Zariski decompositions of numerical cycle classes (Q2832762)

The authors' goal in this paper is to develop a decomposition theory for numerical classes of cycles of any codimension, which generalize the Zariski decomposition of pseudo-effective divisor on a smooth surface. Let \(X\) be a projective variety over an algebraically closed field. Let \(N_k(X)\) denote the \(\mathbb{R}\)-vector space of \(k\)-cycles with \(\mathbb{R}\)-coefficients modulo numerical equivalence. The pseudo-effective cone \(\overline{\text{Eff}}_k(X)\subset N_k(X)\) is the closure of the cone generated by classes of effective cycles. The classes in the interior of \(\overline{\text{Eff}}_k(X)\subset N_k(X)\) is called big classes. The movable cone \(\overline{\text{Mov}}_k(X)\) is the closure in \(N_k(X)\) of the cone generated by members of irreducible families of \(k\)-cycles which dominate \(X\). In [\textit{B. Lehmann}, ``Geometric characterizations of big cycles'', Preprint, \url{arXiv:1309.0880}] a homogeneous continuous function \(m(\alpha)\) on \(N_k(X)\), called the mobility, is introduced and is shown to be positive precisely for big classes. The main theorem of this paper shows that an arbitrary pseudo-effective class \(\alpha\) can be decomposed as a sum \(\alpha=P(\alpha)+N(\alpha)\) where \(P(\alpha)\) is movable, \(N(\alpha)\) is pseudo-effective, and \(\text{mob}(P(\alpha))=\text{mob}(\alpha)\).