The stratified Kisin variety determination conjecture for deformation spaces

Let $D^\psi({\bf v},{\rm t},\overline\rho)$ denote the potentially Barsotti–Tate deformation space, and let $\overline{\mathscr{G\!R}}^\psi({\bf v},{\rm t},\overline\rho)$ be the associated Kisin variety with its stratification. Assume that the Galois type ${\rm t}$ is non-degenerate. Two stratified Kisin varieties are identified up to an isomorphism preserving the stratification, allowing a permutation of the $f$ components. The stratified Kisin variety determination conjecture. The deformation space $D^\psi({\bf v},{\rm t},\overline\rho)$ is completely determined by the isomorphism class of $\overline{\mathscr{G\!R}}^\psi({\bf v},{\rm t},\overline\rho)$ with its stratification. The paper explains that the stratification is essential: the underlying abstract Kisin variety alone does not determine the deformation space. It also proposes a weaker conjecture for the generic fibre of the deformation ring, while the full deformation ring is described separately in terms of a gene.