Yup, falcons, in Germany on the verge of extinction a few decades ago due to DDT overuse in agricultural zones, have moved into German cities, preferably ones with hi-rise buildings because up there on the rooftops its basically a man-free habitat. And pigeons are ample though not as prevalent as they used to be. But they are a hardy species, I'm not worried about their extinction yet. Though there was a time when they were rare forest birds too.
Not sure whether falcons are the more efficiejt pigeon controls, ravens, magpies and crows are effective nest-robbers unless its their own species. They eradicate the pigeon problem at its root.
But I saw once how a falcon (we must have one on the roof of our hi-rise) swooped a pigeon in midair - that was admittedly quite spectacular.