Let $E_{ n}$ be an holomorphic bundle of rank two on an algebraic curve $X$ (the degree of $E_{ n}$ is $n$ apart from an additive constant). Note by $Met(E_{ n})$ the space of hermitian metrics $h$ on $E_{ n}$. Also, consider $Met(W_{ n})$, the space of metrics on $H^{ 0}(X,E_{ n})$. Although $Met(W_{ n})$ is finite dimensional, it's dimension grows with $n$. So we can ask how to obtain a way to describe particular metrics of $Met(E_{ n})$ using $Met(W_{ n})$. First, we link these two spaces by two morphisms $L_{ n}$ and $I_{ n}$. Donaldson gave a criterion for detecting Einstein Hermitian metrics on $Met(E_{ n})$, using a functional $\mathcal M$. We construct another functional on $Met(W_{ n})$, $\mathcal KN_{ n}$, using an algebraic idea of Kempf and Ness. In our investigations we prove that the variation of the analytic torsion, when $h$ varies, becomes small when $n$ grows . However our main result is that $\mathcal M-\mathcal KN\circ L_{ n}$ becomes small when $n$ grows. Moreover, with some hypotheses, we show that Yang-Mills minimum can be characterized(appromimately) by $\mathcal KN_{ n}$. The proof of our main theorem is based on constructing "concentrated" sections of $E_{ n}$, in a way similar to Donaldson's recent work on symplectic varieties.
Categories: math.DG math.AG