Abstract: The smooth solvability of the J-equation on a compact Kähler manifold is closely related to numerical positivity of pairs of Kähler classes. When these numerical conditions degenerate or fail, a natural question is whether the equation admits a canonical singular solution. In this talk, I will discuss recent progress on degenerate J-equations, focusing on the existence, uniqueness, and partial regularity of solutions at the birational minimal slope. I will also describe related results in the J-nef setting, including an analytic characterization of the J-null locus and the finiteness of J-null subvarieties. These results clarify the relationship between numerical obstructions and analytic degeneration.

Brief biography: Junbang Liu is currently a postdoctoral fellow at Hong Kong University of Science and Technology. He obtained his Ph.D. at Stony Brook University in 2025. His research focus on complex differential geometry, and geometric PDEs.

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