Core Media Group, the company behind “American Idol” and “So You Think You Can Dance,” has filed for chapter 11 protection in New York, Billboard reported yesterday. Core was founded a decade ago and was recently brought into a joint venture owned by 21st Century Fox and Apollo Global Management. But as “Idol” has now aired its last show, it appears as though Core envisions some restructuring. In the bankruptcy filing, the company says it owes $398 million to third parties, including from two matured loans from Tennenbaum Capital Partners, Crestview Media Investors, Bayside Capital and Hudson Bay Capital Management. The company reports about $73 million in assets but less than $10 million in cash on hand. “Idol” creator Simon Fuller is the company’s top unsecured creditor and is owed $3.37 million. A payment demand on April 11 from Fuller is one of the factors cited in the bankruptcy papers as contributing to the decision to file for chapter 11.