Logarithmic ramifications of \'etale sheaves by restricting to curves

In this article, we prove that the Swan conductor of an etale sheaf on a smooth variety defined by Abbes and Saito's logarithmic ramification theory can be computed by its classical Swan conductors after restricting it to curves. It extends the same result for rank 1 sheaves due to Barrientos. As an application, we give a logarithmic ramification version of generalizations of Deligne and Laumon's lower semi-continuity property for Swan conductors of etale sheaves on relative curves to higher relative dimensions in a geometric situation.

[1]  Haoyu Hu,et al.  Relative singular support and the semi-continuity of characteristic cycles for étale sheaves , 2017, 1702.06752.

[2]  Yuri Yatagawa.  Equality of Two Non-Logarithmic Ramification Filtrations of Abelianized Galois Group in Positive Characteristic , 2016, Documenta Mathematica.

[3]  S. Saito,et al.  Chow group of $0$-cycles with modulus and higher-dimensional class field theory , 2013, 1304.4400.

[4]  Takeshi Saito.  Wild ramification and the cotangent bundle , 2013, 1301.4632.

[5]  Haoyu Hu,et al.  Semi-continuity for total dimension divisors of \'etale sheaves , 2015, 1511.04866.

[6]  Takeshi Saito.  The characteristic cycle and the singular support of a constructible sheaf , 2015, 1510.03018.

[7]  A. Beilinson.  Constructible sheaves are holonomic , 2015, 1505.06768.

[8]  Yifei Zhao.  Maximally Frobenius-destabilized vector bundles over smooth algebraic curves , 2014, 1408.5117.

[9]  Ivan Barrientos.  Log ramification via curves in rank 1 , 2013, 1307.5814.

[10]  H. Esnault,et al.  A FINITENESS THEOREM FOR GALOIS REPRESENTATIONS OF FUNCTION FIELDS OVER FINITE FIELDS (AFTER DELIGNE) , 2012, 1208.0128.

[11]  L. Xiao.  On ramification filtrations and p-adic differential equations, II: mixed characteristic case , 2008, Compositio Mathematica.

[12]  Takeshi Saito,et al.  Ramification and cleanliness , 2010, 1007.3873.

[13]  K. Kedlaya.  Swan conductors for p-adic differential modules. II Global variation , 2007, Journal of the Institute of Mathematics of Jussieu.

[14]  Takeshi Saito.  Wild ramification and the characteristic cycle of an ℓ-adic sheaf , 2007, Journal of the Institute of Mathematics of Jussieu.

[15]  L. Xiao.  On ramification filtrations and $p$-adic differential modules, I: equal characteristic case , 2008, 0801.4962.

[16]  I. Zhukov.  Ramification of surfaces: sufficient jet order for wild jumps , 2002, math/0201071.

[17]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[18]  Y. André.  Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité , 2007, math/0701894.

[19]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .

[20]  K. Kedlaya.  Swan conductors for p-adic differential modules, I: A local construction , 2006, math/0611835.

[21]  Ahmed Abbes,et al.  Analyse Micro-Locale ℓ-Adique en Caractéristique p > 0 le Cas d’un Trait , 2006, math/0602285.

[22]  Takeshi Saito.  Ramification of local fields with imperfect residue fields III , 2000, math/0010103.

[23]  Shigeki Matsuda.  On the Swan conductor in positive characteristic , 1997 .

[24]  A. D. de Jong.  Smoothness , semi-stability and alterations , 1996 .

[25]  Nicholas M. Katz,et al.  Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116) , 1988 .

[26]  J. Brylinski.  Théorie du corps de classes de Kato et revêtements abéliens de surfaces , 1983 .

[27]  加藤 和也.  A generalization of local class field theory by using K-groups I, II, III = K-群による局所類体論の一般化 , 1980 .

[28]  Kazuya Kato.  A generalization of local class field theory by using $K$-groups, II , 1977 .

[29]  A. Grothendieck.  Étude locale des schémas et des morphismes de schémas , 1964 .