The Millennium Tower in San Francisco is a residential condominium that markets itself as rare and precious. Now two other adjectives also define this luxury high-rise: sinking and tilting.

Residents paid millions of dollars to live in the roughly 400-unit skyscraper. The building boasts an indoor pool, fitness center, wine cellar and private dining room. The tower has attracted Bay Area elites like NFL Hall of Fame quarterback Joe Montana; San Francisco Giants outfielder Hunter Pence; Tibco founder and Sacramento Kings owner Vivek Ranadive; and the late legendary venture capitalist Tom Perkins, who paid more than $9 million for the penthouse.

The tony tower is now attracting attention from a different group of powerful San Franciscans, however: the city's politicians, who want to understand why this 58-story building has sunk 16 inches, and is tilting at least 2 inches to the northwest since it was completed eight years ago.

Thursday, San Francisco city Supervisor Aaron Peskin will hold a hearing on the tower.

"We are in the midst of a huge building boom," Peskin tells CNBC. "We should make sure that we are building to the highest standards, and that we are building the safest buildings that we can in San Francisco, which is a seismically challenged part of the world."