ANNALES
DE LA FACULTÉ
DES SCIENCES
DE TOULOUSE
Mathématiques

ALESSANDRO GHIGI

On the approximation of functions on a Hodge manifold

Tome XXI, n° 4 (2012), p. 769-781.

<http://afst.cedram.org/item?id=AFST_2012_6_21_4_769_0>

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# On the approximation of functions on a Hodge manifold

ALESSANDRO GHIGI(1)

**ABSTRACT.** — If $(M, \omega)$ is a Hodge manifold and $f \in C^\infty(M, \mathbb{R})$ we construct a canonical sequence of functions $f_N$ such that $f_N \to f$ in the $C^\infty$ topology. These functions have a simple geometric interpretation in terms of the moment map and they are real algebraic, in the sense that they are regular functions when $M$ is regarded as a real algebraic variety. The definition of $f_N$ is inspired by Berezin-Toeplitz quantization and by ideas of Donaldson. The proof follows quickly from known results of Fine, Liu and Ma.

**RÉSUMÉ.** — Soit $(M, \omega)$ une variété de Hodge et soit $f \in C^\infty(M, \mathbb{R})$. Nous définissons une suite canonique de fonctions $f_N$ telle que $f_N \to f$ dans la topologie $C^\infty$. Cette construction admet une interprétation très simple du point de vue de l'application moment. En plus les fonctions $f_N$ sont algébriques réelles, c'est-à-dire qu'elles sont des fonctions régulières sur $M$ vue comme variété algébrique réelle. La définition des $f_N$ est inspirée de la quantification de Berezin-Toeplitz et s'appuie sur des idées de Donaldson. La preuve découle très vite de certains résultats dus à Fine, Liu et Ma.

Article proposé par Vincent Guedj.

<footnote>(*) Reçu le 18/08/2011, accepté le 07/03/2012</footnote>

<footnote>(1) Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca
alessandro.ghigi@unimib.it</footnote>
# 1. Introduction

Let $M$ be a complex projective algebraic manifold and let $\omega$ be a Hodge metric, i.e. a Kähler metric such that $[\omega/2\pi]$ lies in the image of $H^2(M, \mathbb{Z})$ inside $H^2(M, \mathbb{R})$. Using ideas of Donaldson [6, §4], Fine, Liu and Ma [7] have already introduced a canonical way to approximate smooth functions on $M$. More precisely, using the Fourier modes of the Szegö kernel they defined a sequence of operators

$$Q_N : C^\infty(M, \mathbb{R}) \to C^\infty(M, \mathbb{R}) \quad N = 1, 2, 3, \dots$$

with the property that for any $f \in C^\infty(M, \mathbb{R})$ one has $Q_N f \to f$ in the topology of $C^\infty(M, \mathbb{R})$, see (3.7) below and Theorem 3.4. The importance of these operators comes from the fact that $Q_N$ is the derivative of the nonlinear map that associates to a Kähler form $\omega$ its $N$-th Bergman metric $\omega_N$, see [7, pp. 495-496].

The purpose of this note is to introduce another sequence of operators, denoted $P_N$, which are a variant of the $Q_N$. Theorem 3.5, which is the main result in the paper, asserts that these operators still have the property that $P_N f \to f$ in $C^\infty(M, \mathbb{R})$. But they have two other nice features. First, if $L$ is a polarization with $2\pi c_1(L) = [\omega]$, the operator $P_N$ admits a simple geometric interpretation in terms of the complete linear system of $L^N$ and the moment map. Next recall that $\mathbb{P}^m(\mathbb{C})$ can be embedded in the space of Hermitian matrices by sending a line in $\mathbb{C}^{m+1}$ to the orthogonal projector onto it. This embedding is just the moment map of $\mathbb{P}^m(\mathbb{C})$ in disguise and it endows $\mathbb{P}^m(\mathbb{C})$ with a canonical structure of (affine) real algebraic variety, see e.g. [1, p. 73]. The same follows for every complex subvariety of $\mathbb{P}^m(\mathbb{C})$. Thus every complex projective variety has a canonical structure of affine algebraic variety over $\mathbb{R}$. The real algebraic functions on $M$ are the functions which are regular with respect to this structure. The second nice feature of the operators $P_N$ is that their range consists of real algebraic functions. Thus an arbitray function $f \in C^\infty(M, \mathbb{R})$ is approximated by functions $P_N f$ which are real algebraic.

The author wishes to thank Roberto Paoletti for various crucial discussions on the Szegö kernel. He also thanks Alberto Della Vedova, Christine Laurent and Xiaonan Ma for useful emails. He also wishes to thank the anonymous referee for a very careful reading of the manuscript, which improved it a lot. Finally he acknowledges financial support from the PRIN 2007 MIUR “Moduli, strutture geometriche e loro applicazioni”.
## 2. The geometric construction

Let $W$ be a finite-dimensional complex vector space. Let $\mathbb{P}(W^*)$ denote the projective space of lines in $W^*$. Any $w \in W$ can be identified with a section $s_w \in H^0(\mathbb{P}(W^*), \mathcal{O}_{\mathbb{P}(W^*)}(1))$ defined by $s_w([\lambda])(t\lambda) = t\lambda(w)$. Here $t\lambda$ is the generic element of $\mathcal{O}_{\mathbb{P}(W^*), [\lambda]}(-1)$. If $\langle \cdot, \cdot \rangle$ is a Hermitian product on $W$, the line bundle $\mathcal{O}_{\mathbb{P}(W^*)}(1)$ inherits a Hermitian metric $h_{FS}$ such that

$$|s_w([\lambda])|_{h_{FS}} = \frac{|\lambda(w)|}{|\lambda|}.$$

(With slight abuse we denote by $|\lambda(w)|$ the absolute value of a complex number and by $|\lambda|$ the norm of a vector in $W^*$.) If $\Theta(h_{FS})$ denotes the curvature of $h_{FS}$, then $\omega_{FS} = i\Theta(h_{FS})$ is by definition the Fubini-Study metric on $\mathbb{P}(W^*)$ induced by the product $\langle \cdot, \cdot \rangle$. The group $SU(W, \langle \cdot, \cdot \rangle)$ acts on $(\mathbb{P}(W^*), \omega_{FS})$ both holomorphically and isometrically and

$$\Phi : \mathbb{P}(W^*) \to \mathfrak{su}(W) \qquad \Phi([\lambda]) = i \left( \frac{\langle \cdot, w_{\lambda} \rangle w_{\lambda}}{|w_{\lambda}|^2} - \frac{\mathrm{id}_W}{\dim W} \right)$$

is the moment map. (Here $w_\lambda$ is the vector such that $\lambda(\cdot) = \langle \cdot, w_\lambda \rangle$..) This means that it is equivariant and that for any $A \in \mathfrak{su}(W)$

$$d(\Phi, A) = -i_{\xi_A} \omega_{FS},$$

where $(X, Y) := -\mathrm{tr} XY$ is the Killing product on $\mathfrak{su}(W)$ and $\xi_A$ is the fundamental vector field corresponding to $A$. Notice that

$$(\Phi([\lambda]), A) = -i \frac{\langle Aw_{\lambda}, w_{\lambda} \rangle}{|w_{\lambda}|^2} \qquad A \in \mathfrak{su}(W). \qquad (2.1)$$

(For the proof see for example [9, p. 24].)

Let $M^m$ be a projective manifold with an ample line bundle $L \to M$ and let $h$ be a Hermitian metric on $L$. Denote by $\Theta(h)$ the curvature of the Chern connection of $(L, h)$ and assume that $\omega := i\Theta(h) \in 2\pi c_1(L)$ is a Kähler form. Let $N$ be a natural number. Using $h$ and the volume form $dV_M := \omega^m/m!$ one can endow the space $V_N := H^0(M, L^N)$ with the $L^2$-Hermitian product

$$\langle s_1, s_2 \rangle_{L^2} := \int_M h^{\otimes N} (s_1(z), s_2(z)) dV_M(z) \quad s_1, s_2 \in V_N.$$

Correspondingly, the projective space $\mathbb{P}(V_N^*)$ is endowed with the Fubini-Study metric induced from $\langle \cdot, \cdot \rangle_{L^2}$. Denote by

$$\varphi_N : M \to \mathbb{P}(V_N^*).$$
the map associated to the complete linear system of $L^N$, by $\omega_N$ the Fubini-Study metric on $\mathbb{P}(V_N^*)$ induced by the Hermitian product $\langle, \rangle_{L^2}$ and by $\Phi_N : \mathbb{P}(V_N^*) \to \mathfrak{su}(V_N)$ the moment map of $(\mathbb{P}(V_N^*), \omega_N)$ as described above.

Let $L^2(M, L^N)$ be the Hilbert space of $L^2$ sections of $L^N$. For any $N$ let $\Pi_N : L^2(M, L^N) \to V_N$ be the orthogonal projector onto $V_N = H^0(M, L^N)$. A function $f \in C^\infty(M, \mathbb{C})$ induces a sequence of Toeplitz operators

$$T_{f,N} : L^2(M, L^N) \to L^2(M, L^N)$$
$$T_{f,Ns} := \Pi_N(f \cdot \Pi_N s) \quad s \in L^2(M, L^N).$$

Since the range of $T_{f,N}$ is contained in $V_N$ and $T_{f,N} \equiv 0$ on $V_N^\perp$, we will identify $T_{f,N}$ with its restriction to $V_N$ considered as an endomorphism of $V_N$. If $s \in V_N$, then $T_{f,N}s := \Pi_N(fs)$. Since $f$ is real valued, the operator $T_{f,N}$ is self-adjoint. Set

$$d_N = h^0(M, L^N) - 1$$

and let $T_{f,N}^0$ be the trace-free part of $T_{f,N}$:

$$T_{f,N}^0 := T_{f,N} - \frac{\text{tr} T_{f,N}}{d_N + 1} \text{id}_{V_N}.$$

Then $iT_{f,N}^0 \in \mathfrak{su}(V_N)$ and $(\cdot, \cdot)_{iT_{f,N}^0}$ is a linear function on $\mathfrak{su}(V_N)$. The pull-back of this function to $M$ via the map $\Phi_N \circ \varphi_N : M \to \mathfrak{su}(V_N)$ is a real valued smooth function on $M$, that we denote by $P_N^0 f$. More explicitly

$$(P_N^0 f)(z) := (\Phi_N(\varphi_N(z)), iT_{f,N}^0). \qquad (2.2)$$

So we have defined a sequence of operators $P_N^0 : C^\infty(M, \mathbb{R}) \to C^\infty(M, \mathbb{R})$. These operators are linear since the map $f \mapsto T_{f,N}$ is linear.

### 3. Relation with the Szegö kernel

The operators $P_N^0$ that we have just defined (and the related operators $P_N$ to be defined below) admit a simple description in terms of the Fourier modes of the Szegö kernel. This will enable us to deduce very quickly some of their analytic properties from known results on the Szegö kernel. We start by recalling some important facts regarding the Szegö kernel. (It should be noted that all the constructions in the paper could be rephrased in terms of the Bergman kernel of the line bundle $L$, as in [4] and [12], rather than using the Szegö kernel of the circle bundle. The two approaches are completely equivalent.)
Set Vol($M$) = $\int_M dV_M = (2\pi)^m c_1(L)^m/m!$. Let $h^*$ denote the Hermitian metric on $L^*$ induced from $h$ and let $|\cdot|_{h^*}$ denote the corresponding norm. Define $\rho: L^* \to \mathbb{R}$ by $\rho(\lambda) := |\lambda|_{h^*}^2 - 1$ and set

$$D := \{\lambda \in L^* : \rho(\lambda) < 0\} = \{\lambda \in L^* : |\lambda|_{h^*} < 1\}$$

$$X := \{\lambda \in L^* : \rho(\lambda) = 0\} = \{\lambda \in L^* : |\lambda|_{h^*} = 1\}.$$

$D$ is a strictly pseudoconvex domain in $L^*$ since $\omega = i\Theta(h)$ is positive. $X = \partial D$ is a smooth hypersurface in $L^*$ and is a principal $S^1$-bundle over $M$. Set $\alpha := i\bar{\partial}\rho|_X$ and

$$dV_X := \frac{\alpha \wedge (d\alpha)^m}{2\pi m!} = \frac{\alpha \wedge \pi^* dV_M}{2\pi}.$$

For $N \in \mathbb{Z}$, let $\rho_N: S^1 \to \text{Gl}(1) = \mathbb{C}^*$ be the representation $\rho_N(e^{i\theta}) = e^{iN\theta}$. Recall that the associated bundle $X \times_{\rho_{-N}} \mathbb{C}$ is the quotient $(X \times \mathbb{C})/S^1$, where $S^1$ acts on $X \times \mathbb{C}$ according to the rule

$$e^{i\theta} \cdot (\lambda, z) = (\lambda e^{i\theta}, \rho_{-N}(e^{-i\theta})z) = (e^{i\theta}\lambda, e^{iN\theta}z).$$

The map

$$[\lambda, z] \mapsto z \cdot (\lambda^{-1}(1))^{\otimes N} \qquad u \mapsto [\lambda, \lambda^{\otimes N}(u)] \qquad (3.1)$$

is an isomorphism of $X \times_{\rho_{-N}} \mathbb{C}$ onto $L^N$. Therefore a section $s$ of $L^N$ corresponds to an equivariant function $\hat{s}: X \to \mathbb{C}$ defined by $\hat{s}(x) = x^{\otimes N}(s(\pi(x)))$. The equivariance means that

$$\hat{s}(e^{i\theta}\lambda) = \rho_{-N}(e^{-i\theta})\hat{s}(\lambda) = e^{iN\theta}\hat{s}(\lambda). \qquad (3.2)$$

Denote by $\mathcal{H}^2(X)$ the space of $L^2$ functions on $X$ that are annihilated by the Cauchy-Riemann operator $\bar{\partial}_b$ (see [19, p. 321]). $\mathcal{H}^2(X)$ coincides with the space of $L^2$-boundary values of holomorphic functions on $D$. It is a closed $S^1$-invariant subspace of $L^2(X)$, hence splits as a direct sum $\mathcal{H}^2(X) = \bigoplus_N \mathcal{H}_{N \in \mathbb{Z}}^2(X)$, where $\mathcal{H}_N^2(X)$ is the set of functions $\hat{s}$ in $\mathcal{H}^2(X)$ that satisfy (3.2). Via the correspondence (3.1) the holomorphic sections of $L^N$ correspond to elements of $\mathcal{H}_N^2(X)$. If $s_1, s_2 \in C^\infty(M, L^N)$, then

$$\langle s_1, s_2 \rangle_{L^2} = \int_X \hat{s}_1 \overline{\hat{s}_2} dV_X = \langle \hat{s}_1, \hat{s}_2 \rangle_{L^2(X)}.$$

If $\pi: X \to M$ denotes the projection of the circle bundle and $x \in X$, then the linear functional

$$V_N \to \mathbb{C} \qquad s \mapsto x^{\otimes N} (s(\pi(x))) = \hat{s}(x)$$
is represented by a section $e_{x,N} \in V_N$. Equivalently, $\hat{s}(x) = \langle s, e_{x,N} \rangle_{L^2} = \langle \hat{s}, \hat{e}_{x,N} \rangle_{L^2(X)}$ for any $s \in V_N$. The sections $e_{x,N}$ are called coherent states in the mathematical physics literature.

We use $\Pi_N$ also to denote the projection $\Pi_N : L^2(X) \to \mathcal{H}_N^2(X)$. Let $\Pi_N(x, x')$ be the Schwartz kernel of $\Pi_N$ (we use $dV_X$ to identify functions with densities). In the following Proposition we gather various well-known and elementary properties of $\Pi_N$.

**PROPOSITION 3.1.** — (a) $\Pi_N \in C^\infty(X \times X)$. (b) $\Pi_N(x, x') = \overline{\Pi_N(x', x)}$. (c) For any $x \in X$, $\hat{e}_{x,N} = \Pi_N(\cdot, x)$. (d) There are functions $K_N \in C^\infty(M \times M)$ and $E_N \in C^\infty(M)$ such that if $x, x' \in X$, $z = \pi(x), z' = \pi(x')$, then

$$K_N(z, z') = |\Pi_N(x, x')|^2 \qquad E_N(z) = \Pi_N(x, x).$$

(e) $K_N(z, z') = K_N(z', z)$. (f) If $\pi(x) = z$ and $\{s_j\}_{j=0}^{d_N}$ is an orthonormal basis of $V_N$, then

$$E_N(z) = \sum_{j=0}^{d_N} |s_j(z)|_h^2 = |e_{x,N}|_{L^2}^2. \qquad (3.3)$$

(g) For $N$ large enough the function $E_N$ is strictly positive.

$(E_N$ coincides with the kernel of the Bergman projector $L^2(M, L^N) \to H^0(M, L^N)$ restricted to the diagonal.)

*Proof.* — Let $\{s_j\}_{j=0}^{d_N}$ be an orthonormal basis of $V_N$. Then for any $x, x' \in X$

$$\Pi_N(x, x') = \sum_{j=0}^{d_N} \hat{s}_j(x) \overline{\hat{s}_j(x')} \qquad (3.4)$$

This proves (a) and (b). If $e_{x,N} = \sum_j a_j s_j$, then

$$\overline{a_j} = \langle s_j, e_{x,N} \rangle_{L^2} = \hat{s}_j(x).$$

Hence $e_{x,N} = \sum_j \overline{\hat{s}_j(x)} s_j$. Together with (3.4) this proves (c). Observe that by (3.2)

$$\Pi_N(e^{i\theta} x, e^{i\theta'} x') = e^{iN(\theta-\theta')} \Pi_N(x, x').$$

Hence the function $|\Pi_N|^2$ is $S^1 \times S^1$-invariant on $X \times X$ and descends to a smooth function $K_N \in C^\infty(M \times M)$. Similarly $\Pi_N(x, x)$ is $S^1$-invariant on $X$ and descends to a smooth function $E_N \in C^\infty(M)$. This proves (d). (e) follows from (b). To prove (f) it is enough to observe that if $x \in X$,
then $|x^{\otimes N}|_{h^*} = 1$, hence $|s_j(z)|_h = |x^{\otimes N}(s_j(z))| = |\hat{s}_j(x)|$. This yields immediately the first equality. For the same reason

$$|e_{x,N}|_{L^2}^2 = \sum_{j=0}^{d_N} |\langle e_{x,N}, s_j \rangle_{L^2}|^2 = \sum_{j=0}^{d_N} |x^{\otimes N}(s_j(z))|^2 = \sum_{j=0}^{d_N} |s_j(z)|_h^2.$$

The positivity of $E_N$ is equivalent to the fact that $L^N$ has no base points, so (g) follows from the ampleness of $L$. $\square$

**THEOREM 3.2.** — If $f \in C^\infty(M, \mathbb{R})$ and $z \in M$, then

$$P_N^0 f(z) = \int_M \frac{K_N(z, z')}{E_N(z)} f(z') dV_M(z') - \frac{\operatorname{tr} T_{f,N}}{d_N + 1}, \quad (3.5)$$

$$\operatorname{tr} T_{f,N} = \int_M E_N(z) f(z) dV_M(z). \quad (3.6)$$

*Proof.* — Let $x \in X$ be such that $z = \pi(x)$. Set $\lambda(\cdot) = \langle \cdot, e_{x,N} \rangle_{L^2}$. Then $\lambda \in V_N^*$ and $\varphi_N(z) = [\lambda]$. By (2.2) and (2.1)

$$\begin{aligned} P_N^0 f(z) &= (\Phi_N(\varphi_N(z)), i T_{f,N}^0) = -i \frac{\langle i T_{f,N}^0 e_{x,N}, e_{x,N} \rangle_{L^2}}{|e_{x,N}|_{L^2}^2} = \\ &= \frac{\langle T_{f,N} e_{x,N}, e_{x,N} \rangle_{L^2}}{|e_{x,N}|_{L^2}^2} - \frac{\operatorname{tr} T_{f,N}}{d_N + 1} \end{aligned}$$

$$\langle T_{f,N} e_{x,N}, e_{x,N} \rangle_{L^2} = \langle f e_{x,N}, e_{x,N} \rangle_{L^2} = \int_M f(z') |e_{x,N}(z')|_h^2 dV_M(z').$$

If $z' = \pi(x')$, then $|e_{x,N}(z')|_h^2 = |\hat{e}_{x,N}(x')|^2 = |\Pi_N(x', x)|^2 = K_N(z', z) = K_N(z, z')$. Hence

$$\langle T_{f,N} e_{x,N}, e_{x,N} \rangle_{L^2} = \int_M K_N(z, z') f(z') dV_M(z').$$

By (3.3) $|e_{x,N}|_{L^2}^2 = E_N(z)$, so

$$\begin{aligned} \frac{\langle T_{f,N} e_{x,N}, e_{x,N} \rangle_{L^2}}{|e_{x,N}|_{L^2}^2} &= \frac{1}{E_N(z)} \int_M K_N(z, z') f(z') dV_M(z') = \\ &= \int_M \frac{K_N(z, z')}{E_N(z)} f(z') dV_M(z'). \end{aligned}$$
This proves (3.5). To prove (3.6) observe that if ${\{s_j\}}_{j=0}^{d_N}$ is an orthonormal basis of $V_N$, then

$$
\begin{align*}
\operatorname{tr} T_{f,N} &= \sum_{j=0}^{d_N} \langle T_{f,N} s_j, s_j \rangle_{L^2} = \sum_{j=0}^{d_N} \langle f s_j, s_j \rangle_{L^2} = \\
&= \sum_{j=0}^{d_N} \int_M f(z') |s_j(z')|^2 h dV_M(z') = \int_M f(z') E_N(z') dV_M(z').
\end{align*}
$$

In the last line we have used (3.3). □

<figure><img src="image_1.png" /></figure>

DEFINITION 3.3. — For $N = 1, 2, 3, \dots$ let $P_N : C^\infty(M, \mathbb{R}) \to C^\infty(M, \mathbb{R})$
be the linear operator defined by

$$
P_N f(z) := \int_M \frac{K_N(z, z')}{E_N(z)} f(z') dV_M(z').
$$

It follows from (3.5) that for any $f \in C^\infty(M, \mathbb{R})$ the functions $P_N f$ and $P_N^0 f$ differ only by a constant (depending on $N$). We will prove now that as $N \to \infty$ this constant approaches the average of $f$, while $P_N f \to f$. It follows that $P_N^0 f \to f$ for any function $f$ with

$$
\int_M f(z) dV_M(z) = 0
$$

This result follows immediately from known results that we now recall.
In [6, §4] Donaldson introduced operators $Q_N : C^\infty(M, \mathbb{R}) \to C^\infty(M, \mathbb{R})$,
defined by

$$
Q_N f(z) = \frac{\operatorname{Vol}(M)}{d_N + 1} \int_M K_N(z, z') f(z') dV_M(z'). \quad (3.7)
$$

(His definition is actually more general since integration is performed with respect to a measure that does not necessarily coincide with $dV_M$.) In [7, Appendix, Thm. 26] Kefeng Liu and Xiaonan Ma proved the following result.

THEOREM 3.4 (LIU-MA). — For every $k$ there is a constant $C_k$ such
that

$$
\|Q_N f - f\|_{C^k(M)} \le \frac{C_k}{N} \|f\|_{C^{k+1}(M)}. \quad (3.8)
$$

In [7, p. 519] the estimate is given in terms of $\|f\|_{C^k(M)}$ on the right
hand side, due to a small inaccuracy in the computations. The correct for-
mulation, as above, is given in [13, p. 1 note 1]. I thank Professor Ma for
pointing this out to me. It is important to notice that the operator $Q_N$
represents the derivative of the map $\omega \mapsto \omega_N$, see [7, pp. 495-496].
THEOREM 3.5. — (a) There is a positive constant C such that

$$ \left| \frac{\operatorname{tr} T_{f,N}}{d_N + 1} - \frac{1}{\operatorname{Vol}(M)} \int_M f(z) dV_M(z) \right| \le \frac{C}{N} \|f\|_{\infty}. $$

(b) For any $k$ there is a positive constant $C_k$ such that

$$ \|P_N f - f\|_{C^k(M)} \le \frac{C_k}{N} \|f\|_{C^{k+1}(M)}. $$

*Proof*. — The first statement follows from a fundamental result of Zelditch [19]: the sequence ${E_N}_N$ admits an asymptotic expansion in $C^\infty(M)$ of the form

$$ E_N = \left(\frac{N}{2\pi}\right)^m + O(N^{m-1}). \qquad (3.9) $$

(Apply [19, Cor. 2] with $G = \operatorname{Ric}(h)$.) By Riemann-Roch

$$ \begin{aligned} d_N + 1 = h^0(M, L^N) &= \frac{c_1(L)^m}{m!} N^m + O(N^{m-1}) = \\ &= \frac{\operatorname{Vol}(M)}{(2\pi)^m} N^m + O(N^{m-1}). \end{aligned} \qquad (3.10) $$

Using (3.9) we get

$$ \frac{E_N}{d_N + 1} = \frac{1}{\operatorname{Vol}(M)} + O\left(\frac{1}{N}\right) $$

in $C^\infty(M)$. Integrating against $f$ yields (a). To prove (b) it is enough to observe that

$$ P_N f = \frac{d_N + 1}{\operatorname{Vol}(M) \cdot E_N} Q_N f. $$

Therefore (3.8) and (3.10) yield the result. $\square$

4. Final remarks

1. Tian approximation theorem [17, 3, 19, 4] asserts that $\varphi_N^*\omega_N/N \to \omega$ in the $C^\infty$ topology. The metrics $\varphi_N^*\omega_N$ are projectively induced hence of algebraic/polynomial character. Theorem 3.5 can be considered as an analogue for functions of Tian theorem. Indeed denote by $\mathcal{A}_N^0$ the set of functions of the form $(\Phi_N \circ \varphi_N)^*\lambda$ where $\lambda$ varies in the dual of $\mathfrak{su}(V_N)$. $\mathcal{A}_N^0$ is a finite dimensional subspace of $C^\infty(M, \mathbb{R})$ whose elements are real algebraic since $\Phi_N$ is a real algebraic mapping. By construction the range of $P_N^0$ is contained in $\mathcal{A}_N^0$ and the range of $P_N$ is contained in

$$ \mathcal{A}_N := \mathbb{R} + \mathcal{A}_N^0 \subset C^\infty(M, \mathbb{R}) $$
where $\mathbb{R}$ denotes the constant functions. Hence for any $f \in C^\infty(M, \mathbb{R})$, the function $P_N f$ is real algebraic function and projectively induced in a sense.

2. The elements of $\mathcal{A}_N^0$ are related to a particular kind of holomorphic functions $L^* \times \bar{L}^*$, the so-called Hermitian algebraic functions, see [18, Def. 2.1 p. 297].

3. The construction of the operators $P_N^0$ and $P_N$ is of course inspired by Berezin-Toeplitz quantization and especially by a paper of Bordemann, Meinrenken, Schlichenmaier [2] (see also [14]). In that paper the authors did not use explicitly the Toeplitz operators to construct new functions on $M$. Nevertheless what they showed in the proof of Theorem 4.1 is equivalent, in our notation, to the fact that $P_N f(z_0) \to f(z_0)$ at points $z_0 \in M$ where $|f|$ attains its maximum.

4. Let $\hat{P}_N : C^\infty(X) \to C^\infty(X)$ be the operator with kernel

$$ \hat{P}_N(x, x') = \frac{|\Pi_N(x, x')|^2}{\Pi_N(x, x)}. $$

Then $\hat{P}_N(\pi^* f) = \pi^*(P_N f)$ for any $f \in C^\infty(M)$ and the relation between $\hat{P}_N$ and $\Pi_N$ is identical with that between the Poisson-Szegö and the Szegö kernels of a domain $D$, see e.g. [8, p. 79] and [11]. Since the Poisson-Szegö operator reproduces holomorphic functions in $\mathcal{H}^2(\partial D)$, one might wonder if something similar holds for $\hat{P}_N$. This is not the case. Indeed the trick used to prove the reproducing property for the Poisson-Szegö operator [10, p. 65] fails in our situation, since for $f \in \mathcal{H}_K^2(X)$ the function

$$ u(x') = f(x') \frac{\overline{\Pi_N(x, x')}}{\Pi_N(x, x)} $$

belongs to $\mathcal{H}_{N+K}^2(X)$. So $\hat{P}_N f = f$ if and only if $f \in \mathcal{H}_0^2(X)$. This means that $\hat{P}_N$ reproduces only the constant functions.

5. One might try to prove (b) in Theorem 3.5 by another path, using the following result. Consider $T_{f,N}$ as an operator on $C^\infty(X)$ and denote by $T_{f,N}(x, x')$ its Schwartz kernel. The function $T_{f,N}(x, x)$ descends to a function $t_N$ on $M$ and

$$ t_N = \left(\frac{N}{2\pi}\right)^m \cdot f + O(N^{m-1}) \quad (4.1) $$

in the topology of $C^\infty(M)$. This follows from the main result in [5] and can also be established using the scaling asymptotics of the Szegö kernel (established in [15, 16]), see [12, Lemma 7.2.4]. (See also [13] for a study of
higher terms in the asymptotic expansion.) If $M_f: L^2(X) \to L^2(X)$ denotes the operator of multiplication by $f$, then $T_{f,N} = \Pi_N M_f \Pi_N$ so

$$ 
\begin{align*}
T_{f,N}(x, x'') &= \int_X \Pi_N(x, x') f(\pi(x')) \Pi_N(x', x'') dV_X(x') \\
t_N(z) &= T_{f,N}(x, x) = \int_X |\Pi_N(x, x')|^2 f(\pi(x')) dV_X(x') = \\
&= \int_M K_N(z, z') f(z') dV_M(z').
\end{align*}
$$

Therefore

$$ P_N f(z) = \frac{t_N(z)}{E_N(z)}. $$

It follows from (3.9) and (4.1) that for every $f \in C^\infty(M)$ and any $k$ there is $C'_k$ such that $||P_N f - f||_{C^k(M)} \le C'_k/N$. The constant $C'_k$ depends on $f \in C^\infty(M)$ and we would like to show that it can be chosen in the form $C'_k = C_k ||f||_{C^{k+1}(M)}$. Unfortunately it is not clear how to accomplish this last step, so this method of proof is incomplete.

**6.** If $M = \mathbb{P}^1, L = \mathcal{O}_{\mathbb{P}^1}(2)$ and $\omega = 2\omega_{FS}$, i.e. if $M = S^2$ with the round metric, it would be interesting to relate this approximation procedure to more classical constructions. The Szegö kernel of $(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(1), \omega_{FS})$ is

$$ \Pi_N(x, x') = \frac{N+1}{2\pi} \langle x, x' \rangle^N $$

where $x, x' \in X = S^3 \subset \mathbb{C}^2$ and $\langle \cdot, \cdot \rangle$ denotes the Hermitian product on $\mathbb{C}^2$ (see e.g. [14, p. 65]). So for $(M, L, \omega) = (\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(2), 2\omega_{FS})$

$$ \Pi_N(x, x') = \frac{2N+1}{4\pi} \langle x, x' \rangle^N $$

where $x, x' \in X = S^3/\{\pm 1\} = \text{SO}(3)$. Hence $P_N = Q_N$ and the Schwartz kernel is

$$ P_N(z, z') = \frac{2N+1}{4\pi} |\langle x, x' \rangle|^{2N} \qquad \pi(x) = z, \pi(x') = z'. $$

If we identify $\mathbb{P}^1$ with the sphere $S^2 \subset \mathbb{R}^3$ in the standard way, this becomes

$$ P_N(y, y') = \frac{2N+1}{4\pi} \left( \frac{1+y \cdot y'}{2} \right)^N \quad y, y' \in S^2 $$

where $y \cdot y'$ denotes the scalar product in $\mathbb{R}^3$. Apart from this expression, which looks quite nice, one might try to express the operators $P_N$ in terms
of spherical harmonics on $S^2$. In fact this computation has already been
carried out by Donaldson in [6, p. 612ff] (recall that $P_N = Q_N$ in this case).
Since $L^2(S^2) = \bigoplus_{m=0}^{\infty} \mathcal{H}_m$, where $\mathcal{H}_m$ is the space of degree $k$ spherical
harmonics, and the operators $P_N$ are SO(3)-equivariant, $P_N(\mathcal{H}_m) \subset \mathcal{H}_m$
and $P_N$ acts on $\mathcal{H}_m$ as multiplication by a scalar $\chi_{m,2N}$. The range of $P_N$
is $\sum_{m=0}^{2N} \mathcal{H}_m$. One might expect (and the writer did hope) that $P_N$ is just
orthogonal projection onto its range. This is equivalent to $\chi_{m,2N} = 1$ for
$m \le 2N$, but Donaldson's computation shows this to be false. The only
thing which is evident from Donaldson's formula is that $\chi_{m,2N} \to 1$ for
$N \to \infty$, which is the equivalent to Theorem 5 in the case at hand.

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