O-singular locus conjecture for quantized symplectic resolutions

Let XX be a conical symplectic resolution with a Hamiltonian torus action TT having finitely many fixed points, and choose a generic one-parameter subgroup nunu. The category Onu(Aλ)\mathcal{O}_nu(\mathcal{A}_\lambda) has Cartan subquotient Cnu(Aλ)\mathsf{C}_nu(\mathcal{A}_\lambda). A parameter λ\lambda is O\mathcal{O}-singular when this Cartan subquotient is not a semisimple commutative algebra of dimension XT|X^T|; denote the locus by cOsing\mathfrak{c}^{\mathcal{O}-\mathrm{sing}}.

O\mathcal{O}-singular locus conjecture. The locus cOsing\mathfrak{c}^{\mathcal{O}-\mathrm{sing}} coincides with the union of singular hyperplanes.

The paper notes that the O\mathcal{O}-singular locus is Zariski closed and contained in a finite union of essential hyperplanes, and proposes equality with the singular-hyperplane locus. The source gives no resolution status.

Sources & referencesView supporting material

Primary source

Ivan Losev, “Localization theorems for quantized symplectic resolutions”, arXiv:2103.11193 (2021).

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