The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the C2-Segal conjecture. We compute the localized slice spectral sequence of the C4-norm of BPR in a range and show that the Hill–Hopkins–Ravenel slice differentials are in one-to-one correspondence with a family of Tate differentials for N12HF2.