Singular supports, characteristic cycles, and microlocalization of constructible sheaves

@misc{saito2026singular,
  title = {{Singular supports, characteristic cycles, and microlocalization of constructible sheaves}},
  author = {Takeshi Saito and Jiangnan Xiong and Enlin Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.06625},
  eprint = {2609.06625},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06625},
  note = {CC0},
}

TY  - UNPB
TI  - Singular supports, characteristic cycles, and microlocalization of constructible sheaves
AU  - Takeshi Saito
AU  - Jiangnan Xiong
AU  - Enlin Yang
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06625
UR  - https://arxiv.org/abs/2609.06625
AB  - For a constructible sheaf ℱ on a smooth scheme X over a perfect field, the singular support SS(ℱ) and the characteristic cycle CC(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(ℱ) and CC_μ(ℱ) of SS(ℱ) and CC(ℱ) respectively, and prove SS_μ(ℱ)= SS(ℱ), CC_μ(ℱ)= clCC(ℱ), where 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 ∞-categories.
ER  -