Log Canonical Counterexamples to the Chen--Jiang Decomposition

Abstract

We construct a projective log-smooth surface pair $(X,\Delta)$ and a morphism $f:X\to E$ to an elliptic curve such that $f_*\OO_X(K_X+\Delta)$ 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}. We also construct a log canonical pair $(X,\Delta)$ and a morphism $f$ to an abelian surface such that $f_*\OO_X(m(K_X+\Delta))$ does not admit a Chen--Jiang decomposition for any integer $m\ge2$. Products yield counterexamples in every higher direct image degree and over abelian bases of arbitrary positive dimension.