A Log Canonical Counterexample to the Chen--Jiang Decomposition

We construct a projective log-smooth surface pair $(X,\mathrm{\Delta })$ and a morphism $f:X\to E$ to an elliptic curve such that $f_{\ast }{\text{OO}}_X(K_X+\mathrm{\Delta })$ is Atiyah’s indecomposable rank-two bundle $F_2$ of degree zero. This gives a negative answer to Question 3.4 of Jiang~\cite{Jiang}. Products yield counterexamples in every higher direct image degree and over abelian bases of arbitrary positive dimension.