Abel定理的推广与Abel-Jacobi映射
A generalization of Abel's Theorem and the Abel--Jacobi map
摘要：
推广了Abel关于黎曼曲面上因子线性等价的经典定理。对于每一个闭子流形$M^d \subset X^n$在一个紧凑的定向黎曼$n$ -流形，或更一般地为任何$d$ -循环$Z$相对于$X$的三角形，我们定义一个（单纯）$(n-d-1)$ -gerbe $Λ_{Z}$，阿贝尔gerbe确定的$Z$，其消失作为Deligne上同调类推广的概念“线性等价于零”。在这种情况下，阿贝尔定理仍然有效。推广了经典的Abel-Jacobi映射的反演定理，从而证明了Abel gerbes的模空间同构于调和Deligne上同调，即具有调和曲率的gerbes.
英文摘要：
We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $Λ_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of `linear equivalence to zero'. In this setting, Abel's theorem remains valid. Moreover we generalize the classical Inversion Theorem for the Abel--Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.
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