发表机构
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
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Simpson关于稳定分次Higgs丛的闭性猜想在任意秩下成立，通过行列式上同调线丛的Quillen几何和行列式联络形式的恰当性，控制了极限滤过并得到Zariski闭性。
AI 中文摘要
对于紧致黎曼曲面$X$，Simpson将每个稳定分次Higgs丛$(E,\theta)$关联到一个轨迹$W_{[(E,\theta)]}^1 \subset M_{\mathrm{dR}}(X,n)$，该轨迹由具有Simpson滤过的平坦丛组成，其相伴分次Higgs丛为$(E,\theta)$。他猜想$W_{[(E,\theta)]}^1$在$M_{\mathrm{dR}}(X,n)$中是Zariski闭的。我们在任意秩下证明了这个猜想。证明基于全纯丛模空间上行列式上同调线丛的Quillen几何。对任何全纯平坦丛族，我们关联一个带有典范全纯行列式联络的行列式上同调线丛，并推导出其曲率的显式公式。对于$W_{[(E,\theta)]}^1$中的平坦丛族，我们随后证明行列式联络形式是恰当的。这种恰当性迫使相关的行列式标架在任意单参数退化中扩展为无处消失的标架，这控制了极限滤过并产生了所期望的闭性。
英文摘要
For a compact Riemann surface $X$, Simpson associated to each stable graded Higgs bundle $(E,θ)$ a locus $W_{[(E,θ)]}^1 \subset M_{\mathrm{dR}}(X,n)$, consisting of flat bundles that admit a Simpson filtration whose associated graded Higgs bundle is $(E,θ)$. He conjectured that $W_{[(E,θ)]}^1$ is Zariski closed in $M_{\mathrm{dR}}(X,n)$. We prove this conjecture in arbitrary rank. The proof is based on the Quillen geometry of the determinant-of-cohomology line bundle over the moduli of holomorphic bundles. To any holomorphic family of flat bundles, we associate a determinant-of-cohomology line bundle equipped with a canonical holomorphic determinant connection, and derive explicit formulas for its curvature. For a family of flat bundles in $W_{[(E,θ)]}^1$, we then prove that the determinant connection form is exact. This exactness forces the associated determinant frame to extend as a nowhere-vanishing frame across any one-parameter degeneration, which controls the limiting filtration and yields the desired closedness.
Comments13 pages, comments are welcome
