2012/06/21 by Mao Sheng, Kang Zuo, Sheng, Mao +1
Mathematics · #14D07 #14F30 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14D07 #msc:14F30
paper · pdf · doi:10.48550/arxiv.1206.4865
This is the updated version of our older manuscript with title 'Periodic Higgs subbundles in mixed characteristic'
arxiv created 2013/02/08 · arxiv updated 2013/02/11
Let k be an algebraically closed field of odd characteristic p and X a proper smooth scheme over the Witt ring W(k). To an object (M,Fil⋅,∇,Φ) in the Faltings category MF∇[0,n](X), n≤ p-2, one associates an étale local system \V over the generic fiber of X and a Higgs bundle (E,θ) over X. Our motivation is to find the analogue of the classical Simpson correspondence for the categories of subobjects of \V and (E,θ). Our main discovery in this paper is the notion of periodic Higgs subbundles, both in positive characteristic and in mixed characteristic. In char p, it relies on the inverse Cartier transform constructed by Ogus and Vologodsky in their work on the char p nonabelian Hodge theory. A lifting of the inverse Cartier transform to mixed characteristic is constructed, which is used for the notion of periodicity in mixed characteristic. We show a one to one correspondence between the set of periodic Higgs subbundles of (E,θ) and the set of étale sub local systems of \V⊗_\Zp\Zpr, where r is a natural number. The notion turns out to be useful in applications. We have proven, among other results, that the reduction (E,θ)0 of (E,θ) modulo p is Higgs stable, if and only if, the corresponding representation \V is absolutely irreducible over k.