Remark on the Off-Diagonal Expansion of the Bergman Kernel on Compact Kähler Manifolds

Original Article

Abstract

In this short note, we compare our previous work on the off-diagonal expansion of the Bergman kernel and the preprint of Lu–Shiffman (arXiv:1301.2166). In particular, we note that the vanishing of the coefficient of p−1/2 is implicitly contained in Dai–Liu–Ma’s work (J. Differ. Geom. 72(1), 1–41, 2006) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels. Progress in Math., vol. 254, 2007).

Keywords

Kähler manifold Bergman kernel of a positive line bundle 

Mathematics Subject Classification (2010)

53C55 53C21 53D50 58J60 

In this short note we revisit the calculations of some coefficients of the off-diagonal expansion of the Bergman kernel from our previous work [4, 5].

Let (X,ω) be a compact Kähler manifold of \(\dim_{\mathbb {C}}X=n\) with Kähler form ω. Let (L,hL) be a holomorphic Hermitian line bundle on X, and let (E,hE) be a holomorphic Hermitian vector bundle on X. Let ∇L, ∇E be the holomorphic Hermitian connections on (L,hL), (E,hE) with curvatures RL=(∇L)2, RE=(∇E)2, respectively. We assume that (L,hL,∇L) is a prequantum line bundle, i.e., \(\omega= \frac{\sqrt{-1}}{2 \pi} R^{L}\). Let Pp(x,x′) be the Bergman kernel of LpE with respect to hL,hE and the Riemannian volume form dvX=ωn/n!. This is the integral kernel of the orthogonal projection from Open image in new window to the space of holomorphic sections H0(X,LpE) (cf. [4, §4.1.1]).

We fix x0X. We identify the ball \(B^{T_{x_{0}}X}(0,\varepsilon)\) in the tangent space \(T_{x_{0}}X\) to the ball BX(x0,ε) in X by the exponential map (cf. [4, §4.1.3]). For \(Z\in B^{T_{x_{0}}X}(0,\varepsilon)\) we identify \((L_{Z}, h^{L}_{Z})\), \((E_{Z}, h^{E}_{Z})\) to \((L_{x_{0}},h^{L}_{x_{0}})\), \((E_{x_{0}},h^{E}_{x_{0}})\) by parallel transport with respect to the connections ∇L, ∇E along the curve \(\gamma_{Z} :[0,1]\ni u \to\exp^{X}_{x_{0}} (uZ)\). Then Pp(x,x′) induces a smooth section \((Z,Z')\mapsto P_{p,x_{0}}(Z,Z')\) of \(\pi^{*} \operatorname{End}(E)\) over {(Z,Z′)∈TX×XTX:|Z|,|Z′|<ε}, which depends smoothly on x0, with π:TX×XTXX the natural projection. If dvTX is the Riemannian volume form on \((T_{x_{0}}X, g^{T_{x_{0}}X})\), there exists a smooth positive function \(\kappa_{x_{0}}:T_{x_{0}}X\to \mathbb {R}\), defined by
$$ dv_X(Z)= \kappa_{x_0}(Z) dv_{TX}(Z),\quad\kappa_{x_0}(0)=1. $$
(1)
For \(Z\in T_{x_{0}}X\cong \mathbb {R}^{2n}\), we denote \(z_{j}=Z_{2j-1}+\sqrt{-1} Z_{2j-1}\) its complex coordinates, and set

The near off-diagonal asymptotic expansion of the Bergman kernel in the form established [4, Theorem 4.1.24] is the following.

Theorem 1

Given\(k,m'\in \mathbb {N}\), σ>0, there existsC>0 such that ifp⩾1, x0X, \(Z,Z'\in T_{x_{0}}X\), \(|Z|,|Z'| \leqslant \sigma/\sqrt{p}\), whereOpen image in new windowis theOpen image in new window-norm with respect to the parameterx0, \(J_{r}(Z,Z')\in \operatorname{End}(E)_{x_{0}}\)are polynomials inZ,Zwith the same parity asrand degJr(Z,Z′)⩽3r, whose coefficients are polynomials inRTX (the curvature of the Levi-Civita connection onTX), REand their derivatives of orderr−2.

Remark 2

For the above properties of Jr(Z,Z′) see [4, Theorem 4.1.21 and end of §4.1.8]. They are also given in [2, Theorem 4.6, (4.107) and (4.117)]. Moreover, by [4, (1.2.19) and (4.1.28)], κ has a Taylor expansion with coefficients the derivatives of RTX. As in [4, (4.1.101)] or [5, Lemma 3.1 and (3.27)] we have Note that a more powerful result than the near-off diagonal expansion from Theorem 1 holds. Namely, by [2, Theorem 4.18′] and [4, Theorem 4.2.1], the full off-diagonal expansion of the Bergman kernel holds (even for symplectic manifolds), i.e., an analogous result to (3) for |Z|,|Z′|⩽ε. This appears naturally in the proof of the diagonal expansion of the Bergman kernel on orbifolds in [2, (5.25)] or [4, (5.4.14), (5.4.23)].

Proposition 3

The coefficientOpen image in new windowvanishes identically: Open image in new windowfor allZ,Z′. Therefore the coefficient ofp−1/2in the expansion ofpnPp(p−1/2Z,p−1/2Z′) vanishes, so the latter converges toOpen image in new windowat ratep−1asp→∞.

Proof

This is [4, Remark 4.1.26] or [5, (2.19)], see also [2, (4.107), (4.117), (5.4)]. □

When \(E=\mathbb {C}\) with trivial metric, the vanishing of Open image in new window was recently rediscovered in [3, Theorem 2.1] (b1(u,v)=0 therein). In [3] an equivalent formulation [6] of the expansion (3) is used, based on the analysis of the Szegö kernel from [1]. In [3, Theorem 2.1] further off-diagonal coefficients Open image in new window, Open image in new window, Open image in new window are calculated in the K-coordinates. From [5, (3.22)], we see that the usual normal coordinates are K-coordinates up to order at least 3. This shows that the vanishing of \(\mathcal{F}_{1}\) given by Proposition 3 implies the vanishing of b1 calculated with the help of in K-coordinates. We wish to point out that we calculated in [5] the coefficients Open image in new window on the diagonal, using the off-diagonal expansion (3) and evaluating Open image in new window for Z=Z′=0. Thus, off-diagonal formulas for Open image in new window are implicitly contained in [5]. We show below how the coefficient Open image in new window can be calculated in the framework of [5].

We use the notation in [5, (3.6)], then \(\boldsymbol {r}=8R_{m\overline{q}q\overline{m}}\) is the scalar curvature.

Proposition 4

The coefficientJ2in (4) is given by

Remark 5

Setting Z=Z′=0 in (6) we obtain the coefficient \(\boldsymbol{b}_{1}(x_{0})=J_{2}(0,0)=\frac{1}{8\pi} \boldsymbol {r}+\frac{1}{\pi}R^{E}_{q\overline{q}}\) of p−1 of the (diagonal) expansion of pnPp(x0,x0), cf. [4, Theorem 4.1.2].

Moreover, in order to obtain the coefficient of p−1 in the expansion (3) we multiply Open image in new window to the expansion of κ(Z)−1/2κ(Z′)−1/2 with respect to the variable \(\sqrt{p}Z\) obtained from (5). If \(E=\mathbb {C}\) the result is a polynomial which is the sum of a homogeneous polynomial of order four and a constant, similar to [3].

Proof of Proposition 4

Set By [4, (4.1.107)] or [5, (2.19)], we have By [5, (4.1a), (4.7)] we have By the symmetry properties of the curvature [5, Lemma 3.1] we have We use throughout that \([g(z,\overline{z}),b_{j}]=2\frac{\partial}{\partial z_{j}}g(z,\overline {z})\) for any polynomial \(g(z,\overline{z})\) (cf. [5, (1.7)]). Hence from (10), we get Thus from (7) and (11), we get Now, Open image in new window, see [4, (4.1.108)] or [5, (4.2)]. Therefore We know that for an operator T we have \(T^{*}(Z,Z')=\overline {T(Z',Z)}\), thus

We have Open image in new window by [4, Theorem 4.1.8], so from (13) and (14), we obtain the factor of \(R_{k\overline{m}\ell\overline{q}}\) in (6).

Let us calculate the contribution of the last term (curvature of E). We have and by (10), we also have The contribution to J2 of the term on E is thus given by the last two terms in (6). □

Notes

Acknowledgements

X. Ma partially supported by Institut Universitaire de France. G. Marinescu partially supported by DFG funded projects SFB/TR 12 and MA 2469/2-2.

References

  1. 1.
    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. In: Journées Équations aux Dérivées Partielles de Rennes (1975). Astérisque, vol. 34–35, pp. 123–164. Soc. Math. France, Paris (1976) Google Scholar
  2. 2.
    Dai, X., Liu, K., Ma, X.: On the asymptotic expansion of Bergman kernel. J. Differ. Geom. 72(1), 1–41 (2006). Announced in C. R. Math. Acad. Sci. Paris 339(3), 193–198 (2004) MATHMathSciNetGoogle Scholar
  3. 3.
    Lu, Z., Shiffman, B.: Asymptotic expansion of the off-diagonal Bergman kernel on compact Kähler manifolds. Preprint arXiv:1301.2166
  4. 4.
    Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Math., vol. 254, Birkhäuser, Basel (2007), xiii, 422 p. MATHGoogle Scholar
  5. 5.
    Ma, X., Marinescu, G.: Berezin-Toeplitz quantization on Kähler manifolds. J. Reine Angew. Math. 662, 1–56 (2012) MATHMathSciNetGoogle Scholar
  6. 6.
    Shiffman, B., Zelditch, S.: Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544, 181–222 (2002) MATHMathSciNetGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.UFR de MathématiquesUniversité Paris Diderot - Paris 7Paris Cedex 13France
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany
  3. 3.Institute of Mathematics ‘Simion Stoilow’Romanian AcademyBucharestRomania

Personalised recommendations