On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

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.