A note on Berezin-Toeplitz quantization of the Laplace operator

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error which tends to zero when taking higher powers of the polarization line bundle.


Introduction
Let M be a n-dimensional projective manifold and let g be a Hodge metric on M . This means that M is equipped with a complex structure J and with a positive Hermitian line bundle (L, h). Denoted by Θ the curvature of the Chern connection, the form ω = 2πiΘ is positive, and it holds g(u, v) = ω(u, Jv). Let ∆ : C ∞ (M ) → C ∞ (M ) be the positive Laplacian associated with the metric g (recall that it is defined by ∆(f )ω n = −n i∂∂f ∧ ω n−1 for any complex-valued smooth function f on M ). In this note it will be shown that ∆ is approximated in a suitable sense by a sequence of self-adjoint positive operators acting on finite dimensional Hermitian vector spaces V m (see definitions at Sections 2 and 4). To be a little more precise, it will be proved that there exist maps in fact adjoint to each other with respect to suitable Hermitian products, such that as m → ∞ for any given smooth function f . For any m > 0 the map T m is the well known Berezin-Toeplitz quantization map, and the operator ∆ m depends only on the projective geometry of the Kodaira embedding of M via L m . Moreover ∆ m is related to the metric g via the Fubini-Study metric induced by the L 2 -inner product on the space of global holomorphic sections of L m (see Section 4). Thanks to results available on asymptotic expansions of Bergman kernel [5] and Toeplitz operators [6], what one can prove is indeed the following result, which obviously implies (1).
There is a complete asymptotic expansion where P r are self-adjoint differential operators on C ∞ (M ). More precisely, for any k, R ≥ 0 there exist constants C k,R,f such that

Moreover one has
P 0 (f ) = ∆f, The construction of the quantized Laplacian ∆ m was inspired by a work of J. Fine on the Hessian of the Mabuchi energy [2]. Even though in principle ∆ m is unrelated to the problem of finding canonical metrics on M , when ω is balanced in the sense of Donaldson (see definition recalled at Section 6) the relation between ∆ m and ∆ is even more evident as shown by the following Thanks to A. Ghigi and A. Loi for some useful discussion on Berezin-Toeplitz quantization. The main part of these note has been written in 2010 while the author was visiting Princeton University, whose hospitality is gratefully acknowledged. At that time the author was partially supported by a Marie Curie IOF (program CAMEGEST, proposal no. 255579). A recent pre-print of J. Keller, J. Meyer, and R. Seyyedali has a substantial overlapping with the present work [4]. The author became aware of that pre-print when it appeared on the arXiv.
for all s, t ∈ H m . Thus V m is a Hermitian vector space with inner product defined by for all A, B ∈ V m . Here B * denotes the adjoint of B with respect to b m .
3 The maps T m and T * m The map T m : C ∞ (M ) → V m is the well known Berezin-Toeplitz quantization operator [5]. Given a smooth function f on M , the operator Proof. It is an easy consequence of general theory. Substituting which gives the thesis by arbitrariness of f after noting that Note that the map T * m takes an endomorphisms A ∈ V m to the restriction to the diagonal of its integral kernel. More precisely, given an orthonormal basis is the metric dual of s α (x) in the fiber of L m over the point x. The restriction of the kernel to the diagonal is (naturally identified with) the smooth function T * m (A) thanks to Lemma 3.1. When A is of the form T m (f ) for some smooth function f , the integral kernel is given by For a constant function f = c ∈ R, one has The right hand side of the equation above can be related to a function on P(H m ) naturally associated to A. Indeed we claim that ν(A) is the gradient of the function µ A defined by This is quite standard, but a proof of that fact is included at the end of the proof for convenience of the reader. Now we go ahead taking the claim for grant. From (6) one gets To this end, let {s α } be an orthonormal basis of H m , so that the pull-back of µ A to M is given by where the ratio sα(x)s β (x) γ |sγ (x)| 2 is well defined and can be computed choosing an arbitrary Hermitian metric on the line bundle L m . In particular, taking h m it becomes , and the identity (7) follows by definition of ρ m and Lemma 3.1. Finally, in order to prove the claim above, let (z α ) be homogeneous coordinates on P(H m ) corresponding to the basis {s α }. The function µ A then takes the form Az t |z| 2 , where now A = (A αβ ) denotes the matrix that represents the endomorphism A with respect the chosen basis. The equality between ν(A) and the gradient of µ A can be proved in local affine coordinates, but here we consider the projection of H m \ {0} on P(H m ), and the fact that ν(A), g F S and µ A lift to C * -invariant objects (which will be denotes with the same symbols). In particular one has which proves the claim.
Next lemma characterizes the kernel of ∆ m . for all A ∈ V m and z ∈ P(H m ). By computation above we proved the following Then it holds Here e ω F S is a mixed-degree form defined by the exponential series. Since ω k F S = 0 for all k ≥ dim H m , one has This implies that Ξ m has mixed degree. More interestingly it depends just on the dimension of P(H m ) (and on choice of homogeneous coordinates) and it is independent of M .
whence the thesis follows since ω m is cohomologous to mω.

Proof of Theorem 1.1
First of all we recall some results on asymptotic expansions in Berezin-Toeplitz quantization.
Theorem 5.1. There is a sequence {b r } of self-adjoint differential operators acting on C ∞ (M ) such that for any smooth function f ∈ C ∞ (M ) one has the asymptotic expansion and for any k, R ≥ 0 there exist constants C k,R,f such that

Moreover one has
Proof. See Ma and Marinescu [5]. The only fact one still needs to show is self-adjointness of operator b r . It follows readily by self-adjointness of T * m • T m and expansion (9). Indeed one has as m → +∞, for all f, g ∈ C ∞ (M ).