Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Given a smooth positive measure $\mu$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate version of the Bochner--Kodaira--Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be K\"ahler nor of"bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schr\"odinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the so-called"twisted basic identities"are standard basic identities with respect to conformally K\"ahler metrics.

(1.1) Under mild assumptions on µ, A 2 (M, µ) is closed in L 2 (M, µ) and actually a reproducing kernel Hilbert space (see Section 2.1). The Bergman kernel K µ : M × M → C is defined by the relation where B µ is the orthogonal projection of L 2 (M, µ) onto A 2 (M, µ). It has been shown (see, e.g., [Chr91,Del98,MOC09,Lin01,MM15,SV14,Dal15]) that, under various assumptions on µ, one can find a Hermitian metric h such that the following Agmon-type pointwise decay estimate holds: |K µ (p, q)|e −ψ(p)−ψ(q) C e −γd(p,q) Vol(p, 1) Vol(q, 1) (p, q ∈ M ). (1.3) Here d(p, q) is the Riemannian distance between p and q, Vol is the Riemannian volume, Vol(p, 1) is the volume of the ball centered at p and of radius 1, and ψ is determined by the relation µ = e −2ψ Vol. The positive constants C and γ do not depend on p and q. We point out that U ψ f := e −ψ f is a unitary isomorphism of L 2 (M, µ) onto L 2 (M, Vol), and U ψ • B µ • U −1 ψ is an orthogonal projector on L 2 (M, Vol). The left-hand side of (1.3) is thus the modulus of the integral kernel of this projector, and the estimate shows that this kernel exhibits an off-diagonal exponential decay, which can be neatly expressed in terms of the metric h. Estimates of the form (1.3) have had numerous applications in complex analysis and geometry (see e.g. [MM15] and [LZ16] and the references therein).
Typically, assumptions for (1.3) to hold can be formulated as conditions on the "curvature form" F µ of µ, which is defined as follows: in local holomorphic coordinates, one writes dµ = ie −2ϕ dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n , where ϕ is smooth and real-valued. Then one can easily check that F µ := i∂∂ϕ is a real (1, 1)-form which does not depend on the choice of the coordinates. Thus, F µ is globally defined and we shall call it the curvature form of µ.
We shall now proceed to describe some of the aforementioned results in a little more details.
(1) In the one-dimensional case M = C, µ = e −2ψ λ, where λ is Lebesgue measure and ψ is subharmonic, the curvature form F µ may be identified with (a multiple of) the measure ∆ψ λ, where ∆ is the usual Laplacian. It was shown by Christ [Chr91] (but see also [MOC09]) that if F µ is doubling and satisfies (2) In [Del98], Delin considers M = C n and µ = e −2ψ λ, where ψ is strictly plurisubharmonic, and proves an estimate that takes the form (1.3) with h the Kähler metric i∂∂ψ, at least when certain quantitative assumptions on F µ are made. These conditions are not explicitly discussed by Delin (see the comment after the statement of Theorem 2 of [Del98]), but it is certainly sufficient that ic∂∂|z| 2 F µ iC∂∂|z| 2 (1.6) holds for some 0 < c and C < +∞, as shown in [Lin01,Proposition 9].
(3) In [Dal15], the second-named author deals with M = C n and µ = e −2ψ λ, where ψ is only assumed to be weakly plurisubharmonic. More precisely, if ∆ψ is in the reverse-Hölder class RH ∞ , and where σ > 1 2 . One can see that F µ = i∂∂ψ − (n + 1)ω in this case (by (1.22) below and the fact that ω is Kähler-Einstein with Ricci curvature Θ = −2(n + 1)ω), and hence (1.10) is equivalent to for some c > −1/2 and C < +∞. (5) In [MM15], Ma and Marinescu prove a pointwise C k estimate for the Bergman kernels in the more general setting of Hermitian line bundles over symplectic manifolds (satisfying appropriate compatibility conditions). Specializing to the present situation, Theorem 1 in that paper requires in particular that the Hermitian manifold (M, h) has "bounded geometry", and that the measure µ = e −2ψ Vol is such that where Vol is the Riemannian volume and ω h the fundamental form) for c > 0 and C < +∞. Then, if k > 0 is large enough, the measure µ (k) = e −2k 2 ψ Vol satisfies with C independent of p, q, and k. Notice that the absence of the volume factors in (1.13) is due to the bounded geometry assumption. In fact, if the volumes of balls with a fixed positive radius is bounded away from zero (which is the case if the sectional curvature is bounded from above (by [GHL04, Theorem 3.101]) then the volume factors can be absorbed into the constant C.
These results, despite being of the same nature, present two different points of view on the problem of establishing exponential decay of Bergman kernels: (1) to (3) start with a measure µ and construct a metric h with respect to which the exponential decay (1.3) holds, while (4) and (5) start with a Hermitian manifold and look for conditions on the density of µ with respect to the Riemannian volume that are sufficient for (1.3) to hold. Moreover, in (1) to (3) a natural candidate for h is the Kähler metric with fundamental form F µ , but the latter form need not be positive, and in fact (1) and (3) consider a sort of regularization of F µ and the resulting metric is typically non-Kähler.
1.2. Our results. To state our results, we shall need to recall and fix some more notation. Let (M, h) be a complete Hermitian manifold with a Hermitian metric h: (1.14) The associated (1, 1)-form ω h := ih jk dz j ∧ dz k is called the fundamental form. As usual, we refer to both h and ω h as a metric on M . It is a classic fact that when h is not Kähler, then the torsion tensor T of the Chern connection is non-trivial: locally, T has components We shall deal with the torsion 1-form, obtained by taking the trace of the torsion: (1.17) The Riemannian metric g := 2 Re h induces a distance d h and a volume Vol. We denote by Vol(p, R) the volume of the metric ball B(p, R) := {q ∈ M : d(p, q) R} of radius R centered at p. By completeness, B(p, R) equals the set of points reached by geodesics starting at p after time at most R, whence we shall refer to these sets as geodesic balls.
As a Hermitian metric h induces inner products for tensors of all ranks, we can consider the space L 2 0,q (M, h, µ) of square-integrable (0, q)-forms on M , with inner product given by (1.18) We denote by ∂ * h,µ the Hilbert space adjoint of the (weak extension of) ∂ with respect to this inner product, and define the complex Laplacian associated to µ and h by This is an unbounded self-adjoint and nonnegative operator that encapsulates the interaction between µ and h. In this paper, we only consider h,µ acting on (0, 1)-forms. We say that in the sense of quadratic forms. We refer to section 2.2 for precise definitions.
We are finally in a position to state our main result.
Moreover, the coercivity condition ((i)) holds if the curvature form F µ satisfies for some σ > 1 2 . If T = 0, the conclusion still holds under the condition F µ 1 2 b 2 ω h .
To compare this result with existing ones in the literature (e.g. [Lin01, SV14, MM15]), it is useful to reformulate (1.21) in terms of the Chern-Ricci form Θ h and i∂∂ψ. Indeed, since Θ h = −i∂∂ log det(h jk ), it follows that if µ = e −2ψ Vol, then Hence, (1.21) is equivalent to (1.23) Observe that the assumptions in Theorem 1.1 are much simplified if h is Kähler. Indeed, if (M, h) is Kähler then T = 0 and hence (1.21) reduces to i∂∂ψ + 1 2 Θ b 2 ω h . On the other hand, since θ = 0 and the Chern-Ricci tensor is bounded from below, condition (2) is implied by the assumption that F µ = i∂∂ψ + 1 2 Θ Bω h < +∞. We obtain the following Corollary which is new already in this special (Kähler) case.
It is worth noticing that under the assumptions of Corollary 1.2, F µ is the fundamental form of a metric h µ that is "comparable" to h, and estimate (1.20) also holds with respect to h µ (with possibly a smaller constant γ).
Also note that when h is the flat metric on C n , the condition (1.24) is equivalent to (1.6), which is considered by Lindholm [Lin01].
It is clear that if h is Kähler, then the inequalities in (1.25) trivially hold for η > 0 arbitrary large, so that (1.27) holds for any γ < 2b (if k is large enough).
Let us discuss a bit the structure of our proof of Theorem 1.1. As a first step, we establish the following exponential decay of the canonical solutions of the ∂-equation, which could be of independent interest.

Theorem 1.4. Let (M, h) be a complete Hermitian manifold and assume that the smooth
Then for every q ∈ M and γ < 4b, the following bound holds: where C depends only on γ, b, and R. If in addition (M, h) has Ricci curvature bounded from below by K with K 0, and µ = e −2ψ Vol satisfies the condition then we have the pointwise bound (γ < 4b as above) where C depends only on γ, b, BR 2 , and R √ −K.
Notice that if u is ∂-closed, then the function f of the statement is the solution of the equation ∂f = u with minimal L 2 (M, µ) norm (see section 2.2 below), that is, the so-called canonical solution.
The first half of Theorem 1.4 states that, under the sole geometric assumption of completeness of (M, h), coercivity of h,µ implies the L 2 exponential decay (1.28) of ∂ * −1 h,µ u off the support of u. Its proof occupies section 3 and is based on a method developed by Agmon to establish exponential decay of eigenfunctions of Schrödinger operators (see, e.g., [Agm82]). The key observation is that h,µ satisfies a localization formula analogous to the simple yet very effective IMS localization formula of Schrödinger operators (see section 3.1).
In a second step, accomplished in section 4, we improve the L 2 decay to an L ∞ decay, exploiting a mean value inequality for nonnegative subsolutions of the heat equation on Riemannian manifolds due to Li and Tam [LT91] (but see also [LS84]), which holds under a lower bound on the Ricci curvature. To apply this inequality in the Hermitian context, we need to control the difference between the Laplacian of the background Riemannian metric and the Laplacian of the Chern connection, which may be expressed in terms of the torsion and ultimately leads to condition (1.29). Thanks to this mean value inequality, we can avoid the "Kerzman trick" (as in [Dal15]) and the "pluriharmonic recentering of the weight" techniques (as in [SV14]). These methods are difficult to implement on manifolds without some sort of "bounded geometry" assumptions.
The analysis just sketched has a conditional nature, resting on the assumption that h,µ is coercive (condition (1) in Theorem 1.1). This condition is made more transparent by a "basic identity" of the type which is typically associated to the names of Bochner, Kodaira, and Nakano in geometry (see, e.g., [Gri66,Dem86]), and Morrey, Kohn, and Hörmander in complex analysis (see, e.g., [Str10,CS01]). Thanks to this identity, we can give a condition that involves only the geometry of the Hermitian metric and the curvature form of the measure for the coercivity to hold (condition (1.21)). For the sake of completeness, in section 5 we give a short proof of the case of interest to us.
Finally, let us remark that Bergman kernels can be fruitfully defined in the more general setting where holomorphic functions are replaced by holomorphic sections of a holomorphic vector bundle on M endowed with a Hermitian metric (see [MM07] for a comprehensive treatment of this matter). We point out that most of our techniques work in this more general framework, but we confine ourselves to the scalar setting for the sake of simplicity.
1.3. Further directions. While the pointwise condition (1.21) is easy to check and sufficient to prove some interesting results, coercivity of h,µ is expected to hold under much weaker conditions (cf. [Dal17]). This is mainly due to the fact that, in loose terms, h,µ is a generalized Schrödinger operator, as made apparent by the basic identity of Proposition 5.2. Condition (1.21) is morally a uniform positive lower bound on the "potential" of h,µ , while coercivity amounts to positivity of the minimal eigenvalue: in the case of ordinary Schrödinger operators it is well-known that the latter condition is much weaker (see, e.g., [MS05]). This idea have an antecedent in [Chr91] and is considered in [Dal17], but, to the authors' knowledge, has never been explored in the general context of Hermitian manifolds (but see Theorem 3 of [Dev14] for a Riemannian counterpart).
We also believe that a better analytical understanding of the quadratic form of h,µ would allow an improvement of Corollary 1.2 in the same vein as the result of [SV14] for the unit ball (see the comment after the statement of the Corollary).

Preliminaries on Bergman kernels and the complex Laplacian on
Hermitian manifolds with measure 2.1. Bergman spaces and Bergman kernels. We recall that in the rather general setting of a complex manifold M equipped with a positive Borel measure µ, one may consider the Bergman space which is a linear subspace of L 2 (M, µ). While in complete generality this is not the case, for many kind of measures the evaluation maps f → f (p) are locally uniformly bounded linear functionals on A 2 (M, µ), i.e., for every compact This condition is sometimes called admissibility of the measure µ (see, e.g., [PW90] and [Zey13]). In this paper we restrict our attention to smooth positive measures, that is, measures having smooth positive density with respect to Lebesgue measure in local coordinates. It is a simple consequence of the mean value property of holomorphic functions that such measures always satisfy the admissibility condition (2.2). In any case, under assumption (2.2), the Bergman space is closed in L 2 (M, µ), so that the associated orthogonal projector is well-defined, and in fact A 2 (M, µ) is a reproducing kernel Hilbert space. Explicitly, there is a function which we call the Bergman kernel, that satisfies the following properties: Moreover, the following Cauchy-Schwarz type inequality holds: For proofs of these properties, see for instance [Ber70,PW90].

The complex Laplacian h,µ and its coercivity. A Hermitian manifold is a complex manifold M endowed with a Hermitian metric
The associated real (1, 1)-form ω h := ih jk dz j ∧ dz k is called the fundamental form (or Kähler form) of h. As usual, we refer to both h and ω h as a metric on M . A Hermitian scalar product ·, · h is induced in the usual way on cotangent spaces: in particular, if u = u j dz j and v k := v k . This Hermitian scalar product can be extended to tensors of all ranks and our convention for the case of covariant tensors of rank 2 is that whenever the η k 's are 1-forms. The associated norms will be denoted by | · | h . We identify differential forms with alternating tensors in such a way that η 1 ∧ η 2 := η 1 ⊗ η 2 − η 2 ⊗ η 1 , when η 1 and η 2 are 1-forms. With this definition, if u and v are (0, 1)-forms, we have Suppose µ is a smooth positive measure on M (we point out that most of the facts discussed below hold under much weaker regularity assumptions). We can define L 2 0,q (M, h, µ) as the Hilbert space of square-integrable (0, q)-forms with respect to µ and h. More explicitly, if u and v are (0, q)-forms, the scalar product on L 2 0,q (M, h, µ) has the expression´M u, v h dµ anticipated in section 1. We restrict our attention to q 2 (recall convention (2.8)). Observe that L 2 0,0 (M, h, µ) = L 2 (M, µ) is the standard L 2 -space of C-valued functions, defined with respect to the measure µ.
We define dom q (∂) := u ∈ L 2 0,q (M, h, µ) : ∂u ∈ L 2 0,q+1 (M, h, µ) , (2.10) where the ∂ in the formula above is to be taken in the sense of distributions (or, more precisely, currents). It is clear that ∂ defines an unbounded operator mapping L 2 0,q (M, h, µ) into L 2 0,q+1 (M, h, µ), whose domain is dom q (∂). This is called the weak extension of the differential operator ∂. We skip any reference in the notation to the degree of forms on which ∂ acts, since this should always be clear from the context. Putting all the operators together, we get the µ-weighted ∂-complex on (M, h): Notice that the operators above are closed, so that (2.11) is a Hilbert complex in the sense of [BL92] (closure follows immediately from the fact that convergence in L 2 0,q (M, h, µ) implies convergence in the sense of currents). Thus, we have the dual complex where every ∂ * h,µ is the Hilbert space adjoint of the corresponding ∂. We decided to use the slightly cumbersome notation ∂ * h,µ to stress the fact that not only the domains, but also the "formulas" of these first-order differential operators depend on the metric h and the measure µ.
We are finally in a position to define the complex Laplacian: The operator h,µ is self-adjoint and nonnegative when considered on the natural domain dom( (2.14) where we used the obvious notation for the domains of the ∂ * h,µ 's. One can analogously define For the purposes of this paper, it is enough to consider the complex Laplacian for q = 1, and we will consequently drop the superscript, putting h,µ := (1) h,µ . As usual, a key role is played by the quadratic form which is well defined whenever u, v ∈ dom 1 (∂) ∩ dom 1 (∂ * h,µ ) =: dom(E h,µ ). Notice that ∂ * h,µ u is a scalar function, while ∂u is a (0, 2)-form. We adopt the convention that E h,µ (u) := E h,µ (u, u). By definition, Our first restriction on the metric h is justified by the following Proposition: Proposition 2.1. If the Hermitian metric h is complete, the space D 0,1 of smooth compactly supported (0, 1)-forms is dense in dom(E h,µ ) with respect to the graph norm. It is also a core of h,µ , and the restriction of h,µ to D 0,1 is essentially self-adjoint.
Proof. See for instance [MM07]. The fact that we do not use the measure induced by the Hermitian metric is of no consequence, since we may rewrite µ = e −2ψ Vol and view E h,µ as the quadratic form of the complex Laplacian on (M, h, Vol) for forms with values in the trivial line bundle on M , with fiber metric given by e −2ψ . For our purposes, the most important consequence of this formula is the well-known Kohn's identity for the Bergman projection:

We say that h,µ is c-coercive
(2.20) See, e.g., [Str10,Ber16]. We point out that while the terms appearing on the right hand side of this identity depend on the metric h, the left hand side depends only on µ.

L 2 exponential decay of canonical solutions of the∂-equation
The goal of this section is to prove the first half of Theorem 1.4, that is, Theorem 3.4 below. In order to do that, we need a localization lemma and a Caccioppoli-type inequality.
3.1. A localization formula for h,µ . Lemma 3.2 below is a localization formula for h,µ that is analogous to the very useful IMS localization formula in the theory of Schrödinger operators. For the latter see, e.g., Lemma 3.1 of [Sim83] or Lemma 11.3 of [Tes14]. Before stating and proving it, we need a few preliminaries.
First, notice that if Lip(M, h) is the class of scalar functions χ : M → R that are Lipschitz with respect to the Riemannian distance, then by Rademacher's theorem, χ is almost everywhere differentiable and where L is the Lipschitz constant of χ. Next, we state the Leibniz rule for ∂ * h,µ for future reference. For this, we employ the notation w ∨ v for the interior product of the forms v and w (with respect to h). This is the form defined by the condition where u is an arbitrary form. Observe that the conjugation on the right hand side makes the interior product bilinear.
Lemma 3.2 (Localization formula). If u ∈ dom( h,µ ) and χ ∈ Lip(M, h) ∩ L ∞ (M ), then χu ∈ dom(E h,µ ) and the following identity holds: Proof. Exactly as in the proof of Lemma 3.1 of [Sim83], we compute in two ways the iterated commutator [χ, [χ, h,µ ]], where χ is identified with a multiplication operator. We will use (3.3) a few times without comment. All the computations below are for u and χ smooth and compactly supported, the statement then follows appealing to Proposition 2.1. We have Thus, Analogously, we get Putting everything together, we get, for all u ∈ dom( h,µ ), On the other hand, we can easily see that (3.11) Combining the two identities we get (3.12) Then (3.5) follows by observing that ∂χ ∨ u = u, ∂χ h and recalling (2.9).

Proposition 3.3. Let u ∈ dom( h,µ ) be such that h,µ u = 0 on a geodesic ball B(p, R). Then, for every R < R,
where d and B are the geodesic distance and balls associated to h, respectively. It is easy to see that χ ∈ Lip(M, h) ∩ L ∞ (M ) and that χ(q) > 0 holds exactly on B(p, R), and that |∂χ| 2 h (R − R ) −1 /2. Applying the localization formula (3.5) to χu one immediately getŝ Recalling the Leibniz rule (3.3), this giveŝ Since χ ≡ 1 on B(p, R ), we are done.
Observe that χ was chosen to be 0 on the support of u, and hence the first term on the right-hand side vanishes. Recalling the coercivity condition (2.17), we obtain b ˆM (3.23) The pointwise bound |∂ d| 2 h 1/2 suggests that we choose a < 2b and reabsorb the rightmost term in the one on the left. By support considerations and the bound Our choice of d guarantees that this function is bounded from below by d(p, q) − 2R on B(q, 2R), and from above by 2R on B(p, 2R).
To complete the proof we combine (3.19) and (3.25), and the observation that −1 h,µ is bounded with operator norm at most b −2 , so that we have´B (p,

From L 2 to pointwise bounds
The key ingredient in the transition to pointwise bounds from the L 2 -bounds of Theorem 3.4 is the following result by Li-Schoen and Li-Tam.
Theorem 4.1. Let (M, g) be a complete Riemannian manifold, p ∈ M and R > 0 be such that the geodesic ball B(p, 2R) does not meet the boundary of M . Suppose that the Ricci curvature of h is bounded below by K with K 0. Let δ ∈ (0, 1 2 ), q > 0, and λ 0. Then there exists a constant C that depends only on δ, q, λR 2 , and R √ −K such that for any nonnegative function f on B(p, 2R) satisfying the differential inequality This is essentially Corollary 3.6 of [JLSS08], which follows easily from the results on subsolutions of the heat equation on Riemannian manifolds of [LT91].
Here, the convention is that ∆ g is nonpositive. To apply this theorem for our purpose, we shall need to compare the Riemannian Laplacian ∆ g f , where g := 2 Re h, with the so-called Chern Laplacian tr ω h (i∂∂f ) of a regular function f . The comparison is well-known and is stated and proved in Proposition 4.2 below for convenience (cf. formula (25) in [Gau84]).

Proposition 4.2. For a regular function f , one has
where θ is the torsion 1-form.
Proof. Let ∇ denote the Levi-Civita connection of g := 2 Re h. Since h is Hermitian, the Christoffel symbols Γ k ij in local holomorphic coordinates reduce to It then follows that where T¯ ī¯ is the torsion (0,1)-form of the Chern connection. Thus Locally, tr ω h (i∂∂f ) = h jk ∂ j ∂kf and therefore As a consequence of Theorem 4.1 and Proposition 4.2, we have the following mean value inequality. Recall that ψ was defined to satisfy µ = e −2ψ Vol.
where θ is the torsion 1-form. If F : B(p, 2R) → C is holomorphic, then where the constant C depends only λR 2 and R √ −K.
Proof. Let f := |F | 2 e −2ψ . First, observe that by the Cauchy-Schwarz inequality Next, we compute Putting the two estimates together and exploiting Proposition 4.2, we obtain, on B(p, 2R), This estimate, together with the lower bound on the Ricci curvature, shows that the hypothesis of Theorem 4.1 are satisfied. Thus, where C depends on λR 2 and R √ −K. This completes the proof.  where C depends on γ, b, BR 2 , and R √ −K.
This completes the proof of Theorem 1.4.

The basic identity for h,µ
We denote by ∇ the Chern connection of h. In local holomorphic coordinates, the only is nonvanishing Christoffel symbols of ∇ are Γ jk and Γ jk = Γ jk , where We shall only need the (0, 1)-part of ∇, which is denoted by ∇. In particular, if u = u k dz k , then the covariant derivative ∇u is the 2-tensor The key to the proof of the basic inequality is an elementary pointwise identity that involves only the metric h. In order to state it, we recall the standard notation u # for the vector field associated to the 1-form u by the metric h. Notice that if u is a (0, 1)-form, then u # is a (1, 0)-vector field, and ∇u # is a 2-tensor with one covariant and one contravariant index.
Proof. Notice that in local coordinates u # = u m ∂ m where u m := h mk u k , and recall that one of the defining properties of the Chern connection is that It is thus clear that the trace in the statement is Now notice that if A = A jk dz j ⊗dz k and A = A kj dz j ⊗dz k , a straightforward computation gives If A = ∇u, by (5.2) we have A − A = ∂u − T u, and the identity above becomes In view of (5.5), this is the formula we set out to prove. and, for any ν > 0, where E h,µ (u) is defined by (2.15) and T u is defined in Lemma 5.1.
Proof. It is enough to prove the identity for u supported on a coordinate chart with coordinates z j . Let ϕ be the real-valued function such that dµ = ie −2ϕ dz 1 ∧ dz 1 ∧ · · · ∧ dz n ∧ dz n . Then the adjoint of ∂ m with respect to dµ is δ m := −∂ m + 2∂ m ϕ. Integrating both sides of the identity of Lemma 5.1, the usual commutation argument yields To complete the proof of (5.8), one may easily check that ∂ m ∂ j ϕ u m u j = F µ , u ∧ u h and that δ m u m = ∂ * h,µ u. The basic inequality (5.9) follows immediately.
We now turn to the proof of Corollary 1.3.

An example: ACH metrics of Bergman-type
In this last section, we discuss in some detail an interesting example. Let D be a precompact strictly pseudoconvex domain in a complex manifold X with smooth boundary. Suppose that D is defined by < 0, with d = 0 on ∂D and is smooth in a neighborhood U of ∂D. We further assume that − log(− ) is strictly plurisubharmonic on U ∩ D. In this case, −i∂∂ log(− ) defines a (asymptotic complex hyperbolic) Kähler metric h on U ∩ D.
Given any Hermitian metric h on D, we can patch, using a partition of unity, h and h to obtain a Hermitian metric h on D such that h = h on U ∩ D. It is well-known that the curvature tensor of h approaches the curvature tensor of constant holomorphic sectional curvature −4, see [Kle78]. In particular, the sectional curvature is bounded from above, Ric h and Θ h are bounded from below. The last fact is easy to see: near the boundary ∂D, in local coordinates Notice that i∂∂ log J[ ], which does not depend on the local coordinates, extends smoothly to a neighborhood of ∂D, and is hence bounded. Moreover, since T h = 0 near the boundary, h must have bounded torsion. Also note that in general the metric h constructed in this way is non-Kähler and need not have bounded geometry.
Suppose that µ is a smooth measure on D such that h,µ is b 2 -coercive and with ∆ h log(dVol h /dµ) bounded from above. Then the Bergman kernel K µ satisfies the exponential decay estimate (1.20), namely |K µ (p, q)| C η(p)η(q) e −γd h (p,q) , (7.3) where η = dVol h /dµ, d h is the Riemannian distance of h, and depends on the coercivity constant b. Observe that the volume factors have been absorbed into the constant since h has sectional curvature bounded from above. Moreover, by Corollary 1.3, if i∂∂η > ω h for some > 0, then for k large enough