Bondal–Orlov–Kawamata DK conjecture

Let $\mu:X_1\longrightarrow X_2$ be a birational map of smooth varieties, resolved by morphisms

$$f_i:X_0\longrightarrow X_i$$

such that $f_1^*K_{X_1}$ and $f_2^*K_{X_2}$ are linearly equivalent; this is a K-equivalence. DK conjecture. Then $X_1$ and $X_2$ are derived equivalent. This conjecture predicts that K-equivalent smooth varieties have equivalent derived categories, and motivates the derived-equivalence result proved in the paper for simple flops of type $D_5$.