The general-rank specialization formula for refined Poincaré polynomials

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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)2g−2.\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.

References

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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