On the Datar-Mete-Song minimal slope conjecture

2026/08/02 by Xin Fu
Mathematics · #math.AG #math.AP #math.CV #msc:32U15 #msc:32W20

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References updated

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We prove a conjecture of Datar-Mete-Song \citeDMS characterizing J-slope semi-stability by the minimal J-slope. More precisely, for a semi-stable pair of Kähler classes (α,β) on a compact Kähler manifold X, every big and nef birational test class has slope at least the topological J-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the J-null locus of a semi-stable pair and prove that it is an analytic subset of X if X is a compact Kähler surface or a compact toric Kähler manifold. In the toric invariant case, we show that Murakami's \citeMurakami weak solution to the J-equation is smooth and Kähler on the dense big torus (ℂ^*)ⁿ of X.

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