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arXiv · 2608.23747

The $J$-equation on K\"ahler manifolds under a smooth boundary cone condition

Junbang Liu
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Abstract

For a $n$-dimensional compact K\"ahler manifold $X$ with a pair of K\"ahler 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 $c_{\alpha,\beta}\omega^{n-1}-(n-1)\omega^{n-2}\wedge\chi\geq 0$ for some K\"ahler 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 K\"ahler space.

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Junbang Liu. 2026-08-24. The $J$-equation on K\"ahler manifolds under a smooth boundary cone condition. https://arxiv.org/abs/2608.23747

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