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Zariski decomposition plays an important role in the theory of algebraicsurfaces due to many applications. For irreducible symplectic manifoldsBoucksom provided a characterization of his divisorial Zariski decomposition interms of the Beauville-Bogomolov-Fujiki quadratic form. Different variants ofsingular holomorphic symplectic varieties have been extensively studied inrecent years. In this note we first show that the "Boucksom-Zariski"decomposition holds for effective divisors in the largest possible framework ofvarieties with symplectic singularities. On the other hand in the case ofprojective surfaces, it was recently shown that there is a strict relationbetween the boundedness of coefficients of Zariski decompositions ofpseudoeffective integral divisors and the bounded negativity conjecture. In thepresent note, we show that an analogous phenomenon can be observed in the caseof projective irreducible symplectic manifolds. We furthermore prove aneffective 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. Fromsuch a bound we easily deduce a result of effective birationality for big linebundles on projective holomorphic symplectic manifolds, answering to a questionasked by F. Charles.

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