  Zariski decomposition plays an important role in the theory of algebraic
surfaces due to many applications. For irreducible symplectic manifolds
Boucksom provided a characterization of his divisorial Zariski decomposition in
terms of the Beauville-Bogomolov-Fujiki quadratic form. Different variants of
singular holomorphic symplectic varieties have been extensively studied in
recent years. In this note we first show that the "Boucksom-Zariski"
decomposition holds for effective divisors in the largest possible framework of
varieties with symplectic singularities. On the other hand in the case of
projective surfaces, it was recently shown that there is a strict relation
between the boundedness of coefficients of Zariski decompositions of
pseudoeffective integral divisors and the bounded negativity conjecture. In the
present note, we show that an analogous phenomenon can be observed in the case
of projective irreducible symplectic manifolds. We furthermore prove an
effective analog of the bounded negativity conjecture in the smooth case.
Combining these results we obtain information on the denominators of
"Boucksom-Zariski" decompositions for holomorphic symplectic manifolds. From
such a bound we easily deduce a result of effective birationality for big line
bundles on projective holomorphic symplectic manifolds, answering to a question
asked by F. Charles.
