There is a series of papers concerning equivalence between $K$-equivalence and $D$-equivalence (DK hypothesis), where two smooth projective varieties $X$ and $Y$ are said to be $K$-equivalent (resp. $D$-equivalent) if they are birationally equivalent and there is a third variety $Z$ together with birational morphisms $f:Z \to X$ and $g:Z\to Y$ such that $f^*K_X$ is linearly equivalent to $g^*K_Y$ (resp. if their bounded derived categories of coherent sheaves are equivalent as triangulated categories).
Clearly, $D$-equivalence is an equivalence relation, so for the DK hypothesis to be true, we also need to have $K$-equivalence to be an equivalence relation, but I am not sure how to show transitivity for that.
So my question is if $K$-equivalence indeed satisfies the transitivity and if so, would you give me a hint/reference?
Thank you in advance.
It is indeed transitive. I will give a hint and leave the rest to you, but I'm happy to explain if this is unclear.
Let $V_1$ define the $K$-equivalence of $X$ and $Y$ and let $V_2$ define the $K$-equivalence of $Y$ and $Z$. Then, there is a natural birational map $V_1 \dashrightarrow V_2$ which can be dominated by a common model $\Gamma$ with projective birational maps $p_1: \Gamma \to V_1$ and $p_2: \Gamma \to V_2$ fitting into a commutative square with $Y$ in the bottom right corner.
This $\Gamma$, being a common model of $X$, $Y$, and $Z$, can then be used to find a $K$ equivalence between $Z$ and $X$