Hilbert's Tenth Problem for function fields over valued fields in characteristic zero

Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(\bar{F}/F) and Gal(\bar{k}/k) have the same 2-cohomological dimension and this dimension is finite. Then Hilbert's Tenth Problem has a negative answer for any function field of a variety over K. In particular, this result proves undecidability for varieties over C((T)).