Uniformly bounded orthonormal sections of positive line bundles on complex manifolds (Q2786825)

Let \((L,h)\) be a positive Hermitian line bundle on a compact Kähler manifold \((M,\omega)\) of dimension \(m\). Consider the space \(H^0(M,L^k)\) of global holomorphic sections of \(L^k=L^{\otimes k}\) endowed with the natural inner product induced by \(h^{\otimes k}\) and volume form \(\omega^m/m!\). The paper under review deals with the interesting open question whether there exist uniformly bounded sequences of orthonormal bases of \(H^0(M,L^k)\), \(k\geq1\). The author gives a nice account of earlier results motivating this question. The main result of the paper gives a partial answer to this question. In the above setting the author proves that there exist constants \(C>0\) and \(0<\beta<1\) such that for every \(k\geq1\) there exist orthonormal sections \(s^k_j\) of \(H^0(M,L^k)\), \(1\leq j\leq n_k\), with \(n_k\geq\beta\dim H^0(M,L^k)\) and \(\|s^k_j\|_\infty\leq C\) for all \(k\geq1\), \(1\leq j\leq n_k\). The author also obtains explicit values \(\beta_m<1\) for the constant \(\beta\) in the above theorem that depend only on the dimension \(m\). Moreover he asks whether these values \(\beta_m\) can be improved by considering sections related to a special collection of points such as Fekete points. Subsequently it was shown by \textit{J. Marzo} and \textit{J. Ortega-Cerdà} that by using Fekete points the above-mentioned result holds for all \(\beta<1\) [Bull. Lond. Math. Soc. 47, No. 5, 883--891 (2015; Zbl 1327.32034)].NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].