A REMARK ON WEIGHTED BERGMAN KERNELS ON ORBIFOLDS. Xianzhe Dai, Kefeng Liu and Xiaonan Ma

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1 Math. Res. Lett. 19 (2012) no c Internatonal Press 2012 A REMARK ON WEIGHTED BERGMAN KERNELS ON ORBIFOLDS Xanzhe Da Kefeng L and Xaonan Ma Abstract. In ths note we explan that Ross Thomas reslt [4 Theorem 1.7] on the weghted Bergman kernels on orbfolds can be drectly dedced from or prevos reslt [1]. Ths reslt plays an mportant role n the companon paper [5] to prove an orbfold verson of Donaldson theorem. In two very nterestng papers [4 5] Ross Thomas descrbe a noton of ampleness for lne bndles on Kähler orbfolds wth cyclc qotent snglartes whch s related to embeddngs n weghted projectve spaces. They then apply [4 Theorem 1.7] to prove an orbfold verson of Donaldson theorem [5]. Namely the exstence of an orbfold Kähler metrc wth constant scalar crvatre mples certan stablty condton for the orbfold. In these papers the reslt [4 Theorem 1.7] on the asymptotc expanson of Bergman kernels plays a crcal role. In ths note we explan how to drectly derve Ross Thomas reslt [4 Theorem 1.7] from Da L Ma [1 (5.25)] provded Ross Thomas condton [4 (1.8)] on c holds. Snce n [1 Secton 5] we state or reslts for general symplectc orbfolds n what follows we wll jst se the verson from [2 Theorem ] where Ma Marnesc wrote them n detal for Kähler orbfolds. We wll se freely the notaton n [2 Secton 5.4]. We assme also the axlary vector bndle E theren s C. Let (X J ω) be a compact n-dmensonal Kähler orbfold wth complex strctrej and wth snglar set X sng.let(l h L ) be a holomorphc Hermtan proper orbfold lne bndle on X. Let L be the holomorphc Hermtan connectons on (L h L )wth crvatre R L =( L ) 2. We assme that (L h L L ) s a preqantm lne bndle.e. (0.1) R L = 2π 1 ω. Let g TX = ω( J ) be the Remannan metrc on X ndced by ω. Let TX be the Lev Cvta connecton on (X g TX ). We denote by R TX =( TX ) 2 the crvatre by r X the scalar crvatre of TX. For x X setd(x X sng ):=nf y Xsng d(x y) the dstance from x to X sng. For p N the Bergman kernel P p (x x )(x x X) s the smooth kernel of the orthogonal projecton from C (X L p )ontoh 0 (X L p ) wth respect to the Remannan volme form dv X (x ). Theorem 0.1 ([1 Theorem 1.4] [2 Theorem ]). There exst smooth coeffcents b r (x) C (X) whch are polynomals n R TX and ts dervatves wth order 2r 2 Receved by the edtors Agst

2 144 XIANZHE DAI KEFENG LIU AND XIAONAN MA at x andc 0 > 0 sch that for any k l N there exst C kl > 0 M N wth 1 k (0.2) p n P p(x x) b r (x)p r C r=0 l C kl (p k 1 + p l/2 (1 + ) pd(x X sng )) M e C 0 pd(xx sng ) for any x X p N. Moreover (0.3) b 0 =1 b 1 = 1 8π rx. In local coordnates there s a more precse form [1 (5.25)] see also [2 Theorem ]. Let {x } I =1 X sng. Foreachpontx we consder correspondng local charts (G x Ũx ) U x wth Ũx C n sch that 0 Ũx s the nverse mage of x U x and 0 s a fxed pont of the fnte stablzer grop G x at x whchactsc-lnearly and effectvely on C n (cf. [2 Lemma 5.4.3]). We assme moreover that BŨx (0 2ε) Ũ x and X sng W := I =1BŨx (0 1 4 ε)/g x. Let Ũ x g be the fxed pont set of g G x n Ũx andletñx g be the normal bndle of Ũ x g n Ũx. For each g G x the exponental map Ñx g x Y expũx x (Y ) dentfes a neghborhood of Ũ x g wth W x g = {Y Ñx g Y ε}. We dentfy wth L L Wx g Ũx g by sng the parallel transport along the above exponental map. Then the g-acton on s the mltplcaton by L Wx e 1θ g andθ g g s locally constant on Ũ x g. Let Ñx g be the connecton on Ñx g ndced by the Lev Cvta connecton va projecton. We trvalze Ñx g Ũ x g C n g by the parallel transport along the crve [0 1] t t Z 1g for Z 1g Ũ x g whch dentfes also the metrc on Ñx g wth the canoncal metrc on C n g.if Z W x g we wll wrte Z =( Z 1g Z 2g )wth Z 1g Ũ x g Z 2g C n g. We wll denote by Z the correspondng pont on the orbfold. Theorem 0.2 ([1 (5.25)] [2 Theorem ]). On Ũx as above there exst polynomals K r ( Z Z1g 2g ) n Z 2g of degree 3r of the same party as r whose coeffcents are polynomals n R TX and ts dervatves of order r 2 and a constant C 0 > 0 sch that for any k l N there exst C kl > 0 N N sch that 1 (0.4) p n P p( Z Z) k r b r ( Z)p r=0 2k p r 2 1θg p K ( r p Z Z1g 2g )e 2πp (1 g 1 ) z 2g z 2g r=0 1 g G x e ( C kl p k 1 l 1 k+ + p 2 (1 + ) pd(z X sng )) N e C 0 pd(zx sng ) for any Z ε/2 p N wthb r ( Z) as n Theorem 0.1 and K 0 Z1g =1. C l

3 A REMARK ON WEIGHTED BERGMAN KERNELS ON ORBIFOLDS 145 Gven a fncton f(p x) np N and x X wewrtef = O C j (p l )fthec j -norm of f s nformly bonded by Cp l. Theorem 0.3. Let (X ω) be a compact n-dmensonal Kähler orbfold wth cyclc qotent snglartes (.e. the stablzer grop G x s a cyclc grop for any x X) and L be a proper orbfold lne bndle on X eqpped wth a Hermtan metrc h L whose crvatre form s 2π 1 ω sch that for any x X the stablzer grop G x acts on L x as Z Gx -order cyclc grop. Fx N 0 andr 0 and sppose c are a fnte nmber of postve constants chosen so that f X has an orbfold pont of order m then 1 (0.5) k c = k c for all and all k =0...N + r. m Then the fncton mod m (0.6) Bp orb (x) := c P p+ (x x) admts a global C 2r -expanson of order N. b 0...b N on X sch that That s there exst smooth fnctons (0.7) B orb p = N b j p n j + O C 2r(p n N 1 ). Frthermore b j are nversal polynomals n the constants c and the dervatves of ω; n partclar b 0 = c b 1 = ( (0.8) c n+ 1 8π rx). Remark 0.1. Theorem 0.3 recovers [4 Theorem 1.7] of Ross Thomas where the remander estmate s O C r(p n N 1 ). We mprove here ther remander estmate to O C 2r(p n N 1 ) and we get Theorem 0.3 drectly from Theorems 0.1 and 0.2. Remark 0.2. By Ma Marnesc [3 (3.30) Remark 3.10] [2 Theorem Remark ] Theorem 0.3 generalzes to any J-nvarant metrc g TX on TX. Set Θ := g TX (J ). The only change s that the coeffcents n the expanson become (0.9) b 0 = ωn Θ n c b 1 = ωn [ Θ n c n+ rx ω 8π 1 4π Δ ω log ( ω n Θ n )] where rω X Δ ω are the scalar crvatre and the Bochner Laplacan assocated to gω TX = ω( J ). Moreover (0.7) can be taken to be nform as (h L g TX ) rns over a compact set. Proof of Theorem 0.3. Recall that now G s a cyclc grop of order m. Let ζ be a generator of G. From the local condton for orb-ample lne bndles ζ acts on L x as

4 146 XIANZHE DAI KEFENG LIU AND XIAONAN MA aprmtvemth root of nty λ. Ths n (0.4) e 1θ ζ = λ. For {1... m 1} set (0.10) η =e 2π (1 ζ ) z 2ζ z 2ζ S ( Z) = m 1 S 2 = c 2N+2r+1 (p + ) n j 2 Kj Z1ζ ( p + Z 2ζ )λ (p+) η p+ N+r c b j ( Z)(p + ) n j. S S 1 = =1 Here Z = z + z andz = z z z = z z when we consder them as vector felds and 2 z = 2 1 z = 2. Smlarly for Z (and those wth sbscrpts). Applyng (0.4) for k = N + r + 1 we obtan for Z ε/2 (0.11) B orb p ( Z) S 1 S 2 C l C l p n N r 2 (1 + p l +1 2 (1 + pd(z X sng )) M e C 0 pd(zx sng ) ) + c (p + ) n N r 1 ( m 1 ) K ( 2N+2r+2 p + Z Z1ζ 2ζ )η p+ + bn+r+1 ( Z). C l C l =1 In what follows we wrte for smplcty Z 1ζ as Z 1 and Z 2ζ as Z 2. For a fncton f(p Z) wthp N and Z ε/2 wewrtef = O C j (g(p Z)) f the C j -norm of f n Z can be nformly controlled by C g(p Z). Note that K jz1 (Z 2 ) s a polynomal n Z 2 wth the same party as j and deg K jz1 3j. Denote by K jz1 l the l-homogeneos part of K jz1. Then K jz1 l =0fl and j are not n the same party or l>3j. By (0.10) (0.12) S (Z) = 2N+2r+1 c (p + ) 2N+2r+1 = λ p l l j 2n ( ) p n j 2 q n j l 2 q = ( N+r γ q=0 pγ q q followng relatons for r r N r l Here we sed (p + ) γ n j l n j l 2 q=0 ) q 2 KjZ1 l(z 2 )λ (p+) η p+ + l<j 2n q=0 c q λ η p+ N+r K jz1 l( pz 2 ) + O C 2r(pn N 1 ). + O(p γ N r 1 ) for γ < 0 and the (0.13) K jz1 l( pz 2 )η p r = O C r (p 2 η p/2 ) K jz1 l( pz 2 )=O C r (p l 2 Z2 l r ).

5 A REMARK ON WEIGHTED BERGMAN KERNELS ON ORBIFOLDS 147 In order to prove (0.7) t s sffcent to show that for 0 l N + r r 2r (0.14) w lp := c l λ η p+ = O C r (pl N r 1+ r 2 η p/2 ). r In fact we wll prove that w lp = O C r (pl N r 1+ Snce dw lp = dη η absorbed by η 1 8r p prove w lp = O C 0(p l N r 1 η 3 4 p (0.15) 2 η ( 3 4 r 8r )p ) for r 2r. (pw lp + w l+1p ) and dη η has a term z 2 or z 2 whch can be to get a factor p 1/2 we see by ndcton that t s sffcent to ). To ths end wrte [ w lp = c l λ η ] (η 1) N+r l+1 (η 1) N+r l+1 η. p Snce λ s a prmtve mth root of nty λ 1f {1...m 1}. From [4 Lemma 3.5] nder the condton (0.5) the fncton η c l λ η has a root of order N + r l +1atη = 1 and so the term n sqare brackets s bonded. For z 2 ε we have by (0.10) (0.16) η 1 C z 2 2. By sng (0.10) and (0.16) and the fact that [0 ) x x s e x s bonded for any s 0 we get (0.17) (η 1) s η p/4 = O(p s ) for s 1. Ths w lp = O C 0(p l N r 1 η 3 4 p ) and (0.14) follows. Back n (0.12) for q>n+ r the correspondng contrbton s certanly p n j 2 q O C 2r(p r η p/2 )=O C 2r(p n N 1 η p/2 ) by (0.13). On the other hand f 0 q N + r then by (0.13) and (0.14) the correspondng contrbton s p n j 2 q O C 2r(p q N r 1+r (1 + p Z 2 ) 6N+6r+3 η p/2 )=O C 2r(p n N 1 η p/4 ) agan. Ths S = O C 2r(p n N 1 ). From (0.10) and the above argment S 2 = O C 2r(p n N 1 ). Combnng wth (0.10) (0.11) and (0.13) we get (0.7) and (0.8). References [1] X. Da K. LandX. Ma On the asymptotc expanson of Bergman Kernel J. Dfferental Geom. 72 (2006) [2] X. Ma and G. Marnesc Holomorphc Morse neqaltes and Bergman kernels progress n mathematcs 254 Brkhäser Boston Inc. Boston MA pp. [3] X. Ma and G. Marnesc Generalzed Bergman kernels on symplectc manfolds Adv. Math. 217(4) (2008) [4] J. Ross and R. Thomas Weghted Bergman kernels on Orbfolds J. Dfferental Geom. 88 (2011) [5] J. Ross and R. Thomas Weghted projectve embeddngs stablty of orbfolds and constant scalar crvatre Kähler metrcs J. Dfferental Geom. 88 (2011) Department of Mathematcs UCSB CA USA E-mal address: da@math.csb.ed

6 148 XIANZHE DAI KEFENG LIU AND XIAONAN MA Center of Mathematcal Scence Zhejang Unversty and Department of Mathematcs UCLA CA USA E-mal address: Unversté Pars Dderot - Pars 7 UFR de Mathématqes Case 7012 Ste Chevaleret Pars Cedex 13 France E-mal address: ma@math.jsse.fr