Hilbert schemes of points on Calabi-Yau 4-folds via wall-crossing (Q6547688)

\textit{S. K. Donaldson} [J. Differ. Geom. 18, 279--315 (1983; Zbl 0507.57010)] defined invariants counting antiself-dual connections on a real 4-manifold. \textit{S. K. Donaldson} and \textit{R. P. Thomas} [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)] proposed a holomorphic version of this construction for a Calabi-Yau fourfold. To give a rigorous formulation of their ideas, a new approach to sheaf counting was pioneered by \textit{D. Borisov} and \textit{D. Joyce} [Geom. Topol. 21, No. 6, 3231--3311 (2017; Zbl 1390.14008)] using derived differential geometry and \textit{J. Oh} and \textit{R. P. Thomas} [Duke Math. J. 172, No. 7, 1333--1409 (2023; Zbl 1525.14052)]. They constructed new virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau 4-folds, which are not canonically determined by their deformation and obstruction theory.\N\N\textit{J. Gross} et al. [SIGMA, Symmetry Integrability Geom. Methods Appl. 18, Paper 068, 61 p. (2022; Zbl 1505.14026)] proposed a wall-crossing conjecture for Calabi-Yau fourfolds. Assuming this conjecture, the author proves the conjecture for 0-dimensional sheaf-counting invariants on projective Calabi-Yau 4-folds. He also extracts the full topological information contained in the virtual fundamental classes of Hilbert schemes of points, which turns out to be equivalent to the data of all descendent integrals. One can then express many generating series of invariants in terms of explicit universal power series.\N\NOn \(\mathbb{C}^4\), Nekrasov proposed invariants with a conjectured closed form. The author shows that an analog of his formula holds for compact Calabi-Yau 4-folds satisfying the wall-crossing conjecture. There is also a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre-Verlinde correspondence for Calabi-Yau fourfolds.