Cohomological -independence for Higgs bundles and Gopakumar–Vafa invariants
Tasuki Kinjo
The University of Tokyo, Tokyo, Japan; Kyoto University, Kyoto, JapanNaoki Koseki
University of Liverpool, Liverpool, UK
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Abstract
The aim of this paper is two-fold. Firstly, we prove Toda’s -independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank and Euler characteristic which are not necessary coprime, and show that it does not depend on . This result extends the Hausel–Thaddeus conjecture on the -independence of E-polynomials proved by Mellit, Groechenig–Wyss–Ziegler and Yu in two ways: We obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.
Cite this article
Tasuki Kinjo, Naoki Koseki, Cohomological -independence for Higgs bundles and Gopakumar–Vafa invariants. J. Eur. Math. Soc. (2024), published online first
DOI 10.4171/JEMS/1487