AI 中文总结
本文构造了一个对数光滑曲面偶及到椭圆曲线的态射，使其直像为Atiyah不可分解丛，否定了Jiang问题3.4，并通过乘积推广到高直像及阿贝尔基。
AI 中文摘要
我们构造了一个射影对数光滑曲面偶$(X,\Delta)$以及一个到椭圆曲线的态射$f:X\to E$，使得$f_*\OO_X(K_X+\Delta)$是Atiyah的不可分解的秩二丛$F_2$，其度为0。这给出了Jiang~\cite{Jiang}中问题3.4的否定回答。通过乘积，我们在每个高直像次数以及任意正维数的阿贝尔基上得到反例。
英文摘要
We construct a projective log-smooth surface pair $(X,Δ)$ and a morphism $f:X\to E$ to an elliptic curve such that $f_*\OO_X(K_X+Δ)$ 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 $(X,Δ)$ and a morphism $f$ to an abelian surface such that $f_*\OO_X(m(K_X+Δ))$ 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.
Comments15 pages
