圆的复变形：群上同调与维拉索洛统一化
Complex deformations of the circle: Group cohomology and Virasoro uniformization
AI总结：
本文研究圆微分同胚群的复化问题，通过考虑从圆到复平面上 punctured 点的实解析映射，定义了具有 winding number +1 的复变形，这些变形构成一个无限维流形，并利用 Witt 代数关系研究其上同调群，最终证明了 Segal 模空间的维拉索洛统一化定理。
摘要：
我们通过考虑从圆到复平面上 punctured 点的实解析映射，研究圆微分同胚群的复化问题。这些复变形构成一个无限维流形，具有部分定义的逆和组合操作，具有 Frölicher 结构的光滑性，并在单位元处由 Witt 代数给出李代数关系。考虑到共形场论的应用，我们计算了第二群上同调群，发现扩展了 Bott-Thurston 群上同调与 Gelf'and-Fuks 群上同调相关的维拉索洛代数群上同调，并发现一个自然的相对群上同调，结合了复变形的旋转数和共形半径。复变形自然作用于具有解析参数化边界组件的 (无限维) Segal 模空间上。这些作用使所述模空间具有光滑 Frölicher 结构。我们证明了维拉索洛统一化定理：Segal 模空间的切空间由 Witt 代数诱导的向量场张成。最后，我们将复变形的作用与芬奇-尼尔森坐标和有限维超几何面模空间上的 Schiffer 变分联系起来。
英文摘要：
We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an infinite-dimensional manifold with partially defined inversion and composition operations, smooth in the sense of Frölicher structures, and with Lie algebra relations at the identity given by the Witt algebra. We compute their second group cohomology with real coefficients. It is generated by two cocycles: One cocycle extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and another Euler class cocycle combining mean rotation and conformal radius. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with enumerated and analytically parametrized boundary components, equipping them with smooth Fr{ö}licher structures. We prove a Virasoro uniformization theorem: The tangent spaces of the Segal moduli spaces are spanned by vector fields induced by the Witt algebra. Finally, we relate the actions of complex deformations to Fenchel-Nielsen coordinates and Schiffer variation on finite-dimensional moduli spaces of bordered hyperbolic surfaces with one marked point on each boundary component.
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