Balanced metrics on complex manifolds

Let be $g$ a Kähler metric which admits a polarization $L \rightarrow M$, i.e. $L$ is a holomorphic line bundle over $M$ such that $\omega\in [c_1(L)]$, where $\omega$ is the Kähler form associated to the metric $g$. Fixed an Hermitian metric $h$ for $L$ whose Ricci curvature equals $\omega$, consider the Hilbert space $\mathcal H$ consisting of global holomorphic sections of $L$ whose $L^2$-norm is bounded. The Kähler metric $g$ is said to be {\it balanced} if the function, $\,x \mapsto \sum_j h(s_j(x), s_j(x))$ is constant, where $\{s_j\}_{j\inJ}$ is an orthonormal basis of $\mathcal H$. In the first part of the seminar I will recall the main results due to S. Donaldson about the existence and uniqueness of balanced metrics in the compact case. The second part of the seminar is dedicated to the noncompact case, in particular I will describe all balanced metrics on a bounded homogeneous domain and the link between the balanced condition and Berezin's quantization of a Kähler manifold.