, there exists C l,m,γ > 0 such that for any > 1, we have (5.6) ρ(x) γ ρ(y) γ K (x, y) C m (h ) C l,m,γ l, in the sense of (.14). Proof. We roceed as follows. Take 1, and ick a and b two real arameters to 6