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