In this note we verify certain statement about the operator $Q\_K$
constructed by Donaldson in [3] by using the full asymptotic expansion of
Bergman kernel obtained in [2] and [4].Comment: 7 pages, modified the relation on $K\_p$ and $K\_{\omega,p}

Liu, Kefeng

Ma, Xiaonan

English

arXiv

7 pages, modified the relation on $K_p$ and $K_{\omega,p}$.In this note we verify certain statement about the operator $Q_K$ constructed by Donaldson in [3] by using the full asymptotic expansion of Bergman kernel obtained in [2] and [4]

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A remark on ’Some numerical results in complexdifferential geometry’Kefeng Liu, Xiaonan MaTo cite this version:Kefeng Liu, Xiaonan Ma. A remark on ’Some numerical results in complex differential geome-try’. 7 pages, modified the relation on K p and K . 2006. <hal-00016826v2>HAL Id: hal-00016826https://hal.archives-ouvertes.fr/hal-00016826v2Submitted on 18 Jan 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00016826, version 2 - 18 Jan 2006A REMARK ON ’SOME NUMERICAL RESULTS IN COMPLEXDIFFERENTIAL GEOMETRY’KEFENG LIU AND XIAONAN MAAbstract. In this note we verify certain statement about the operatorQK constructedby Donaldson in [3] by using the full asymptotic expansion of Bergman kernel obtainedin [2] and [4].In order to find explicit numerical approximation of Ka¨hler-Einstein metric of projec-tive manifolds, Donaldson introduced in [3] various operators with good properties toapproximate classical operators. See the discussions in Section 4.2 of [3] for more detailsrelated to our discussion. In this note we verify certain statement of Donaldson aboutthe operator QK in Section 4.2 by using the full asymptotic expansion of Bergman kernelderived in [2, Theorem 4.18] and [4, §3.4]. Such statement is needed for the convergenceof the approximation procedure.Let (X,ω, J) be a compact Ka¨hler manifold of dimC X = n, and let (L, hL) be a holo-morphic Hermitian line bundle on X. Let ∇L be the holomorphic Hermitian connectionon (L, hL) with curvature RL. We assume that√−12piRL = ω.(1)Let gTX(·, ·) := ω(·, J ·) be the Riemannian metric on TX induced by ω, J . Let dvXbe the Riemannian volume form of (TX, gTX), then dvX = ωn/n!. Let dν be any volumeform on X. Let η be the positive function on X defined bydvX = η dν.(2)The L2–scalar product 〈 〉ν on C∞(X,Lp), the space of smooth sections of Lp, is givenby〈σ1, σ2〉ν :=∫X〈σ1(x), σ2(x)〉Lp dν(x) .(3)Let Pν,p(x, x′) (x, x′ ∈ X) be the smooth kernel of the orthogonal projection from(C∞(X,Lp), 〈 〉ν) onto H0(X,Lp), the space of the holomorphic sections of Lp on X,with respect to dν(x′). Note that Pν,p(x, x′) ∈ Lpx ⊗ Lp∗x′ . Following [3, §4], setKp(x, x′) := |Pν,p(x, x′)|2hLpx ⊗hLp∗x′ , Rp := (dimH0(X,Lp))/Vol(X, ν),(4)here Vol(X, ν) :=∫Xdν. Set Vol(X, dvX) :=∫XdvX .Let QKp be the integral operator associated to Kp which is defined for f ∈ C∞(X),(5) QKp(f)(x) :=1Rp∫XKp(x, y)f(y)dν(y).Date: January 18, 2006.12 KEFENG LIU AND XIAONAN MALet ∆ be the (positive) Laplace operator on (X, gTX) acting on the functions on X.We denote by | |L2 the L2-norm on the function on X with respect to dvX .Theorem 1. There exists a constant C > 0 such that for any f ∈ C∞(X), p ∈ N,∣∣∣∣(QKp − Vol(X, ν)Vol(X, dvX)η exp(− ∆4pip))f∣∣∣∣L2≤ Cp|f |L2 ,∣∣∣∣(∆p QKp − Vol(X, ν)Vol(X, dvX)∆pη exp(− ∆4pip))f∣∣∣∣L2≤ Cp|f |L2 .(6)Moreover, (6) is uniform in that there is an integer s such that if all data hL, dν runover a set which are bounded in C s and that gTX, dvX are bounded from below, then theconstant C is independent of hL, dν.Proof. We explain at first the full asymptotic expansion of Pν,p(x, x′) from [2, Theorem4.18′] and [4, §3.4]. For more details on our approach we also refer the readers to therecent book [5].Let E = C be the trivial holomorphic line bundle on X. Let hE the metric on Edefined by |1|2hE = 1, here 1 is the canonical unity element of E. We identify canonicallyLp to Lp ⊗E by Section 1.As in [4, §3.4], let hEω be the metric on E defined by |1|2hEω = η−1, here 1 is the canonicalunity element of E. Let 〈 〉ω be the Hermitian product on C∞(X,Lp⊗E) = C∞(X,Lp)induced by hL, hEω , dvX as in (3). Then by (2),(7) (C∞(X,Lp ⊗ E), 〈 〉ω) = (C∞(X,Lp), 〈 〉ν).Observe that H0(X,Lp⊗E) does not depend on gTX, hL or hE. If Pω,p(x, x′), (x, x′ ∈X) denotes the smooth kernel of the orthogonal projection Pω,p from (C∞(X,Lp ⊗E), 〈 ·, ·〉ω) onto H0(X,Lp ⊗ E) = H0(X,Lp) with respect to dvX(x), from (2), as in[4, (3.38)], we have(8) Pν,p(x, x′) = η(x′)Pω,p(x, x′).For f ∈ C∞(X), setKω,p(x, x′) = |Pω,p(x, x′)|2(hLp⊗hEω )x⊗(hLp∗⊗hE∗ω )x′ ,(Kω,pf)(x) =∫XKω,p(x, y)f(y)dvX(y).(9)By the definition of the metric hE , hEω , if we denote by 1∗ the dual of the section 1 ofE, we know(10) 1 = |1⊗ 1∗|2hE⊗hE∗(x, x′) = |1⊗ 1∗|2hEω⊗hE∗ω (x, x′)η(x)η−1(x′).Recall that we identified (Lp, hLp) to (Lp ⊗ E, hLp ⊗ hE) by Section 1. Thus from (4),(8) and (10), we get(11) Kp(x, x′) = |Pν,p(x, x′)|2(hLp⊗hE)x⊗(hLp∗⊗hE∗)x′ = η(x) η(x′)Kω,p(x, x′),and from (2), (5) and (11),(12) QKp(f)(x) =1Rp∫XKω,p(x, y)η(x)f(y)dvX(y).A REMARK ON ’SOME NUMERICAL RESULTS IN COMPLEX DIFFERENTIAL GEOMETRY’ 3Now for the kernel Pω,p(x, x′), we can apply the full asymptotic expansion [2, Theorem4.18′]. In fact let ∂Lp⊗E,∗ωbe the formal adjoint of the Dolbeault operator ∂Lp⊗Eon theDolbeault complex Ω0,•(X,Lp⊗E) with the scalar product induced by gTX, hL, hEω , dvXas in (3), and set(13) Dp =√2(∂Lp⊗E+ ∂Lp⊗E,∗ω).Then H0(X,Lp ⊗ E) = KerDp for p large enough, and Dp is a Dirac operator, asgTX(·, ·) = ω(·, J ·) is a Ka¨hler metric on TX.Let ∇E be the holomorphic Hermitian connection on (E, hEω ). Let ∇TX be the Levi-Civita connection on (TX, gTX). Let RE, RTX be the corresponding curvatures.Let aX be the injectivity radius of (X, gTX). We fix ε ∈]0, aX/4[. We denote byBX(x, ε) and BTxX(0, ε) the open balls in X and TxX with center x and radius ε. Weidentify BTxX(0, ε) with BX(x, ε) by using the exponential map of (X, gTX).We fix x0 ∈ X. For Z ∈ BTx0X(0, ε) we identify (LZ , hLZ), (EZ , hEZ) and (Lp ⊗ E)Zto (Lx0 , hLx0), (Ex0, hEx0) and (Lp ⊗ E)x0 by parallel transport with respect to the con-nections ∇L, ∇E and ∇Lp⊗E along the curve γZ : [0, 1] ∋ u → expXx0(uZ). Thenunder our identification, Pω,p(Z,Z′) is a function on Z,Z ′ ∈ Tx0X, |Z|, |Z ′| ≤ ε, wedenote it by Pω,p,x0(Z,Z′). Let pi : TX ×X TX → X be the natural projection fromthe fiberwise product of TX on X. Then we can view Pω,p,x0(Z,Z′) as a smooth func-tion on TX ×X TX (which is defined for |Z|, |Z ′| ≤ ε) by identifying a section S ∈C∞(TX×X TX, pi∗End(E)) with the family (Sx)x∈X , where Sx = S|pi−1(x), End(E) = C.We choose {wi}ni=1 an orthonormal basis of T (1,0)x0 X, then e2j−1 = 1√2(wj + wj) ande2j =√−1√2(wj − wj) , j = 1, . . . , n forms an orthonormal basis of Tx0X. We use thecoordinates on Tx0X ≃ R2n where the identification is given by(14) (Z1, · · · , Z2n) ∈ R2n −→∑iZiei ∈ Tx0X.In what follows we also introduce the complex coordinates z = (z1, · · · , zn) on Cn ≃R2n. By [2, (4.114)] (cf. [4, (1.91)]), setPN(Z,Z ′) = exp(− pi2∑i(|zi|2 + |z′i|2 − 2ziz′i)).(15)Then PN is the classical Bergman kernel on Cn (cf. [4, Remark 1.14]) and(16) |PN(Z,Z ′)|2 = e−pi|Z−Z′|2.By [2, Proposition 4.1], for any l,m ∈ N, ε > 0, there exists Cl,m,ε > 0 such that forp ≥ 1, x, x′ ∈ X,|Pω,p(x, x′)|Cm(X×X) ≤ Cl,m,ε p−l if d(x, x′) ≥ ε.(17)Here the Cm-norm is induced by ∇L, ∇E , ∇TX and hL, hE , gTX.By [2, Theorem 4.18′], there exist Jr(Z,Z ′) polynomials in Z,Z ′, such that for anyk,m,m′ ∈ N, there exist N ∈ N, C > 0, C0 > 0 such that for α, α′ ∈ Nn, |α|+ |α′| ≤ m,4 KEFENG LIU AND XIAONAN MAZ,Z ′ ∈ Tx0X, |Z|, |Z ′| ≤ ε, x0 ∈ X, p ≥ 1,(18)∣∣∣∣∣ ∂|α|+|α′|∂Zα∂Z ′α′(1pnPω,p,x0(Z,Z′)−k∑r=0(JrPN)(√pZ,√pZ ′)p−r/2)∣∣∣∣∣Cm′ (X)≤ Cp−(k+1−m)/2(1 + |√pZ|+ |√pZ ′|)N exp(−C0√p|Z − Z ′|) + O(p−∞).Here Cm′(X) is the Cm′norm for the parameter x0 ∈ X. The term O(p−∞) meansthat for any l, l1 ∈ N, there exists Cl,l1 > 0 such that its C l1-norm is dominated byCl,l1p−l. (In fact, by [2, Theorems 4.6 and 4.17, (4.117)] (cf. [4, Theorem 1.18, (1.31)]),the polynomials Jr(Z,Z′) have the same parity as r and deg Jr(Z,Z ′) ≤ 3r, whosecoefficients are polynomials in RTX , RE and their derivatives of order 6 r − 1).Now we claim that in (18),(19) J0 = 1, J1(Z,Z′) = 0.In fact, let dvTx0X be the Riemannian volume form on (Tx0X, gTx0X), and κ be thefunction defined by(20) dvX(Z) = κ(x0, Z)dvTx0X(Z).Then (also cf. [4, (1.31)])(21) κ(x0, Z) = 1 +16〈RTXx0 (Z, ei)Z, ei〉x0+ O(|Z|3).As we only work on C∞(X,Lp ⊗E), by [2, (4.115)], we get the first equation in (19).Recall that in the normal coordinate, after the rescaling Z → Z/t with t = 1√p, we getan operator Lt from the restriction of D2p on C∞(X,Lp ⊗ E) which has the followingformal expansion (cf. [2, (1.104)], [4, Theorem 1.4]),(22) Lt = L +∞∑r=1Qrtr.Now, from [2, Theorem 5.1] (or [4, (1.87), (1.97)]),L =n∑j=1(−2 ∂∂zi+ pizi)(2∂∂zi+ pizi), Q1 = 0.(23)(In fact, PN(Z,Z ′) is the smooth kernel of the orthogonal projection from L2(R) ontoKer(L )). Thus from [2, (4.107)] (cf. [4, (1.111)]), (21) and (23) we get the secondequation of (19).Note that |Pω,p,x0(Z,Z ′)|2 = Pω,p,x0(Z,Z ′)Pω,p,x0(Z,Z ′), thus from (9), (18) and (19),there exist J ′r(Z,Z′) polynomials in Z,Z ′ such that(24)∣∣∣∣∣ 1p2n+1∆Z(Kω,p,x0(Z,Z′)−(1 +k∑r=2p−r/2J ′r(√pZ,√pZ ′))e−pip|Z−Z′|2)∣∣∣∣∣≤ Cp−(k+1)/2(1 + |√pZ|+ |√pZ ′|)N exp(−C0√p|Z − Z ′|) + O(p−∞).For a function f ∈ C∞(X), we denote it as f(x0, Z) a family (with parameter x0) offunction on Z in the normal coordinate near x0. Now, for any polynomial Qx0(Z′), weA REMARK ON ’SOME NUMERICAL RESULTS IN COMPLEX DIFFERENTIAL GEOMETRY’ 5define the operator(Qpf)(x0) = pn∫|Z′|≤εQx0(√pZ ′)e−pip|Z′|2f(x0, Z ′)dvX(x0, Z ′).(25)Then we observe that there exists C1 > 0 such that for any p ∈ N, f ∈ C∞(X), we have|Qpf |L2 ≤ C1|f |L2.(26)In fact,(27) |Qpf |2L2 ≤∫XdvX(x0){pn(∫|Z′|≤ε|Qx0(√pZ ′)|e−pip|Z′|2dvX(x0, Z ′))× pn(∫|Z′|≤ε|Qx0(√pZ ′)|e−pip|Z′|2|f(x0, Z ′)|2dvX(x0, Z ′))}≤ C ′∫XdvX(x0)pn∫|Z′|≤ε|Qx0(√pZ ′)|e−pip|Z′|2|f(x0, Z ′)|2dvX(x0, Z ′)≤ C1|f |2L2.Observe that in the normal coordinate, at Z = 0, ∆Z = −∑2nj=1∂2∂Z2j. Thus(28) (∆Ze−pip|Z−Z′|2)|Z=0 = 4pip(n− pip|Z ′|2)e−pip|Z′|2.Thus from (16), (18), (19), (24) and (26), we get∣∣∣∣p−nKω,pf − pn∫|Z′|≤εe−pip|Z′|2f(x0, Z′)dvX(x0, Z′)∣∣∣∣L2≤ Cp|f |L2 ,∣∣∣∣p−n−1∆Kω,pf − 4pipn∫|Z′|≤ε(n− pip|Z ′|2)e−pip|Z′|2f(x0, Z ′)dvX(x0, Z ′)∣∣∣∣L2≤ Cp|f |L2 .(29)SetKη,ω,p(x, y) = 〈dη(x), dxKω,p(x, y)〉gT∗X ,(Kη,ω,pf)(x) =∫XKη,ω,p(x, y)f(y)dvX(y).(30)Then from (18), (19) and (26), we get∣∣∣∣∣p−n−1Kη,ω,pf − 2pipn∫|Z′|≤ε2n∑i=1(∂∂Ziη)(x0, 0)Z′ie−pip|Z′|2f(x0, Z ′)dvX(x0, Z ′)∣∣∣∣∣L2≤ Cp|f |L2 .(31)Let e−u∆(x, x′) be the smooth kernel of the heat operator e−u∆ with respect to dvX(x′).Let d(x, y) be the Riemannian distance from x to y on (X, gTX). By the heat kernelexpansion in [1, Theorems 2.23, 2.26], there exist Φi(x, y) smooth functions on X × Xsuch that when u→ 0, we have the following asymptotic expansion∣∣∣∣∣ ∂l∂ul(e−u∆(x, y)− (4piu)−nk∑i=0uiΦi(x, y)e− 14ud(x,y)2)∣∣∣∣∣Cm(X×X)= O(uk−n−l−m2+1),(32)6 KEFENG LIU AND XIAONAN MAandΦ0(x, y) = 1.(33)If we still use the normal coordinate, then by (32), there exist φi,x0(Z′) := Φi(0, Z ′)such that uniformly for x0 ∈ X, Z ′ ∈ Tx0X, |Z ′| ≤ ε, we have the following asymptoticexpansion when u→ 0,∣∣∣∣∣ ∂l∂ul(e−u∆(0, Z ′)− (4piu)−n(1 +k∑i=1uiφi,x0(Z′))e−14u|Z′|2)∣∣∣∣∣ = O(uk−n−l+1),(34)and(35)∣∣∣〈dη(x0), dx0e−u∆〉gT∗X (0, Z ′)− (4piu)−n2n∑i=1(∂∂Ziη)(x0, 0)Z ′i2u(1 +k∑i=1uiφi,x0(Z′)))e−14u|Z′|2− (4piu)−nk∑i=1ui〈dη(x0), (dx0Φi)(0, Z ′)〉e−14u|Z′|2∣∣∣ = O(uk−n+ 12 ).Observe that1p∆exp(− ∆4pip)= −1p( ∂∂ue−u∆)|u= 14pip.(36)Now from (26), (29)–(36), we get∣∣∣∣(p−nKω,p − exp(− ∆4pip))f∣∣∣∣L2≤ Cp|f |L2 ,∣∣∣∣1p(p−n∆Kω,p −∆exp(− ∆4pip))f∣∣∣∣L2≤ Cp|f |L2 .(37)and ∣∣∣∣1p(p−nKη,ω,p − 〈dη, d exp(− ∆4pip)〉)f∣∣∣∣L2≤ Cp|f |L2 .(38)Note that(39) (∆ηKω,p)(x, y) = (∆η)(x)Kω,p(x, y) + η(x)∆xKω,p(x, y)− 2〈dη(x), dxKω,p(x, x′)〉gT∗X ,and Rp =Vol(X,dvX )Vol(X,ν)pn + O(pn−1). From (12), (37)-(39), we get (6).To get the last part of Theorem 1, as we noticed in [2, §4.5], the constants in (18)will be uniformly bounded under our condition, thus we can take C in (6), (37)and (38)independent of hL, dν. Acknowledgments. We thank Professor Simon Donaldson for useful communications.A REMARK ON ’SOME NUMERICAL RESULTS IN COMPLEX DIFFERENTIAL GEOMETRY’ 7References[1] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundl. Math. Wiss.Band 298, Springer-Verlag, Berlin, 1992.[2] X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, C. R. Math.Acad. Sci. Paris 339 (2004), no. 3, 193–198. The full version: J. Differential Geom. to appear,math.DG/0404494.[3] S. K. Donaldson, Some numerical results in complex differential geometry, math.DG/0512625.[4] X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad.Sci. Paris 339 (2004), no. 7, 493–498. The full version: math.DG/0411559.[5] , Holomorphic Morse Inequalities and Bergman Kernels, book in preparation, (2006).Center of Mathematical Science, Zhejiang University and Department of Mathemat-ics, UCLA, CA 90095-1555, USA (liu@math.ucla.edu)Centre de Mathe´matiques Laurent Schwartz, UMR 7640 du CNRS, Ecole Polytech-nique, 91128 Palaiseau Cedex, France (ma@math.polytechnique.fr)

A remark on 'Some numerical results in complex differential geometry'

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A remark on 'Some numerical results in complex differential geometry'

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