Semistable $J$-equation on Kähler threefold

Abstract

For a $3$-dimensional compact Kähler manifold $X$ with a pair of Kähler class $(α,β)$ which is $J$-semistable, we show that the $J$-null locus of $(α,β)$ forms a subvariety of $X$. We also give an analyic characterization of the $J$-null locus using Kähler currents with analytic singularities analogous to the theorem of Collins-Tosatti \cite{CT15}. As an application, we show the smooth convergence of the $J$-flow off the $J$-null locus. It gives higer order partial regularity for weak solutions of semistable $J$-equations obtained in \cite{M26b}.