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

Original Article


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).


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,h L ) be a holomorphic Hermitian line bundle on X, and let (E,h E ) be a holomorphic Hermitian vector bundle on X. Let ∇ L , ∇ E be the holomorphic Hermitian connections on (L,h L ), (E,h E ) with curvatures R L =(∇ L )2, R E =(∇ E )2, respectively. We assume that (L,h L ,∇ L ) is a prequantum line bundle, i.e., \(\omega= \frac{\sqrt{-1}}{2 \pi} R^{L}\). Let P p (x,x′) be the Bergman kernel of L p E with respect to h L ,h E and the Riemannian volume form dv X =ω n /n!. This is the integral kernel of the orthogonal projection from Open image in new window to the space of holomorphic sections H 0(X,L p E) (cf. [4, §4.1.1]).

We fix x 0X. We identify the ball \(B^{T_{x_{0}}X}(0,\varepsilon)\) in the tangent space \(T_{x_{0}}X\) to the ball B X (x 0,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 P p (x,x′) induces a smooth section \((Z,Z')\mapsto P_{p,x_{0}}(Z,Z')\) of \(\pi^{*} \operatorname{End}(E)\) over {(Z,Z′)∈TX× X TX:|Z|,|Z′|<ε}, which depends smoothly on x 0, with π:TX× X TXX the natural projection. If dv TX 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
$$ dv_X(Z)= \kappa_{x_0}(Z) dv_{TX}(Z),\quad\kappa_{x_0}(0)=1. $$
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 exists C>0 such that if p⩾1, x 0X, \(Z,Z'\in T_{x_{0}}X\), \(|Z|,|Z'| \leqslant \sigma/\sqrt{p}\), where Open image in new window is the Open image in new window -norm with respect to the parameter x 0, \(J_{r}(Z,Z')\in \operatorname{End}(E)_{x_{0}}\) are polynomials in Z,Zwith the same parity as r and degJ r (Z,Z′)⩽3r, whose coefficients are polynomials in R TX (the curvature of the Levi-Civita connection on TX), R E and their derivatives of orderr−2.

Remark 2

For the above properties of J r (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 R TX . 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 coefficient Open image in new window vanishes identically: Open image in new window for all Z,Z′. Therefore the coefficient of p −1/2 in the expansion of p n P p (p −1/2 Z,p −1/2 Z′) vanishes, so the latter converges to Open image in new window at rate p −1 as p→∞.


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] (b 1(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 b 1 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 coefficient J 2 in (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 p n P p (x 0,x 0), 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)−1/2 κ(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 J 2 of the term on E is thus given by the last two terms in (6). □



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.


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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

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