The general-rank specialization formula for refined Poincaré polynomials

Let Nr,dN_{r,d} be the moduli space of stable rank-rr bundles of degree dd on a smooth projective curve of genus gg, and let Ω(Nr,d,q,t)\Omega(N_{r,d},q,t) denote its refined Poincaré polynomial. The specialization formula.

Ω(Nr,d,q,1)=k=2r(1(q)k)2g2.\Omega(N_{r,d}, q, -1)=\prod_{k=2}^r(1-(-q)^k)^{2g-2}.

This is proposed as a formula for the t=1t=-1 specialization in general rank; the surrounding discussion presents it as conjectural, and no general proof or resolution is supplied.

Sources & referencesView supporting material

Primary source

Woonam Lim, Miguel Moreira and Weite Pi, “On the Chern filtration for the moduli of bundles on curves”, arXiv:2410.24008 (2024).

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