Real-normalized differentials: limits on stable curves [Вещественно-нормированные дифференциалы: пределы на стабильных кривых]
Real-normalized differentials: limits on stable curves [Вещественно-нормированные дифференциалы: пределы на стабильных кривых]
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实数归一化微分：稳定曲线的限制 [ÐеѪеÐÑвеннÐ⁄-ниÑÐ⁄иÑиваннÑе диÑÑеÑен呸呸呸呸呸呸
DOI：
10.1070/rm9877
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发表时间：
2019
影响因子：
0.9
通讯作者：
Norton, Chaya
中科院分区：
文献类型：
--
作者：
Grushevsky, Samuel;Krichever, Igor Moiseevich;Norton, Chaya
研究了黎曼曲面上的实正规亚纯微分在退化下的行为。我们描述了RN微分在任何稳定曲线上的所有可能的极限。特别是，我们证明了在节点处的剩余是一个合适的基尔霍夫问题的曲线的对偶图的解决方案。我们进一步表明，RN微分的零点的极限是一个扭曲的微分-一个明确构造的RN微分的不可约组件的稳定曲线，在某些节点上的高阶极点的集合的零点的除数。我们的主要工具是一种新的方法，用于构造微分（在本文中，RN微分，但该方法更一般）光滑的黎曼曲面，在一个给定的稳定曲线的铅垂邻域。为了实现这一点，我们认为一个光滑的黎曼曲面是一个稳定曲线中节点邻域的补，边界圆成对标识。然后，在具有规定奇异性的光滑表面上构造微分可以简化为沿着所识别的圆（接缝）沿着具有规定“跳跃”（不匹配）的合适的归一化全纯微分的构造。我们解决这个加法模拟的乘法Riemann-Hilbert问题的一种新的方式，通过迭代使用柯西积分内核的不可约组件的稳定曲线，而不是使用柯西内核的铅垂光滑表面。由于稳定曲线是固定的，这为构造的微分提供了明确的估计，并允许精确的退化分析。
We study the behaviour of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeros of RN differentials are the divisor of zeros of a twisted differential—an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighbourhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighbourhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduced to a construction of a suitable normalized holomorphic differential with prescribed'jumps'(mismatches) along the identified circles (seams). We solve this additive analogue of the multiplicative Riemann–Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis.