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Although the analogue of the theorem of Neukirch-Uchida for $p$-adic localfields fails to hold as it is, Mochizuki proved a certain analogue of thistheorem for the absolute Galois groups with ramification filtrations of$p$-adic local fields. Moreover, Mochizuki and Hoshi gave various (necessaryand) sufficient conditions for homomorphisms between the absolute Galois groupsof $p$-adic local fields to be "geometric" (i.e., to arise from homomorphismsof fields). In the present paper, we consider similar problems for generalmixed-characteristic complete discrete valuation fields with perfect residuefields. One main result gives (necessary and) sufficient conditions forhomomorphisms between the absolute Galois groups of mixed-characteristiccomplete discrete valuation fields with residue fields algebraic over the primefields to be geometric. We also give a "weak-Isom" anabelian result forhomomorphisms between the absolute Galois groups of these fields.

Geometric properties of homomorphisms between the absolute Galois groups  of mixed-characteristic complete discrete valuation fields with perfect  residue fields