AI 中文总结
研究紧致凯勒流形\(X\)上希格斯丛和平坦丛的微分几何模栈，通过构建相对解析理论，证明稳定希格斯族存在全局光滑调和度量，刻画扩张生成栈，分析热流数据损失并构造光滑霍奇\(\lambda\)族。
AI 中文摘要
设\(X\)为紧致凯勒流形。在先前工作中，我们构建了\(X\)上希格斯丛和平坦丛的微分几何模栈，通过光滑调和族的扩张完备相关联。在此，我们发展相对解析理论。在任意图表上的索伯列夫完备化上，我们证明每个满足数值条件的光滑稳定希格斯族都允许一个全局光滑调和度量。固定一个厄米 - 爱因斯坦行列式度量消除标量自由度，然后椭圆正则性和归一化胶合产生逐图表光滑性。该定理在每个有限参数正则性\(C^d\)以及具有环境扩张的约化奇异参数空间上成立。对于希格斯形变\(\eta\)，归一化度量变化满足\(L_hs = -\mathcal S_h(\eta)\)且\(s = -G_h\mathcal S_h(\eta)\)，直至\(\mathrm{GL}_r\)的一个独立的一阶行列式项。这计算了逐图表微分并恢复了经典比较。局部分裂、常类型多稳定族允许光滑调和度量。实解析例子表明一般多稳定族可能既没有连续调和度量也没有相对调和滤层，并且可能位于每个\(C^d\)扩张生成的轨迹之外。在一个例子中，一个奇异调和约化产生一个连续伴随希格斯场和一个具有半单切片的平坦族。这定义了一个弱\(C^0\)算子级调和调解器，严格大于度量正则的调解器，其在有限扩张完备化和栈化后的端点图像满足\(\mathscr M_{\mathrm{Dol},0}^{\mathrm{wk}\mathcal H}(X)\simeq\mathscr M_{\mathrm{dR},0}^{\mathrm{wk}\mathcal H}(X)\)。我们通过相对调和滤层刻画扩张生成的栈，发展它们的障碍理论，分析热流下扩张数据的损失，并在稳定轨迹上构造光滑霍奇\(\lambda\)族。
英文摘要
Let $X$ be a compact Kähler manifold. In prior work, we constructed diffeological moduli stacks of Higgs and flat bundles on $X$, related by extension completion of smooth harmonic families. Here, we develop the relative analytic theory. On Sobolev completions over arbitrary plots, we prove that every smooth stable Higgs family satisfying the numerical conditions admits a global smooth harmonic metric. Fixing a Hermitian--Einstein determinant metric removes scalar freedom, and then elliptic regularity and normalized gluing yield plotwise smoothness. The theorem holds at every finite parameter regularity $C^d$ and on reduced singular parameter spaces with ambient extensions. For a Higgs deformation $η$, the normalized metric variation satisfies $L_hs=-\mathcal S_h(η)$ and $s=-G_h\mathcal S_h(η)$ up to an independent rank-one determinant term for $\mathrm{GL}_r$. This computes the plotwise differential and recovers the classical comparison. Locally split, constant-type polystable families admit smooth harmonic metrics. Real-analytic examples show general polystable families may have neither continuous harmonic metrics nor relative harmonic filtrations and may lie outside every $C^d$ extension-generated locus. In one example a singular harmonic reduction produces a continuous adjoint Higgs field and a flat family with semisimple slices. This defines a weak $C^0$ operator-level harmonic mediator, strictly larger than the metric-regular one, whose endpoint images after finite extension completion and stackification satisfy $\mathscr M_{\mathrm{Dol},0}^{\mathrm{wk}\mathcal H}(X)\simeq\mathscr M_{\mathrm{dR},0}^{\mathrm{wk}\mathcal H}(X)$. We characterize the extension-generated stack by relative harmonic filtrations, develop their obstruction theory, analyze the loss of extension data under heat flow, and construct the smooth Hodge $λ$-family on the stable locus.
Comments103 pages, 1 table
