In Juneau peregrine falcons have found the local pigeons. On the other hand since that is a natural process rather than an imposed one I suppose that presupposes some kind of balance, rather than eradication.

I HATE adhesives in rodent and vermin control.

One possibility might lie in walling off- as with the steel wool o chicken wire that Wyzzi suggested - any ledges and protrected nooks that pigeons use for nesting.