That email warning turned a continuing attack on Sony by hackers from a matter of theft to one of terrorism. A spokeswoman for the F.B.I. said it "is aware of the threat and is continuing to investigate the attack on Sony."
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Late Tuesday, a person with knowledge of Sony's dealings said that the theater chain Carmike Cinemas had canceled its showings of the movie. Carmike could not be reached for comment.
A Homeland Security official said the department was analyzing the threat but as yet had found no clear indication of an active plot against theaters. Sony had no comment.
Tuesday's development posed an ugly dilemma for Sony and exhibitors: whether to pull "The Interview," caving to hackers who have wreaked havoc with Sony's digital systems for weeks in an attempt to block the release,or to forge ahead, risking possible violence and potential legal liability. In an already-fragile industry, studio executives privately voiced concern that any theater violence could swing the market further toward home viewing.
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Several people briefed on Sony's internal discussions on Tuesday said the studio was not withdrawing the film but had told theater owners that it would not object if they canceled or avoided booking "The Interview." Those people spoke on condition of anonymity. Theater owners have been particularly pressed by the operators of malls and stores within them to avoid the film, two of those people said.
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An executive for one of the theater chains, who also spoke on condition of anonymity, called the invocation of the 9/11 terror attacks by hackers "a game changer." The executive last week dismissed the notion that theater owners might shy away from "The Interview" over earlier, more general threats by North Korean officials and pressure from the hackers, who have called themselves the Guardians of Peace. Nobody yet knows the hackers' true identity.
Representatives of AMC Entertainment, Regal Entertainment and Cinemark, North America's three largest theater chains, did not respond to queries. A spokesman for the National Association of Theater Owners, which represents exhibitors, declined to comment.
Pressure to pull the "The Interview," which stars Seth Rogen and James Franco and is directed by Mr. Rogen and Evan Goldberg, has centered on its depiction of Mr. Kim's assassination. To depict the killing of a sitting world leader, comically or otherwise, is virtually without precedent in major studio movies, film historians say. Mr. Rogen canceled planned publicity interviews on Tuesday.
But a broad threat of theater violence, following a sustained attack on Sony's digital existence, is also without precedent, and opens a new range of worry for Hollywood.
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As Sony and exhibitors spoke in a 2 p.m. conference call on Tuesday, they faced the concerns of competing studios, whose important holiday films will be playing side-by-side with "The Interview" in multiplexes nationwide.
A further complication is a general reluctance, even after the 2012 mass shooting at an Aurora, Colo., theater, to visibly increase security, which might create an impression that multiplexes in general are not safe and might complicate dealings with their own insurers.