Artin and Swan Conductors via Nearby Cycles 	for Strictly Semistable Varieties over Local Fields

Rahul  Kundnani,ShriKant  Ojha,Khursheed  Alam

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Artin and Swan Conductors via Nearby Cycles for Strictly Semistable Varieties over Local Fields

R. Kundnani,

Sh. Ojha,

K. Alam

2026-03-10

https://doi.org/10.24108/preprints-3113859

Let $K$ be a non-archimedean local field with finite residue field of characteristic $p$, and let $\ell \neq p$ be a prime. We develop a cohomological framework for analyzing Artin and Swan conductors associated with strictly semistable varieties over $K$. Using the formalism of nearby and vanishing cycles, we relate the ramification behavior of the $\ell$-adic cohomology of the generic fiber to the geometry of the special fiber. In the strictly semistable (simple normal crossings) case with $\ell\neq p$, we give explicit formulas for (i) inertia invariants and the unramified local factor via Frobenius acting on nearby-cycles cohomology, and (ii) the tame/unipotent (monodromy) contribution to the Artin conductor in terms of the monodromy operator on the associated Weil--Deligne representation. Outside the strictly semistable range, we isolate the precise mechanism by which additional vanishing-cycle terms contribute to ramification: the failure of specialization, and any genuinely wild contribution, are detected on the vanishing-cycle complex. This perspective clarifies how local $\ell$-adic cohomological invariants reflect the combinatorial and geometric structure of the special fiber. The resulting formulas provide a transparent description of conductor behavior within the strictly semistable range and identify the cohomological obstructions that arise beyond it. As applications, we obtain a structural decomposition of local zeta factors and a refined interpretation of wild ramification phenomena in arithmetic geometry over local fields.

Ссылка для цитирования:

Kundnani R., Ojha Sh., Alam K. 2026. Artin and Swan Conductors via Nearby Cycles for Strictly Semistable Varieties over Local Fields. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113859

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