论文标题
超链齿轮在仿射复合曲线上的类似物
Analogues of hyperlogarithm functions on affine complex curves
论文作者
论文摘要
对于$ c $的光滑仿射复合曲线，代数$ \ Mathcal o__ {hol}（\ tilde c）$ holomorthic函数在其通用封面$ \ tilde c $上，在所有操作$ f \ int $ f \ int $ f f.ind $ f f.inde c $上都有独特的最低属于$ \ tilde c）$ $ f \ int $ f for p \ int $ ffff，而$ f \ int $ ffforcy（c）$ f \ int $ ffforc， $ c $的差异。我们将$ a_c $带有迭代集成图$ i_ {x_0}的图像：\ Mathrm {sh}（ω（c））\ to \ Mathcal o_ {hol}（\ tilde c）$基于任何点$ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $ \\ tilde c $（$ \ \ tilde c $）矢量空间以及与单位部分有关，就$ \ mathrm {aut}（\ tilde c/c）$（$ \ mathcal o_ {hol}（h hol}（\ tilde c）$中等增长功能的动作）。我们表明，$ c $上的任何常规Maurer-Cartan（MC）元素$ j $ in $ c $具有拓扑上免费的lie代数超过$ \ mathrm h^1 _ {\ mathrm {\ mathrm {dr}}}（c）^*$，引起了$ a_c $ a_c $ a_c $的同态，$ \ nmatcal o（c）c） h^1 _ {\ mathrm {dr}}（c））$，其中$ \ mathcal o（c）$是$ c $上的常规函数的代数，从而导致subalgebra $ \ mathcal h_c（j）$ a_c $ a_c $ of $ a_c $（isomorphic of $ \ a_c $（isomorphic） H^1 _ {\ Mathrm {dr}}（c））$。我们还将MC元素$J_σ$与投影$ω（c）\的每个部分$σ$相关联与\ Mathrm H^1 _ {\ Mathrm {dr}}}（c）$;当$ c $具有$ 0 $的属时，我们将展示特定的$σ_0$，其中$ \ MATHCAL H_C（J_ {σ_0}）$是Hyperlogarithm函数的代数（Poincaré，Lappo-Danilevsky）。
For $C$ a smooth affine complex curve, there is a unique minimal subalgebra $A_C$ of the algebra $\mathcal O_{hol}(\tilde C)$ of holomorphic functions on its universal cover $\tilde C$, which is stable under all the operations $f\mapsto \int fω$, for $ω$ in the space $Ω(C)$ of regular differentials on $C$. We identify $A_C$ with the image of the iterated integration map $I_{x_0} : \mathrm{Sh}(Ω(C))\to\mathcal O_{hol}(\tilde C)$ based at any point $x_0$ of $\tilde C$ (here $\mathrm{Sh}(-)$ denotes the shuffle algebra of a vector space), as well as with the unipotent part, with respect to the action of $\mathrm{Aut}(\tilde C/C)$, of a subalgebra of $\mathcal O_{hol}(\tilde C)$ of moderate growth functions. We show that any regular Maurer-Cartan (MC) element $J$ on $C$ with values in the topologically free Lie algebra over $\mathrm H^1_{\mathrm{dR}}(C)^*$ gives rise to an isomorphism of $A_C$ with $\mathcal O(C) \otimes\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$, where $\mathcal O(C)$ is the algebra of regular functions on $C$, leading to the assignment of a subalgebra $\mathcal H_C(J)$ of $A_C$ (isomorphic to $\mathrm{Sh}(\mathrm H^1_{\mathrm{dR}}(C))$) to any MC element. We also associate a MC element $J_σ$ to each section $σ$ of the projection $Ω(C)\to \mathrm H^1_{\mathrm{dR}}(C)$; when $C$ has genus $0$, we exhibit a particular section $σ_0$ for which $\mathcal H_C(J_{σ_0})$ is the algebra of hyperlogarithm functions (Poincaré, Lappo-Danilevsky).
