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

Rahul  Kundnani,ShriKant  Ojha,Khursheed  Alam

ПРЕПРИНТ

Эта статья является препринтом и не была отрецензирована.
О результатах, изложенных в препринтах, не следует сообщать в СМИ как о проверенной информации.

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

R. Kundnani,

Sh. Ojha,

K. Alam

2026-03-02

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 obstruction to 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

Список литературы

1. P. Deligne and N. Katz (eds.), Groupes de monodromie en géométrie algébrique (SGA 7 II), Lecture Notes in Mathematics 340, Springer, 1973.

2. M. Rapoport and T. Zink, Period Spaces for p-Divisible Groups, Annals of Mathematics Studies 141, Princeton University Press, Princeton, 1996.

3. T. Saito, Weight spectral sequences and independence of ℓ, Journal of the Institute of Mathematics of Jussieu 2 (2003), no. 4, 583–634.

4. P. Deligne, Le formalisme des cycles proches et des cycles évanescents (Exposé XIII), in Groupes de monodromie en géométrie algébrique (SGA 7 I), Lecture Notes in Mathematics 288, Springer, 1972, Exp. XIII.

5. M. Artin, Théorie de Picard–Lefschetz (Exposé XV), in Groupes de monodromie en géométrie algébrique (SGA 7 II), Lecture Notes in Mathematics 340, Springer, 1973, Exp. XV.

6. L. Illusie, Autour du théorème de monodromie locale, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 9–57.

7. M. Artin, A. Grothendieck, and J.-L. Verdier (eds.), Théorie des topos et cohomologie étale des schémas (SGA 4), Lecture Notes in Mathematics, vols. 269, 270, 305, Springer, 1972–1973.

8. A. Grothendieck (ed.), Cohomologie ℓ-adique et fonctions L (SGA 5), Lecture Notes in Mathematics 589, Springer, 1977.

9. P. Deligne and N. Katz (eds.), Groupes de monodromie en géométrie algébrique (SGA 7), Lecture Notes in Mathematics, vols. 288, 340, Springer, 1972–1973.

10. P. Deligne, La conjecture de Weil II, Publications Mathématiques de l’IHÉS 52 (1980), 137–252.

11. J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, 1980.

12. G. Faltings, Crystalline cohomology and p-adic Galois representations, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), Johns Hopkins University Press, 1989, 25–80.

13. L. Illusie and O. Gabber, Notes on vanishing cycles and the theorem of the fixed part, unpublished manuscript, circulated notes.

14. P. Deligne, Weil II: La conjecture de Weil II revisited, Publications Mathématiques de l’IHÉS 52 (1980), 137–252. [Same as no. 10, cited separately when historical context requires.]

15. T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Inventiones Mathematicae 137 (1999), no. 2, 233–411.

16. W. Nizioł, Semistable conjecture via p-adic Hodge theory, Annals of Mathematics (2) 168 (2008), no. 2, 485–545.

17. K. Fujiwara, A proof of the absolute purity conjecture (after Gabber), in Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics 36, Mathematical Society of Japan, Tokyo, 2002, 153–183.

18. L. Illusie (ed.), Cohomologie ℓ-adique et cohomologie étale (SGA 4½), Lecture Notes in Mathematics 569, Springer, 1977.

19. L. Illusie, Théorie de Hodge III, Publications Mathématiques de l’IHÉS 54 (1981), 73–212.

20. C. Nakayama and T. Saito, Weight spectral sequences and independence of ℓ, Inventiones Mathematicae 127 (1997), no. 1, 73–83.

21. L. Illusie, Complexe de de Rham–Witt II, Publications Mathématiques de l’IHÉS 40 (1971), 5–57.

ПРЕПРИНТ

ИНФОРМАЦИЯ

Использование куки-файлов