  Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are
called {\it $D$-equivalent} if their derived categories of bounded complexes of
coherent sheaves are equivalent as triangulated categories, while {\it
$K$-equivalent} if they are birationally equivalent and the pull-backs of their
canonical divisors to a common resolution coincide. We expect that the two
equivalences coincide at least for birationally equivalent varieties. We shall
provide a partial answer to the above problem in this paper.
