AI 中文总结
该研究探究紧Kähler流形的超越数值维性质，建立其与纤维化退化除子的联系，结合Cao-Păun定理回答有理商相关问题，得到伪有效线丛的数值维不等式。
AI 中文摘要
我们研究紧Kähler流形上超越数值维的性质，尤其建立该不变量与纤维化中退化除子的联系，拓展了射影情形下的若干结果。还研究具有有理连通一般纤维的紧Kähler流形间的纤维化，证明余切丛张量幂中包含的每个伪有效线丛，在涉及退化除子的显式关系下，均来自基上对应张量幂中的伪有效线丛。最后结合这些结果与Cao-Păun定理，回答了他们提出的关于有理商的问题：设\
英文摘要
We study properties of the transcendental numerical dimension on compact Kähler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact Kähler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao--Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(Ω_X^1)^{\otimes m},~m \geq 1\). We prove that \[ ν(L,X)\leqν(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.
Comments47 pages, 1 figure. AI-assisted proof. Comments are welcome!
