Modern enumerative geometry studies invariants defined through virtual cycles. The main examples are counting curves via Gromov-Witten theory and counting sheaves via Donaldson-Thomas theory. There are two powerful tools developed to compute such virtual invariants: (1) Manolache's virtual pullbacks and (2) Kiem-Li's cosection localization.

Recently, Borisov-Joyce and Oh-Thomas introduced a new type of virtual cycles for Donaldson-Thomas theory of Calabi-Yau 4-folds (in short DT4 theory). In the first part of this talk, we generalize the above two tools to DT4 theory. The three main applications are Lefschetz principle, Pairs/Sheaves correspondence, and a foundation of surface counting theory. The first two applications prove various conjectures in DT4 theory. The third application connects DT4 theory with the variational Hodge conjecture.

In the second part of this talk, we revisit the above two tools via the Kimura sequence for Artin stacks, derived algebraic geometry, and algebraic cobordism. As applications, we completely remove the technical assumptions in Graber-Pandharipande's torus localization and develop cosection localization over a field of positive characteristic.

This is based on arXiv:2110.03631, joint work with Young-Hoon Kiem in arXiv:1908.03340, arXiv:2012.13167, joint work in progress with Younghan Bae and Martijn Kool, and joint work in progress with Dhyan Aranha, Adeel Khan, Alexei Latyntsev, and Charanya Ravi.