Derived Θ-stratifications and the D-equivalence conjecture

The theory of \Θ-stratifications generalizes a classical stratification the moduli of vector bundles on a smooth curve, the Harder-Narasimhan-Shatz, to any moduli problem that can be represented by an algebraic. Using derived algebraic geometry, we develop a structure theory, which a refinement of the theory of local cohomology, for the derived category of-coherent complexes on an algebraic stack equipped with a\Θ-stratification. We then apply this to the D-equivalence conjecture, predicts that birationally equivalent Calabi-Yau manifolds have derived categories of coherent sheaves. We prove that any two Calabi-Yau manifolds that are birationally equivalent to a smooth space of Gieseker semistable coherent sheaves on a $K3$ surface have derived categories. This establishes the first known case of the-equivalence conjecture for a birational equivalence class in dimension than three.