The $J$-equation on Kähler manifolds under a smooth boundary cone condition

Abstract

For a $n$-dimensional compact Kähler manifold $X$ with a pair of Kähler classes $(\alpha ,\beta )$, we show that the algebraically defined $J$-null locus is equal to the analytically defined $J$-non-ample locus under the assumption of $J$-bigness(which is automatic for semistable pair up to dimension $3$ \cite{3d}), and boundary cone condition $c_{\alpha ,\beta }{\omega }^{n-1}-(n-1){\omega }^{n-2}\wedge \chi \ge 0$ for some Kähler form $\omega \in \alpha ,\chi \in \beta$. The key tool is the generalized Khovanskii-Teissier inequality associated to $J$ equation formulated by Collins \cite{CT21} and its extension to singular Kähler space.