Year: 2026
Type: preprint
Abstract: For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C^2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L^\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C^2$-bounds on compact subsets of the regular locus.
Source: arXiv (Cornell University)
Authors: Fang, Hao, Ma, Biao
Institutions: University of Iowa, East China Normal University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G377084566
OpenAlex ID: W7214127454