  Let $X$ denote the total space of cotangent bundle of projective plane. This
is a non-compact Calabi-Yau $4$-fold (also called local Calabi-Yau variety in
physics literature). The aim of this paper is to use tilting objects to
characterize a large class of t-structures in the relevant Calabi-Yau category
and try to calculate the combinatorics of them.
