发表机构
Chern Institute of Mathematics Nankai University（南开大学陈省身数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Hitchin模空间中测地球体积的渐近公式，结合超kähler体积形式的齐次性与度量渐近性，并利用辛约化和Prym极化确定系数。
AI 中文摘要
设$X$为亏格$g\geq2$的紧黎曼曲面，令$\M$为固定奇次行列式的秩二无迹Higgs丛的模空间。对于本文所使用的Hitchin度量的归一化，我们证明对于每个$p\in\M$，Hitchin模空间中测地球的体积为$$ \Vol_{g_{L^2}}B_{L^2}(p,R) =\frac{2^{4g-3}\pi^{9g-9}}{(3g-3)!}\\,R^{6g-6}+o(R^{6g-6}) $$。我们还确定了Hamiltonian子水平集体积和指数加权体积的领先渐近行为。证明将超kähler体积形式的齐次性与正则Hitchin轨迹上的度量渐近性相结合。辛约化和Prym极化计算了系数，该系数也由等变局部化独立得到。
英文摘要
Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g_{L^2}}B_{L^2}(p,R) =\frac{2^{4g-3}π^{9g-9}}{(3g-3)!}\,R^{6g-6}+o(R^{6g-6}) $$ for every $p\in\M$. We also determine the leading asymptotics of Hamiltonian sublevel volumes and exponentially weighted volumes. The proof combines homogeneity of the hyperkähler volume form with metric asymptotics on the regular Hitchin locus. Symplectic reduction and the Prym polarization evaluate the coefficient, which is independently recovered by equivariant localization.
