Derived $\Theta$-stratifications and the $D$-equivalence conjecture

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