Zhou valuations and jumping numbers

Abstract

In this article, we prove that for any Zhou valuation $\nu$, there exists a graded sequence of ideals $\mathfrak{a}_{\bullet}$ and a nonzero ideal $\mathfrak{q}$ such that $\nu$ $\mathscr{A}-$computes the jumping number $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$, and that for the subadditive sequence $\mathfrak{b}^{\varphi}_{\bullet}$ related to a plurisubharmonic function $\varphi$, there exists a Zhou valuation which $\mathscr{A}-$computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})$, where the ``$\mathscr{A}-$compute'' coincides with the ``compute'' in Jonsson-Musta\c{t}\u{a}'s Conjecture when the Zhou valuation $\nu$ is quasimonomial. There are also some results obtained for Zhou valuations, including a characterization for a valuation being a Zhou valuation, and a denseness property of the cone of Zhou valuations.