Positivity for arbitrary coherent quotients on Deligne–Mumford stacks

The problem asks whether the positivity theorem extends from torsion-free quotients to arbitrary coherent quotients. A 2026 paper claims an affirmative answer under the original geometric hypotheses.

Known results

• The earlier theorem proves pseudo-effectivity of $c_1(\mathcal Q)$ for every torsion-free coherent quotient of positive tensor powers of $\Omega^1_{\mathcal X}(\log\Delta)$.
• The arbitrary-quotient case was previously known when the torsion subsheaf is a quotient of a torsion-free sheaf, including certain quotient stacks.
• The obstruction was the determinant and effectivity of the torsion subsheaf.

2026 claimed extension

The new paper decomposes $\mathcal Q$ into its torsion subsheaf and torsion-free quotient, asserts that the torsion part is a quotient of a torsion-free sheaf with effective first Chern class, and combines this with the earlier theorem. This yields pseudo-effectivity for every coherent quotient, but the claim currently lacks independent mathematical corroboration.

Current status (as of July 2026): The extension is claimed in a new paper, while the earlier torsion-free theorem is established; independent verification of the new argument remains open.