Analytic Zariski decomposition (Q1803563)

Let \(X\) be a projective variety and let \(D\) be an \(\mathbb{R}\)-Cartier divisor on \(X\). The expression \(D=P+N\) \((P,N\in\text{Div}(X)\otimes\mathbb{R})\) is called a Zariski decomposition of \(D\), if the following conditions are satisfied: 1. \(P\) is nef, 2. \(N\) is effective, 3. \(H^ 0(X,{\mathcal O}_ X([\nu P]))\simeq H^ 0(X,{\mathcal O}_ X([\nu D]))\) holds for all \(\nu\in\mathbb{Z}_{\geq 0}\) where [ \ ]'s denote the integral parts of the divisors. Although many useful applications of this decomposition have been known, as for the existence, very little has been known. There is the following (rather optimistic) conjecture: Let \(X\) be a normal projective variety and let \(D\) be a pseudoeffective \(\mathbb{R}\)-Cartier divisor on \(X\). Then there exists a modification \(f:Y\to X\) such that \(f^*D\) admits a Zariski decomposition. In this paper, I would like to announce a ``weak solution'' to the conjecture. Details will be published elsewhere.