T-structures on a local 4-Calabi–Yau variety

Let \(X= {{\,\textrm{Tot}\,}}\Omega _{\mathbb {P}^2}\) $X=\text{Tot}{\mathrm{\Omega }}_{{\mathbb{P}}^2}$ be the total space of cotangent bundle of \(\mathbb {P}^2\) ${\mathbb{P}}^2$ . 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 the tilting objects to characterize a large class of t-structures in the relevant Calabi–Yau category and try to calculate the tilting of those t-structures in the sense of Happel–Reiten–Smalø.