Bergman kernel functions associated to measures supported on totally real submanifolds

Let $K$ be a non-pluripolar compact subset in $\mathbb{C}^{n}$, i.e, $K$ is not contained in $\{\varphi=-\infty\}$ for some plurisubharmonic (psh) function $\varphi$ on $\mathbb{C}^{n}$. Let $\mu$ be a probability measure whose support is non-pluripolar and is contained in $K$, and $Q$ be a real continuous function on $K$. Let $\mathcal{P}_{k}$ be the space of restrictions to $K$ of complex polynomials of degree at most $k$ on $\mathbb{C}^{n}$. The scalar product

$$\langle s_{1},s_{2}\rangle_{L^{2}(\mu,kQ)}:=\int_{K}s_{1}\overline{s}_{2}e^{-2kQ}d\mu$$

induces the $L^{2}(\mu,kQ)$-norm on $\mathcal{P}_{k}$. The Bergman kernel function of order $k$ associated to $\mu$ with weight $Q$ is defined by

$$B_{k}(x):=\sup_{s\in\mathcal{P}_{k}}|s(x)e^{-kQ(x)}|^{2}/\|s\|^{2}_{L^{2}(\mu,kQ)}$$

for $x\in K$. Equivalently if $(s_{1},\ldots,s_{d_{k}})$ (here $d_{k}$ denotes the dimension of $\mathcal{P}_{k}$) is an orthonormal basis of $\mathcal{P}_{k}$ with respect to the $L^{2}(\mu,kQ)$-norm, then

$$B_{k}(x)=\sum_{j=1}^{d_{k}}|s_{j}(x)|^{2}e^{-2kQ(x)}.$$

When $Q\equiv 0$, we say that $B_{k}$ is unweighted. In this case ($Q\equiv 0$) the inverse of $B_{k}$ is known as the Christoffel function in the literature on orthogonal polynomials. In practice we also use a modified version of the Bergman kernel function as follows:

$$\tilde{B}_{k}(x):=\sup_{s\in\mathcal{P}_{k}}|s(x)|^{2}/\|s\|^{2}_{L^{2}(\mu,kQ)}$$

for $x\in\mathbb{C}^{n}$. The advantage of $\tilde{B}_{k}$ is that it is well defined on $\mathbb{C}^{n}$.

The asymptotics of the Bergman kernel function (or its inverse, the Christoffel function) is essential for many applications in (higher dimensional or not) real analysis including approximation theory, random matrix theory, etc. There is an immense literature on such asymptotics. We refer to [7, 9, 16, 24, 26, 35, 44, 47, 60, 65], to cite a just few, for an overview on this very active research field.

Most standard settings are measures supported on concrete domains on $\mathbb{R}^{n}\subset\mathbb{C}^{n}$ (such as balls or simplexes in $\mathbb{R}^{n}$) or in the unit ball in $\mathbb{C}^{n}$. Considering measures on $\mathbb{C}^{n}$ whose support are not necessarily in $\mathbb{R}^{n}$ are also important in many applications; e.g., one can consult [7, 39, 61, 60] where the authors consider measures supported on finite unions of piecewise smooth Jordan curves $\mathbb{C}$ or domains in $\mathbb{C}$ bounded by Jordan curves. We refer to the end of this section for a concrete application to the equidistribution of zeros of random polynomials.

All of settings mentioned above are particular cases of a more natural situation where our measures are supported on piecewise-smooth domains in a generic Cauchy-Riemann submanifold $K$ in $\mathbb{C}^{n}$. This is the context in which we will work on in this paper.

We underline that in view of potential applications, it is important to work with piecewise-smooth compact sets $K$ (rather than only smooth ones). In what follows, by a (convex) polyhedron in $\mathbb{R}^{M}$, we mean a subset in $\mathbb{R}^{M}$ which is the intersection of a finite number of closed half-hyperplanes in $\mathbb{R}^{M}$.

A point $p\in K$ is said to be a regular point of $K$ if the above local chart $(W_{p},\Psi_{p})$ can be chosen such that $\Psi_{p}(K\cap W_{p})$ is the intersection of the unit ball with an $m$-dimensional vector subspace in $\mathbb{R}^{M}$, in other words, $K$ is a $m$-dimensional submanifold locally near $p$. The regular part of $K$ is the set of regular points of $K.$ The singular part of $K$ is the complement of the regular part of $K$ in $K.$ Hence if $K$ is a smooth manifold with boundary, then the boundary of $K$ is the singular part of $K$ and its complement in $K$ is the regular part of $K$.

Now let $K$ be a nondegenerate $\mathcal{C}^{5}$ piecewise-smooth submanifold of $X.$ Since $X$ is a complex manifold, its real tangent spaces have a natural complex structure $J$. We say that $K$ is (Cauchy-Riemann or CR for short) generic in the sense of Cauchy-Riemann geometry if for every $p\in K$ and every sequence of regular points $(p_{m})_{m}\subset K$ approaching to $p$, any limit space of the sequence of tangent spaces of $K$ at $p_{m}$ is not contained in a complex hyperplane of the (real) tangent space at $p$ of $X$ (equivalently, if $E$ is a limit space of the sequence $(T_{p_{m}}K)_{m\in\mathbb{N}}$ of tangent spaces at $p_{m}$, then we have $E+JE=T_{p}X$, where $T_{p}X$ is the real tangent space of $X$ at $p$).

For a CR generic $K$, note that the space $T_{p}K\cap JT_{p}K$ ($p$ is a regular point in $K$) is invariant under $J$ and hence has a complex structure induced by $J$. In this case, the complex dimension of $T_{p}K\cap JT_{p}K$ is the same for every $p$ and is called the CR dimension of $K$. If $r$ denotes the CR dimension of $K$, then $r=\dim K-n$. Thus the dimension of a generic $K$ is at least $n.$

If $Y$ is CR generic and $\dim Y=n$, then $Y$ is said to be (maximally) totally real, and it is locally the graph of a smooth function over a small ball centered at $0\in\mathbb{R}^{n}$ which is tangent at $0$ to $\mathbb{R}^{n}$. Examples of piecewise-smooth totally real submanifolds are polygons in $\mathbb{C}$ or boundaries of polygons in $\mathbb{C}$, and polyhedra of dimension $n$ in $\mathbb{R}^{n}\subset\mathbb{C}^{n}$.

A notion playing an important role in the study of Bergman kernel functions is the following extremal function

$$V_{K,Q}:=\sup\{\psi\in\mathcal{L}(\mathbb{C}^{n}):\,\psi\leq Q\quad\text{ on }K\},$$

where $\mathcal{L}(\mathbb{C}^{n})$ is the set of psh functions $\psi$ on $\mathbb{C}^{n}$ such that $\psi(z)-\log|z|$ is bounded at infinity on $\mathbb{C}^{n}$.

Since $K$ is non-pluripolar, the upper semi-continuous regularisation $V_{K,Q}^{*}$ of $V_{K,Q}$ belongs to $\mathcal{L}(\mathbb{C}^{n})$. If $V_{K,Q}=V^{*}_{K,Q}$, then we say $(K,Q)$ is regular. A stronger notion is the following: we say that $K$ is locally regular if for every $z\in K$ there is an open neighborhood $U$ of $z$ in $\mathbb{C}^{n}$ such that for every increasing sequence of psh functions $(u_{j})_{j}$ on $U$ with $u_{j}\leq 0$ on $K\cap U$, then

$$(\sup_{j}u_{j})^{*}\leq 0$$

on $K\cap U$. One can see that if $K$ is a locally regular set, then $(K,Q)$ is regular for every continuous function $Q$ on $K$. The following result answers the question raised in [9, Remark 1.8].

Note that it was known that $K$ is locally regular if $K$ is smooth real analytic; see, e.g., [9, Corollary 1.7].

The measure $\mu$ is said to be a Bernstein-Markov measure (with respect to $(K,Q)$) if for every $\epsilon>0$, there exists $C>0$ such that

$\displaystyle\sup_{K}|s|^{2}e^{-2kQ}\leq Ce^{\epsilon k}\|s\|^{2}_{L^{2}(\mu,kQ)}$

for every $s\in\mathcal{P}_{k}$. In other words, the Bergman kernel function of order $k$ grows at most subexponentially, i.e., $\sup_{K}B_{k}=O(e^{\epsilon k})$ as $k\to\infty$ for every $\epsilon>0$.

For some examples of Bernstein-Markov measures and criteria checking this condition, we refer to [16]. However, apart from few explicit geometric situations, there are not many (geometric) examples of Bernstein-Markov measures in higher dimensions. This is the motivation for our next main result giving a large geometric class of Bernstein-Markov measures.

To the best of our knowledge, the above result was only known when $K$ is real analytic. We are not aware of results of this type in previous literature for maximally totally real submanifolds. A measure $\mu_{0}$ on $K$ is said to be a smooth volume form on $K$ if $\mu_{0}$ is given by a smooth volume form locally at every regular point in $K$, and if for every singular point $p$ in $K$, there is a local chart $(\Psi,W_{p})$ as in Definition 1.1, such that $\Psi(K\cap W_{p})=\bigcup_{j=1}^{s}P_{j}$, where $P_{j}$’s are polyhedra in $\mathbb{C}^{n}$ of the same dimension for every $1\leq j\leq s$, and the restriction of $\mu_{0}$ to $P_{j}$ is a smooth volume form on $P_{j}$ for $1\leq j\leq s$.

Let $\mathop{\mathrm{Leb}}\nolimits_{K}$ be now a smooth volume form on $K$. Then for any $M>0$ the measure $\mu=|z-z_{0}|^{M}\mathop{\mathrm{Leb}}\nolimits_{K}$ satisfies the hypothesis of Theorem 1.3. Here is our next main result.

We would like to point out that [11, Remark 3.2] shows that the regularity of weights affects considerably the growth of the Bergman kernel function. One can also consult the last reference for upper bounds for $K=\mathbb{C}^{n}$ and $\mu=\mathop{\mathrm{Leb}}\nolimits_{\mathbb{C}^{n}}$ (the Lebesgue measure on $\mathbb{C}^{n}$).

There have been very few works concerning polynomial growth of Bergman kernel functions associated to measures supported on real submanifolds in higher dimensions. With an exception of [12], known upper bounds on $B_{k}$ were proved mostly based on special geometric structures of the compact $K\subset\mathbb{R}^{n}$ (see, e.g.,[44, 47]). Such a method is not useful in dealing with general situations as in Theorem 1.4. In [12] it was supposed that $K$ is smooth real algebraic in $\mathbb{R}^{m}$ or the closure of a bounded convex open subset in $\mathbb{R}^{m}$, and their arguments use this hypothesis in an essential way.

Note that $\mu=|z-z_{0}|^{M}\mathop{\mathrm{Leb}}\nolimits_{K}$ for any constant $M>0$ satisfies the hypothesis of Theorem 1.4. In general it is not possible to bound from below $B_{k}$ by a polynomial in $k$, see Remark 3.7.

By [12, Theorems 2 and 4], if $K$ is the closure of a bounded open convex subset in $\mathbb{R}^{n}$ and $Q\equiv 0$ and $\mu$ is the restriction of the Lebesgue measure on $\mathbb{R}^{n}$ to $K$, then $k^{-n}B_{n}$ is bounded and bounded away from $0$ on a fixed compact subset in the interior of $K$ (the behaviour of $B_{k}$ at boundary points is more complicated). On the other hand, for general $\mu$ on such $K$, by [24], the upper bound for $B_{k}$ on $K$ can not be $O(k^{n})$ in general. To be precise, it was proved there that if $K$ is a smooth Jordan curve in $\mathbb{C}$, and $\mu_{0}$ is the arc measure on $K$, and $\mu=(z-z_{0})^{\alpha}\mu_{0}$ for some constant $\alpha>0$, then $B_{k}(z_{0})\approx k^{1+\alpha}$ as $k\to\infty$. One can see also [45] for a similar asymptotic in the case where $K$ is the closure of the unit ball in $\mathbb{R}^{n}$.

We note that by [31, Corollary 2.13], if $(K,\mu,Q)$ is as in the hypothesis of Theorem 1.4), then the triple $(K,\mu,Q)$ is $1$-Bernstein-Markov in the sense that for every constant $0<\delta\leq 1$, there exists a constant $C>0$ such that

$\displaystyle\sup_{K}|s|^{2}e^{-2kQ}\leq Ce^{Ck^{1-\delta}}\|s\|^{2}_{L^{2}(\mu,kQ)}$

for every $s\in\mathcal{P}_{k}$. This is much weaker than our bound. Nevertheless, [31, Corollary 2.13] is applicable to a broader class of $K$.

When $K$ is smooth (no boundary) and $Q\in\mathcal{C}^{1,\delta}(K)$ for some constant $\delta>0$ (e.g, $K$ is the unit circle in $\mathbb{C}$ as in a classical setting), we obtain sharp bounds which have potential applications in studying sampling or interpolation problems of multivariate polynomials on maximally totally real sets in $\mathbb{C}^{n}$. The case where $K$ is compact smooth real algebraic was considered in [12]. Here is our next main result.

When $K$ is smooth compact real algebraic of dimension $n$ in $\mathbb{R}^{m}$, it was proved in [12] that $B_{k}/k^{n}\approx 1$. The proof of the upper bound for $B_{k}$ in [12] relies crucially on the algebraicity of $K$. Our approach to Theorems 1.4 and 1.5 is different and is based on constructions of analytic discs partly attached to $K$, subharmonic functions on unit discs, and fine regularity of extremal plurisubharmonic envelopes associated to $K$.

In dimension one, we refer to [39, Theorem 4.3] for a similar bound when the case where $K$ is analytic, and to [24, 61] and references therein for asymptotics of $B_{k}$ (it behaves like $k$ at regular points, but the asymptotic of $B_{k}$ at singular points is more complicated).

We are not aware of any estimates of flavor similar to Theorem 1.5 in the previous literature for general smooth maximally totally real submanifolds in higher dimensions (except the real algebraic case mentioned above in [12]). We refer to [1, 43, 55] for more precise bounds when $K$ is a convex subset in $\mathbb{R}^{n}$.

Our next result is the following convergence which is a consequence of Theorem 1.4.

Finally we note that we also obtain a version of (Bernstein-)Markov inequality for maximally totally real submanifolds which might be useful elsewhere; see Theorem 3.13 below.

Let $K$ be a non-pluripolar set in $\mathbb{C}^{n}$ and let $\mu$ be a probability measure on $\mathbb{C}^{n}$ such that the support of $\mu$ is contained in $K$ and is non-pluripolar. Let $Q$ be a continuous weight on $K$. Let $\mathcal{P}_{k}(K)$ be the space of restrictions of complex polynomials of degree at most $k$ in $\mathbb{C}^{n}$ to $K$. Let $d_{k}:=\dim\mathcal{P}_{k}(K)$, and let $p_{1},\ldots,p_{d_{k}}$ be an orthonormal basis of $\mathcal{P}_{k}(K)$ with respect to the $L^{2}(\mu,kQ)$-scalar product. Consider the random polynomial

$\displaystyle p:=\sum_{j=1}^{d_{k}}\alpha_{j}p_{j},$

(1.2)

where $\alpha_{j}$’s are complex i.i.d. random variables. The study of zeros of random polynomials has a long history. The most classical example may be the Kac polynomial where $n=1$, and $p_{j}=z^{j}$.

The distribution of zeros of more general random polynomials associated to orthonormal polynomials (as in (1.2)) was considered in [58] by observing that $1,z,\ldots,z^{k}$ are an orthonormal basis of the restriction of the space of polynomials in $\mathbb{C}$ to $\mathbb{S}^{1}$ with respect to the $L^{2}$-norm induced by the Haar measure $\mu_{0}$ on $\mathbb{S}^{1}$. In this setting, the necessary and sufficient conditions for the distribution of $\alpha_{j}$’s so that the zeros of $p$ is equidistributed almost surely or in probability with respect to the Lebesgue measure $\mu_{0}$ on the unit circle as $k\to\infty$ are known; see [14, 25, 40, 42].

There are many works (in one or higher dimension) following [58], to cite just a few, [13, 15, 5, 4, 6]. In all of these works, it seems to us that the question of large deviation type estimates for the equidistribution of zeros of random polynomials has not been investigated in details. As it will be clear in our proof below, the new ingredient needed for such an estimate is an quantitative rate of convergence between $1/(2k)\log\tilde{B}_{k}$ and the extremal function associated to $K$. This is what we obtained in Theorem 1.6. To state our result we need some hypothesis on $\mu$ and the distribution of $\alpha_{j}$’s.

Assume now that the distribution of $\alpha_{j}$’s is $f\mathop{\mathrm{Leb}}\nolimits_{\mathbb{C}}$, where $f$ is a nonnegative bounded Borel function on $\mathbb{C}$ satisfying the following mild regularities:

$\displaystyle\int_{|z|>r}|f|\mathop{\mathrm{Leb}}\nolimits_{\mathbb{C}}\leq C/r^{2}$

(1.3)

for some constant $C>0$ independent of $r$. This condition was introduced in [13, 15]. We want to study the distribution of zeros of $p$ as $k\to\infty$. We denote by $[p=0]$ the current of integration along the zero divisor $\{p=0\}$ of $p$. Note that if $n=1$, then $[p=0]$ is the sum of Dirac masses at zeros of $p$.

If $(K,Q,\mu)$ is Bernstein-Markov, it was proved in [15, Theorem 4.2], that almost surely

$\displaystyle k^{-1}[p=0]\to dd^{c}\log|V_{K,Q}^{*}|$

(1.4)

as $k\to\infty$, where the convergence is the weak one between currents. In other words, for every smooth form $\Phi$ of degree $(2n-2)$ with compact support in $\mathbb{C}^{n}$, one has

$$k^{-1}\int_{\{p=0\}}\Phi\to\int_{\mathbb{C}^{n}}dd^{c}\log|V_{K,Q}^{*}|\wedge\Phi$$

as $k\to\infty$. Theorem 1.3 above thus provides us a large class of measures for which the equidistribution of zeros of $p$ holds.

Our goal now is to obtain a rate of convergence in (1.4). To this end, it is reasonable to ask for finer regularity on $\mu$ and of the distribution of $\alpha_{j}$. We don’t try to make the most optimal condition. Here is our hypothesis:

(H1) $|f(z)|\leq|z|^{-3}$ for $|z|$ sufficiently large.

(H2) let $K$ be a non-degenerate $\mathcal{C}^{5}$ piecewise-smooth generic Cauchy-Riemann submanifold of $\mathbb{C}^{n}$, and $Q$ be a Hölder continuous function on $K$. Let $\mu=\rho\mathop{\mathrm{Leb}}\nolimits_{K}$, where $\rho^{-\lambda}\in L^{1}(\mathop{\mathrm{Leb}}\nolimits_{K})$ for some constant $\lambda>0$.

The condition (H1) ensures that (1.3) holds, and the joint-distribution of $\alpha_{1},\ldots,\alpha_{d_{k}}$ is dominated by the Fubini-Study volume form on $\mathbb{C}^{d_{k}}$ (by definition the Fubini-Study volume form is equal to $\omega_{FS}^{d_{k}}$, where $\omega_{FS}$ is the Fubini-Study form on $\mathbb{P}^{d_{k}}\supset\mathbb{C}^{d_{k}}$). Clearly the Gaussian random variables satisfy this condition.

The condition (H2) is a natural generalization of the classical setting with Kac polynomials where $K$ is the unit circle in $\mathbb{C}$. Indeed in [58] the authors considered the setting in which $\mu$ is the surface area on a closed analytic curve in $\mathbb{C}$ bounding a simply connected domain $\Omega$ in $\mathbb{C}$ or $\mu$ is the restriction of the Lebesgue measure on $\mathbb{C}$ to $\Omega$. This setting is relevant to the theory of random matrix theory as already pointed out in [58]. We refer to [14, 53, 54] for partial generalizations (without quantitative estimates) to domains with smooth boundary in $\mathbb{C}$. We would like to mention also that in some cases, certain large deviation type estimates for random polynomials in dimension one were known; see [37, Theorem 10] for polynomial error terms, and [28, Theorem 1.1], and [34, Theorem 3.10] for exponential error terms.

To our best knowledge there has been no quantitative generalization of results in [58] to higher dimension. It was commented in the last paper that their method seems to have no simple generalization to the case of higher dimension.

We now recall distances on the space of currents. For every constant $\beta\geq 0$, and $T,S$ closed positive currents of bi-degree $(m,m)$ on the complex projective space $\mathbb{P}^{n}$, define

$$\mathop{\mathrm{dist}}\nolimits_{-\beta}(T,S):=\sup_{\Phi:\,\|\Phi\|_{\mathcal{C}^{[\beta],\beta-[\beta]}}\leq 1}|\langle T-S,\Phi\rangle|,$$

where $[\beta]$ denotes the greatest integer less than or equal to $\beta$, and $\Phi$ is a smooth form of degree $(2n-m)$ on $\mathbb{P}^{n}$.

It is a standard fact that the distance $\mathop{\mathrm{dist}}\nolimits_{-\beta}$ for $\beta>0$ induces the weak topology on the space of closed positive currents (see for example [33, Proposition 2.1.4]). We have the following interpolation inequality: for $0<\beta_{1}\leq\beta_{2}$, there is a constant $c_{\beta_{1},\beta_{2}}$ such that

$\displaystyle\mathop{\mathrm{dist}}\nolimits_{-\beta_{2}}\leq\mathop{\mathrm{dist}}\nolimits_{-\beta_{1}}\leq c_{\beta_{1},\beta_{2}}[\mathop{\mathrm{dist}}\nolimits_{-\beta_{2}}]^{\beta_{1}/\beta_{2}};$

(1.5)

see [33, Lemma 2.1.2] or [48, 62].

Note that the currents $[p=0]$ and $dd^{c}V_{K,Q}$ extends trivially through the hyperplane at infinity $\mathbb{P}^{n}\backslash\mathbb{C}^{n}$ to be closed positive currents of bi-degree $(1,1)$ on $\mathbb{P}^{n}$ (this is due to the correspondence (2.1) above). Hence one can consider $\mathop{\mathrm{dist}}\nolimits_{-\beta}$ between $k^{-1}[p=0]$ and $dd^{c}V_{K,Q}$ as closed positive currents on $\mathbb{P}^{n}$.

By (1.5), one obtains similar estimates for $\mathop{\mathrm{dist}}\nolimits_{-\beta}$ with $0<\beta\leq 2$ as in Theorem 1.7. We don’t know if the right-hand side of (1.6) can be improved. We state now a direct consequence of Theorem 1.7 which gives a higher dimensional generalization of [58, Theorems 1 and 2] (except that we only obtain the error term $O(\frac{\log k}{k})$ instead of $O(k^{-1})$); see also Theorem 1.9 below. Denote by ${\mathbb{E}}_{k}(k^{-1}[p=0])$ the expectation of the random normalized currents of zeros $k^{-1}[p=0]$.

Even when the $\alpha_{j}$’s are Gaussian variables, we underline that the decay obtained in [58] is only $O(k^{-1})$, this error term is optimal in dimension 1 (one can see it by a careful examination of computations in [58, Proposition 3.3]).

Since zeros of random polynomials in higher dimension form no longer a discrete set, one might be somehow not at ease to speak of questions like correlation of zeros. To remedy this problem, one can reformulate the equidistribution of zeros of random polynomials in the following way. Let $L$ be a complex line in $\mathbb{C}^{n}$ or an (complex) algebraic curve in $\mathbb{C}^{n}$. Sine generic polynomials intersect transversely $L$, almost surely the number of intersection points (without counting multiplicities) of the random hypersurface $\{p=0\}$ and $L$ is exactly $k\deg L$ by Bezout’s theorem. Define

$$\mu_{k,L}:=\frac{1}{k\deg L}\sum_{j=1}^{k\deg L}\delta_{z_{j}},$$

where $z_{1},\ldots,z_{k\deg L}$ are zeros of $p$ on $L$. Let $[L]$ be the current of integration along $L$. Since $V_{K,Q}$ is bounded, the product

$$\mu_{L}:=\frac{1}{\deg L}dd^{c}V_{K,Q}\wedge[L]$$

is a well-defined measure supported on $L$ (it is simply $dd^{c}(V_{K,Q}|_{L})$ if $L$ is smooth).

Now since zeros of $p$ on $L$ is discrete and is equidistributed as $k\to\infty$, one can ask as in [58] how zeros of $p$ on $L$ (if scaled appropriately) are correlated. Nevertheless such questions seem to be still out of reach in the higher dimensional setting. Finally we note that one can even consider $L$ to be a transcendental curve in $\mathbb{C}^{n}$. In this case generic polynomials $p$ still intersect $L$ transversely asymptotically (see [41]); the question of equidistribution is however more involving.

The results mentioned in the Introduction have their direct generalizations in the context of complex geometry where $\mathbb{C}^{n}$ is replaced by a compact Kähler manifold. Working in such a generality will make the presentation more clear and enlarge the range of applicability of the theory. We will now describe the setting.

Let $X$ be a projective manifold of dimension $n$. Let $(L,h_{0})$ be an ample line bundle equipped with a Hermitian metric $h_{0}$ whose Chern form $\omega$ is positive. Let $K$ be a compact non-pluripolar subset in $X$. Let $\mu$ be a probability measure on $X$ such that the support of $\mu$ is non-pluripolar and is contained in $K$. Let $h$ be a Hermitian metric on $L|_{K}$ such that $h=e^{-2\phi}h_{0}$, where $\phi$ is a continuous function on $K$. For $s_{1},s_{2}\in H^{0}(X,L)$, we define

$$\langle s_{1},s_{2}\rangle:=\int_{X}\langle s_{1},s_{2}\rangle_{h}d\mu.$$

Since ${\rm Supp}\mu$ is non-pluripolar, the last scalar product defines a norm called $L^{2}(\mu,h)$-norm on $H^{0}(X,L)$. Let $k\in\mathbb{N}$. We obtain induced Hermitian metric $h^{k}$ on $L^{k}$ and a similar norm $L^{2}(\mu,h^{k})$ on $H^{0}(X,L^{k})$. Put $d_{k}:=\dim H^{0}(X,L^{k})$. Let $\{s_{1},\ldots,s_{d_{k}}\}$ be an orthonormal basis of $H^{0}(X,L^{k})$ with respect to $L^{2}(\mu,h^{k})$. The Bergman kernel function of order $k$ associated to $(L,h,\mu)$ is

$$B_{k}(x):=\sum_{j=1}^{d_{k}}|s_{j}(x)|_{h^{k}}^{2}=\sup\big{\{}|s(x)|_{h^{k}}^{2}:\quad s\in H^{0}(X,L^{k}),\quad\|s\|_{L^{2}(\mu,h^{k})}=1\big{\}}$$

for $x\in K$.

When $\mu$ is a volume form on $X$ and $h=h_{0}$, the Bergman kernel function is an object of great importance in complex geometry, for example see [49] for a comprehensive study.

The setting considered in Introduction corresponds to the case where $X=\mathbb{P}^{n}$ and $(L,h_{0})=(\mathcal{O}(1),h_{FS})$ is the hyperplane line bundle on $\mathbb{P}^{n}$ endowed with the Fubini-Study metric. We consider $\mathbb{C}^{n}$ as an open subset in $\mathbb{P}^{n}$ and the weight $Q$ corresponds to $\phi-\frac{1}{2}\log(1+|z|^{2})$. Recall that there is a natural identification between $\mathcal{L}(\mathbb{C}^{n})$ and the set of $\omega_{FS}$-psh functions on $\mathbb{P}^{n}$ (where $\omega_{FS}$ denotes the Fubini-Study form on $\mathbb{P}^{n}$) given by

$\displaystyle u\,\longleftrightarrow\,u-\frac{1}{2}\log(1+|z|^{2})$

(2.1)

for $u\in\mathcal{L}(\mathbb{C}^{n})$.

Another well-known example is the case where $K$ is the unit sphere in $\mathbb{R}^{n}$ (here $n\geq 2$; see, e.g, [50]) and $X$ is the complexification of $K$, i.e, $K=\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$ which is considered as usual a compact subset of $X:=\{z_{0}^{2}+z_{1}^{2}+\cdots+z_{n}^{2}=1\}\subset\mathbb{P}^{n}$. The line bundle $L$ on $X$ is the restriction of the hyperplane bundle $\mathcal{O}(1)\to\mathbb{P}^{n}$ to $X$. We remark that in this case $H^{0}(X,L^{k})$ is equal to the restriction of the space of $H^{0}(\mathbb{P}^{n},\mathcal{O}(k))$ to $X$. Hence the restriction of $H^{0}(X,L^{k})$ to $K$ is that of the space of complex polynomials in $\mathbb{C}^{n}$ to $K$. To see this, notice that $X$ is a smooth hypersurface in $\mathbb{P}^{n}$. Consider the standard exact sequence of sheaves:

$$0\to\mathcal{O}(k-\deg X)\to\mathcal{O}(k)\to\mathcal{O}(k)|_{X}\rightarrow 0,$$

where the second arrow is the multiplication by a section of $\mathcal{O}(\deg X)$ whose zero divisor is equal to $X$. We thus obtain a long exact sequence of cohomology spaces:

$$0\to H^{0}(\mathbb{P}^{n},\mathcal{O}(k-\deg X))\to H^{0}(\mathbb{P}^{n},\mathcal{O}(1))\to H^{0}(\mathbb{P}^{n},\mathcal{O}(k)|_{X})\\ \to H^{1}(\mathbb{P}^{n},\mathcal{O}(k-\deg X))\to\cdots$$

In the last sequence, $H^{0}(\mathbb{P}^{n},\mathcal{O}(k)|_{X})$ is exactly equal to $H^{0}(X,\mathcal{O}(k)|_{X})$, and $H^{1}(\mathbb{P}^{n},\mathcal{O}(k-\deg X))=0$ by Kodaira-Nakano vanishing theorem; see [38, p. 156]. As above, the weight $Q$ on $K$ in the spherical model corresponds to $\phi-\frac{1}{2}\log(1+|z|^{2})|_{X}$ in the setting $(K,X,\mathcal{O}(1)|_{X})$.

The measure $\mu$ is said to be a Bernstein-Markov measure (with respect to $(K,\phi,L)$) if for every $\epsilon>0$ there exists $C=C(\epsilon)>0$ such that

$\displaystyle\sup_{K}|s|_{h^{k}}^{2}\leq Ce^{\epsilon k}\|s\|^{2}_{L^{2}(\mu,h^{k})}$

(2.2)

for every $s\in H^{0}(X,L^{k})$. In other words, the Bergman kernel function of order $k$ grows at most subexponentially, i.e, $\sup_{K}B_{k}=O(e^{\epsilon k})$ as $k\to\infty$ for every $\epsilon>0$. Theorem 1.3 is a particular case of the following result.

Let

$$\phi_{K}:=\sup\{\psi\in{\rm PSH}(X,\omega):\,\psi\leq\phi\>\text{ on }K\}.$$

(2.3)

Since $K$ is non-pluripolar, the function $\phi_{K}^{*}$ is a bounded $\omega$-psh function. If $\phi_{K}=\phi^{*}_{K}$, then we say $(K,\phi)$ is regular. A stronger notion is the following: we say that $K$ is locally regular if for every $z\in K$ there is an open neighborhood $U$ of $z$ such that for every increasing sequence of psh functions $(u_{j})_{j}$ on $U$ with $u_{j}\leq 0$ on $K\cap U$, then

$$(\sup_{j}u_{j})^{*}\leq 0$$

on $K\cap U$. The following result answers the question raised in [9, Remark 1.8].

Note that Theorem 1.2 is a direct consequence of the above result. It was shown in [9, Corollary 1.7] that $K$ is locally regular if $K$ is smooth real analytic. Theorem 2.1 is actually a direct consequence of Theorem 2.2 and the criterion [16, Proposition 3.4] giving a sufficient condition for measures being Bernstein-Markov.

The Monge-Ampère current $(dd^{c}\phi_{K}^{*}+\omega)^{n}$ is called the equilibrium measure associated to $(K,\phi)$. It is well-known that the last measure is supported on $K$. By [9, Theorem B] one has

$\displaystyle d_{k}^{-1}B_{k}\mu\to(dd^{c}\phi_{K}^{*}+\omega)^{n}\,,\quad k\to\infty,$

(2.4)

provided that $\mu$ is a Bernstein-Markov measure associated to $(K,\phi,L)$. The last property suggests that the Bergman kernel function $B_{k}$ cannot behave too wildly at infinity.

Note that by the proof of [29, Theorem 1.3] or [30, Theorem 3.6], for every Hölder continuous function $\phi_{1}$ on $X$, and $\mu_{1}:=\omega^{n}$, the Bergman kernel function of order $k$ associated to $(X,\mu_{1},\phi_{1})$ grows at most polynomially on $K$ as $k\to\infty$; see also [8, Theorem 3.1] for the case where $\phi_{1}$ is smooth.

Consider the case when $X=\mathbb{P}^{n}$, $L:=\mathcal{O}(1)$, $h_{0}=h_{FS}$ is be the Fubini-Study metric on $\mathcal{O}(1)$, and $K$ is a smooth maximally totally real compact submanifold in $\mathbb{C}^{n}\subset\mathbb{P}^{n}$, and $h$ is the trivial line bundle on $K$. It is clear that $\phi:=Q-\frac{1}{2}\log(1+|z|^{2})$ is in $\mathcal{C}^{1,\delta}(K)$ if $Q$ is so. In this case the hypothesis of Theorem 2.4 are fulfilled. Thus Theorem 2.4 implies Theorem 1.5.

As a consequence of Theorem 2.3, we obtain the following estimate generalizing Theorem 1.6 in Introduction.

In the first part of this section we prove Theorem 2.2, and hence Theorem 2.1 as commented in the paragraph after Theorem 2.2. In the second part of the section, assuming Theorem 2.3, we prove Theorem 2.5.