Fourier--Mukai partners of elliptic surfaces

For smooth projective varieties $X$ and $Y$, when their derived categories are equivalent we say that $X$ is a Fourier--Mukai partner of $Y$. We study the set of isomorphism classes of Fourier--Mukai partners of elliptic surfaces with negative Kodaira dimensions. As an application, we give a counterexample to Kawamata's D-K conjecture and a counterexample to the birational Torelli probrem for minimal $3$-folds.