The conjectural dotted-arrow identification for seminilpotent quiver varieties

Let QQ be the quiver, let dNQ0\mathbf{d}\in\mathbb{N}^{Q_0}, and let NΠQ,dSSN\mathfrak{N}_{\Pi_Q,\mathbf{d}}^{\mathcal{SSN}} be the seminilpotent stack, with irreducible components ZIrr(NΠQ,dSSN)Z\in\operatorname{Irr}(\mathfrak{N}_{\Pi_Q,\mathbf{d}}^{\mathcal{SSN}}). Let fZf_Z denote the element associated with ZZ by the cited lemma, and let [Z][Z] be its fundamental class. Dotted-arrow identification. The assignment

fZ(1)d,d[Z]f_Z\longmapsto (-1)^{\langle\mathbf{d},\mathbf{d}\rangle}[Z]

should define the dotted arrow in the realization diagram in such a way that the entire diagram commutes. This would identify the basis indexed by irreducible components with the corresponding fundamental classes and complete the compatibility among the algebraic, constructible-function, and Borel--Moore homology realizations. The source does not provide evidence that this conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Lucien Hennecart, “Geometric realisations of the unipotent enveloping algebra of a quiver”, arXiv:2209.06552 (2024).

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