Uniformly bounded orthonormal sections of positive line bundles on complex manifolds

Abstract

We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers $L^k$ of a positive holomorphic line bundle $L$ on a compact Kähler manifold $M$. In particular, we construct for each positive integer $k$, orthonormal sections $s_1^k,\dots ,s_{n_k}^k$ in $H^0(M,L^k)$, $n_k\ge \beta \dim H^0(M,L^k)$, such that $\{s_j^k\}$ is a uniformly bounded family, where $\beta$ is an explicit positive constant depending only on the dimension of $M$. For $m=1$, we can take $\beta =.99564$.

13 pages