Several Broadway shows are set to close this summer, bringing major productions and star-studded runs to an end across New York City. The first closure comes June 14, when the dark comedy “Becky Shaw” ends its run at the Helen Hayes Theatre. The Broadway revival of “Chess” closes June 21. One week later, “Death Becomes Her,” “Giant,” and “Romeo and Juliet” all take their final bows. July brings even more departures, including “Every Brilliant Thing,” “Oh, Mary!,” “Dog Day Afternoon,” “Proof,” and “The Rocky Horror Show.” The biggest closing may be “Moulin Rouge! The Musical,” which ends its award-winning Broadway run on August 30. “Celebrity Autobiography” follows on September 6.