Real points of coarse moduli schemes of vector bundles on a real algebraic curve
Real points of coarse moduli schemes of vector bundles on a real algebraic curve
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实代数曲线上向量丛粗模方案的实点
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发表时间：
2010
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通讯作者：
Florent Schaffhauser
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作者：
Florent Schaffhauser
研究了具有固定实结构的光滑复射影曲线上实数和四元矢量束的模问题，给出了具有固定拓扑型的半稳定实数和四元矢量束的模空间的规范理论构造。这些空间嵌入到复射影变内实数点的连通子集上。我们将这一观点与Biswas， Huisman和Hurtubise （arxiv:0901.3071）先前的工作联系起来，并利用这一观点研究了曲线上给定实结构对稳定全纯束模变所诱导的伽罗瓦作用。我们特别给出了一个harnack型定理，用$2^g +1$限定了该作用的不动点集的连通分量的数目，其中$g$是曲线的属。实际上，考虑到实结构的所有拓扑不变量，我们给出了连通分量的精确计数，从而将Gross和Harris关于实代数曲线的Picard格式的结果推广到秩r > 1$。
We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semi-stable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas, Huisman and Hurtubise (arxiv:0901.3071), and we use this to study the Galois action induced on moduli varieties of stable holomorphic bundles on a complex curve by a given real structure on the curve. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by $2^g +1$, where $g$ is the genus of the curve. In fact, taking into account all the topological invariants of the real structure, we give an exact count of the number of connected components, thus generalising to rank $r > 1$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.