AI 中文总结
本文证明Saito微局部奇异支集包含于Beilinson奇异支集，回答Saito问题，并解决相关支集估计猜想，证明特征类猜想。
AI 中文摘要
设$X$是特征$p>0$的完美域上的光滑概形，$F$是有限Tor维数的可构造复形，其有限系数特征与$p$互素。我们证明\\[ {\rm SS}_\mu(F)\subseteq {\rm SS}(F), \\]其中${\rm SS}_\mu(F)$是Saito的微局部奇异支集，${\rm SS}(F)$是Beilinson的奇异支集。这回答了Saito的一个问题。对于perverse $\ell$-adic层，该包含关系成为等式。作为应用，我们解决了Saito关于沿光滑闭子概形微局部化的支集估计的另一个问题，并证明了Saito关于特征类的猜想（\emph{Invent. Math. 207: 597-695, 2017}），表明Abbes和Saito的上同调特征类是对应特征循环的循环类。
英文摘要
For a constructible sheaf $\mathcal{F}$ on a smooth scheme $X$ over a perfect field, the singular support $SS(\mathcal{F})$ and the characteristic cycle $CC(\mathcal{F})$ are defined on the cotangent bundle. In analogy with the construction of Kashiwara--Schapira via microlocalization in the transcendental setting, we construct the microlocal versions ${ SS}_μ(\mathcal{F})$ and $CC_μ(\mathcal{F})$ of $SS(\mathcal{F})$ and $CC(\mathcal{F})$ respectively, and prove \[ SS_μ(\mathcal{F})= SS(\mathcal{F}),\qquad CC_μ(\mathcal{F})= {\rm cl}CC(\mathcal{F}), \] where $\operatorname{cl}$ is the cycle class map. As a consequence of the second equality, we show that the cohomological characteristic classes are the cycle classes associated with the corresponding characteristic cycles, as conjectured in (\emph{Invent.~Math.~207: 597--695, 2017}). As further applications, we give new proofs of the index formula and Bloch's conductor formula in the proper case, independently of Abe's homotopical approach via $\infty$-categories.
CommentsThis version merges the previous version of this paper with arXiv:2607.17026, and supersedes both earlier manuscripts
