Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles
Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles
复制标题
三阶抛物线希格斯丛的模空间的贝蒂数
DOI：
10.1090/memo/0879
复制
发表时间：
2004
影响因子：
2.6
通讯作者：
V. Muñoz
中科院分区：
文献类型：
--
作者：
O. Garc'ia;P. Gothen;V. Muñoz
黎曼曲面上的抛物型希格斯丛由于许多原因而引起人们的兴趣，其中之一是它们在研究
复一般线性群中穿孔曲面的基本群。本文计算了秩为3的抛物模空间的Betti数
用莫尔斯理论计算行列式固定和不固定的希格斯丛。一个关键点是，某些临界子流形的莫尔斯函数可以确定与模空间的抛物三元组。这些模空间的家庭取决于一个真实的参数，我们进行了仔细的分析，他们通过研究他们的变化与此参数。因此，我们获得特别是有关抛物型三重模空间的拓扑结构的参数的值相关的抛物型希格斯包的研究信息。其余的临界子流形也被描述：其中之一是模空间的抛物丛，而其余的有一个描述方面的对称产品的黎曼曲面。作为我们的莫尔斯理论分析的另一个结果，我们得到了Laumon定理的抛物形式的证明，该定理指出幂零锥（零在Hitchin映射下的原像）是抛物模空间的拉格朗日子簇。
希格斯粒子束。
Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the
fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic
Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic
Higgs bundles.