$\chi$-Independence for BPS Invariants of Warped Local Curves

A well-known conjecture of Toda’s asserts that the BPS invariants associated to a Calabi–Yau threefold are independent of the chosen Euler characteristic. We answer the question in the affirmative for a certain class of local surfaces, which we call warped local curves. Along the way we show that the moduli stack of sheaves on a warped local curve can be written as the global critical locus of a function. We also show that the moduli stack of sheaves on a curve fibration over ${\mathbb{A}}^1$ is the global critical locus of a function.