CHICAGO — A high-ranking Chicago police officer was fatally shot on Tuesday afternoon within sight of City Hall, turning some of downtown’s busiest blocks into a chaotic crime scene and adding to a grim tally of officers killed across the country in recent days.

The officer, Cmdr. Paul R. Bauer, 53, was chasing a man near a state government building across the street from City Hall in the early afternoon when he was shot. Within minutes, major streets in the Loop were blocked with police tape and filled with squad cars and sirens.