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Fekete points and convergence towards equilibrium measures on complex manifolds. (English) Zbl 1241.32030

Let \(L\) be a holomorphic line bundle on a compact complex manifold \(X\) of complex dimension \(n\). Let \((K,\phi)\) be a weighted compact subset, i.e., \(K\) is a non-pluripolar compact subset \(K\) of \(X\) and \(\phi\) is a continuous Hermitian metric on \(L\). Let \(\mu\) be a probability measure on \(K\). If \(s\) is a section of \(kL=L^{\otimes k}\), we denote the corresponding pointwise length function by
\[ |s|_{k\phi} = |s| e^{-k\phi}. \]
It is then an interesting and intensively discussed problem to study the asymptotic behavior of the spaces of global sections \(s\in H^0=(X,kL)\) endowed with either the \(L^2\)-norm
\[ \|s\|^2_{L^2(\mu,k\phi)} := \int_X |s|^2_{k\phi} d\mu \]
or the \(L^\infty\)-norm
\[ \|s\|_{L^\infty(K,k\phi)} := \sup_K |s|_{k\phi}. \]
We introduce the equilibrium weight of \((K,\phi)\) as
\[ \phi_K := \sup \big\{\psi\text{ psh weight on }L: \psi \leq \phi\text{ on }K\big\}, \]
and denote by \(\phi_K^*\) its upper semi-continuous regularization which is a psh weight on \(L\) since \(K\) is non-pluripolar.
The volume \(\mathrm{vol}(L)\) of \(L\) is defined by
\[ N_k := \dim H^0(X,kL) = \mathrm{vol}(L) \frac{k^n}{n!} + o(k^n). \]
Assume that \(L\) is a big line bundle, i.e., assume that \(\mathrm{vol}(L)>0\). Then the equilibrium measure of \((K,\phi)\) is defined as the Monge-Ampère measure of \(\phi_K^*\) normalized to unit mass:
\[ \mu_{eq}(K,\phi):=\frac{1}{\mathrm{vol}(L)} (dd^c \phi_K^*)^n. \]
This makes sense as it was shown by the first two authors in [“Growth of balls of holomorphic sections and energy at equilibrium”, Invent. Math. 181, No. 2, 337–394 (2010; Zbl 1208.32020)] that \[ \mathrm{vol}(L) = \int_X (dd^c \phi_K^*)^n. \]
The main goal of the paper under review is to give a general criterion involving spaces of global sections that ensures convergence of probability measures on \(K\) of Bergman-type towards the equilibrium measure \(\mu_{eq}(K,\phi)\). To describe that, we also need the notion of a Fekete configuration, that is a finite subset of points in \(K\) maximizing the interpolation problem. Let \(N:=\dim H^0(L)\) and \(P=(x_1, \dots, x_N)\in K^N\) be a configuration of points in \(K\). Then \(P\) is said to be a Fekete configuration for \((K,\phi)\) if it maximizes the determinant of the evaluation operator
\[ \mathrm{ev}_P: H^0(L) \rightarrow \bigoplus_{j=1}^N L_{x_j} \]
with respect to a given basis \(s_1, \dots, s_N\) of \(H^0(L)\), i.e., the determinant
\[ \big|\det\big(s_i(x_j)\big)\big| e^{-(\phi(x_1) + \dots + \phi(x_N))}. \] This condition does not depend on the choice of the basis \((s_i)\). For each \(P=(x_1, \dots, x_N)\in X^N\) let \[ \delta_P := \frac{1}{N}\sum_{j=1}^N \delta_{x_j} \] be the averaging measure along \(P\).
The first main result of the paper under review is an equidistribution theorem for Fekete configurations:
Theorem A. Let \((X,L)\) be a compact complex manifold equipped with a big line bundle. Let \(K\) be a non-pluripolar compact subset of \(X\) and \(\phi\) a continuous weight on \(L\). For each \(k\) let \(P_k\in K^{N_k}\) be a Fekete configuration for \((K,k\phi)\). Then the sequence \(P_k\) equidistributes towards the equilibrium measure as \(k\rightarrow \infty\), that is
\[ \lim_{k\rightarrow \infty} \delta_{P_k} = \mu_{eq}(K,\phi) \]
holds in the weak topology of measures.
There is another way to represent \(\mu_{eq}(K,\phi)\) in terms of Bernstein-Markov measures. We first need the notion of the Bergman measure. The distortion between the natural \(L^2\) and \(L^\infty\)-norms on \(H^0(L)\) is locally accounted for by the distortion function \(\rho(\mu,\phi)\) whose value at \(x\in X\) is defined by \[ \rho(\mu,\phi)(x) = \sup_{\|s\|_{L^2(\mu,\phi)}=1} |s(x)|^2_\phi, \] the squared norm of the evaluation operator at \(x\in X\). Then the probability measure \[ \beta(\mu,\phi) := \frac{1}{N} \rho(\mu,\phi)\mu, \] where \(N=\dim H^0(L)\), is called the Bergman measure.
The measure \(\mu\) is called Bernstein-Markov if the growth of the distortion between \(L^2(\mu,k\phi)\) and \(L^\infty(K,k\phi)\) norms is subexponential as \(k\rightarrow \infty\), that is
\[ \sup_K \rho(\mu,k\phi) = O\big(e^{\epsilon k}\big)\text{ for all } \epsilon>0. \] Now then, the Bergman measures converge to the equilibrium measure:
Theorem B. Let \((X,L)\) be a compact complex manifold equipped with a big line bundle. Let \(K\) be a non-pluripolar compact subset of \(X\) and \(\phi\) be a continuous weight on \(L\). Let \(\mu\) be a Bernstein-Markov measure for \((K,\phi)\). Then \[ \lim_{k\rightarrow \infty} \beta(\mu,k\phi) = \mu_{eq}(K,\phi) \] holds in the weak topology of measures.
Both Theorem A and B are obtained as special cases of Theorem C below which gives a more general criterion ensuring convergence of Bergman measures to equilibrium in terms of \(\mathcal{L}\)-functionals, first introduced by S. K. Donaldson [“Scalar curvature and projective embeddings. II”, Q. J. Math. 56, No. 3, 345–356 (2005; Zbl 1159.32012); “Some numerical results in complex differential geometry”, Pure Appl. Math. Q. 5, No. 2, 571–618 (2009; Zbl 1178.32018)].
The \(L^2\) and \(L^\infty\) norms on \(H^0(kL)\) are described geometrically by their unit balls, which will be denoted by \[ \mathcal{B}^\infty(K,k\phi) \subset \mathcal{B}^2(\mu,k\phi) \subset H^0(kL). \] Fix a reference weighted compact set \((K_0,\phi_0)\) and a probability measure \(\mu_0\) on \(K_0\) which is Bernstein-Markov with respect to \((K_0,\phi_0)\). Normalize the Haar measure vol on \(H^0(kL)\) by \[ \mathrm{vol}\mathcal{B}^2(K_0,k\phi_0)=1, \]
and introduce the \(\mathcal{L}\)-functionals \[ \mathcal{L}_k(\mu,\phi) := \frac{1}{2kN_k} \log\mathrm{vol} \mathcal{B}^2(\mu,k\phi) \] and \[ \mathcal{L}_k(K,\phi) := \frac{1}{2 k N_k} \log\mathrm{vol} \mathcal{B}^\infty(K,k\phi). \]
For a psh function with minimal singularities \(\psi\), let \(\mathcal{E}(\psi)\) be the Monge-Ampère energy, characterized as the primitive of the Monge-Ampère operator
\[ \frac{d}{dt}|_{t=0^+} \mathcal{E}(t\psi_1 + (1-t)\psi_2)= \int_X (\psi_1 -\psi_2) (dd^c\psi_2)^n \]
normalized by \(\mathcal{E}(\phi^*_{0,K_0})\).
Then the energy at equilibrium of \((K,\phi)\) is \[ \mathcal{E}_{eq}(K,\phi):= \frac{1}{\mathrm{vol}(L)} \mathcal{E}(\phi_K^*) \]
and by Theorem A in [the first two authors, loc. cit.] we have:
\[ \lim_{k\rightarrow \infty} \mathcal{L}_k(K,\phi) = \mathcal{E}_{eq}(K,\phi). \]
If \(\mu\) satisfies the Bernstein-Markov property described above, then also \[ \lim_{k\rightarrow \infty} \mathcal{L}_k(\mu,\phi) = \mathcal{E}_{eq}(K,\phi). \]
The key result (implying Theorem A and Theorem B) of the paper under review is as follows:
Theorem C. Let \((\mu_k)\) be a sequence of probability measures on \(K\) such that \[ \lim_{k\rightarrow \infty} \mathcal{L}_k(\mu_k,\phi) = \mathcal{E}_{eq}(K,\phi). \] Then the associated Bergman measures satisfy \[ \lim_{k\rightarrow \infty} \beta(\mu_k,k\phi) = \mu_{eq}(K,\phi) \] in the weak topology of measures.
This pretty general statement comprises a lot of classical and more special results and yields some nice applications to interpolation. We refer to the introduction of the paper under review for more details.

MSC:

32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
32L05 Holomorphic bundles and generalizations
32U15 General pluripotential theory
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
31C15 Potentials and capacities on other spaces
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