Slice spectral sequences through synthetic spectra

We define a $t$-structure on the category of filtered $G$-spectra such that for a Borel $G$-spectrum $X$ the slice filtration of $X$ is the connective cover of the homotopy fixed-point filtration of $X$. Using this, we show that the slice spectral sequence for the norm $N_{C_2}^{G}M{U}_{\mathbb{R}}$ of Real bordism theory refines canonically to a ${\mathbb{E}}_{\mathrm{\infty }}$-algebra in $MU$-synthetic spectra, when $G$ is a cyclic $2$-group. Concretely, this gives a map of multiplicative spectral sequences from the classical Adams–Novikov spectral sequence of $\mathbb{S}$ to the slice spectral sequence for $N_{C_2}^{G}M{U}_{\mathbb{R}}$ that respects the higher ${\mathbb{E}}_{\mathrm{\infty }}$ structure, such as Toda brackets and power operations. We give a conjecture on the existence of vanishing lines in the equivariant Adams–Novikov spectral sequence based at tom Dieck’s homotopical complex bordism $M{U}_G$. Conditional on this conjecture, our $t$-structure implies that the slice filtration for $N_{C_2}^{G}M{U}_{\mathbb{R}}$ lifts further to an $\mathcal{O}$-algebra in $M{U}_G$-synthetic spectra, where $\mathcal{O}$ is the ${\mathbb{N}}_{\mathrm{\infty }}$-operad with all norms from nontrivial subgroups of $G$.