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

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## 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 58J60In 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]).

*x*

_{0}∈

*X*. 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 }

*TX*→

*X*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

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*

_{0}∈

*X*, \(Z,Z'\in T_{x_{0}}X\), \(|Z|,|Z'| \leqslant \sigma/\sqrt{p}\),

*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*,

*Z*′

*with the same parity as*

*r*

*and*deg

*J*

_{ r }(

*Z*,

*Z*′)⩽3

*r*,

*whose coefficients are polynomials in*

*R*

^{ TX }(

*the curvature of the Levi*-

*Civita connection on*

*TX*),

*R*

^{ E }

*and their derivatives of order*⩽

*r*−2.

## Remark 2

*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*→∞.

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

## 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

*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).

## 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

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