AI 中文总结
该研究证明陈类非零阻碍解析准周期配边的可测特征截面，并在秩二丢番图情形给出精确判据，通过磁荷与协变性方法实现。
AI 中文摘要
我们研究由酉缝合矩阵描述的厄米向量丛上二维环面的遍历平移作用的线性配边。对于连续不变复线子丛 $L$，我们证明在可测相位共边界条件下，$c_1(L) \ e 0$ 阻碍所有非零可测特征截面。该抽象障碍对任意秩成立，而对于具有对角占优分裂的秩二解析配边在丢番图平移上，相位条件是自动满足的。在该情形下，判据是精确的：$L$ 携带非零可测特征截面当且仅当 $c_1(L)=0$ 且乘子相位的绕转向量为零。当这些条件满足时，实现的特征值在圆上形成稠密陪集；对于每个这样的特征值，由 $L$ 承载的特征空间是一维的，并且具有实解析无处为零的生成元。我们还构造了一个刘维尔例子，表明丢番图假设不能被移除。证明将陈数转化为万有覆盖上的磁荷，并应用无理磁平移的协变性。
英文摘要
We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle $L$, we prove that $c_1(L) \ne 0$ obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: $L$ carries a nonzero measurable eigensection if and only if $c_1(L)=0$ and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by $L$ is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.
Comments38 pages, 49 references
