well-known case of a surface carries on to the case where $X$ is a compact hyper-Kähler manifold. Using the quadratic Beauville-Bogomolov form on ${H^{1,1}(X,{\mathbf{R}})}$ and deep results due to D.Huybrechts, we can prove the following facts: a family of primes is exceptional in our sense iff their Gram matrix is negative definite. In particular, a prime is exceptional iff it has negative square, and this forces it to be uniruled. The modified nef cone of a hyper-Kähler manifold is just the dual cone to the pseudoeffective cone, which is also the closure of the so-called birational (or bimeromorphic) Kähler cone. Finally, the divisorial Zariski decomposition is the unique orthogonal decomposition into a modified nef class and an exceptional divisor. In particular, the divisorial Zariski decomposition is also rational in that case.\
In a last part, we explain how to tackle the above constructions in a more algebraic fashion. When $L$ is a big divisor on a projective manifold, we prove that the divisorial Zariski projection of $L$ is the only decomposition $L=P+N$ into real divisors with $P$ modified nef and $H^0(\lfloor kP\rfloor)=H^0(kL)$ for every $k$. The minimal multiplicities of $\{L\}$ (and thus its negative part) can be recovered from the 