Grade: B+

Given his nationality, Yorgos Lanthimos has more right than most to draw on the themes of Greek tragedy, and as the title of “The Killing of a Sacred Deer” indicates (along with a direct reference late in the film, in case you’re not up on your mythology), here he is using the legend of Iphigenia, the daughter King Agamemnon had to sacrifice in order to placate Artemis and gain a favorable wind for embarking to Troy, as a template for his modern fable of retribution. Agamemnon, of course, had earned the goddess’ wrath by killing one of the animals sacred to her, and had to atone for doing so.

Lanthimos’ Agamemnon is a Cincinnati cardiologist named Steven Murphy (Colin Farrell), a man of rigid professional control that extends into his home life with wife Anna (Nicole Kidman), an ophthalmologist who is similarly given to strict decorum. (Even their sex life is determined by prearranged pattern.) Both express admiration, and even a bit of chilly affection, for their children—fourteen-year old Kim (Raffey Cassidy), who is just on the cusp of young womanhood, and her younger brother Bob (Sunny Suljic)—although they also set rules for them.

Steven meets on a fairly regular basis with an oddly punctilious sixteen-year old named Martin (Barry Keoghan), whom he gives gifts along with conversation as friendly as he can muster. During one such encounter he invites Martin to dinner with his family, and during the course of the evening the boy and Kim develop a bond which she will come to think of in romantic terms.

The connection between Steven and Martin is withheld from the audience for a considerable time, but it eventually emerges that the young man’s father had died during a surgical procedure and the doctor feels guilt over it, especially since—we will later learn from Steven’s anesthesiologist colleague (Bill Camp) that he had been drinking before the operation—something that was apparently not unusual for Steven at the time.

Ultimately Martin reveals to Steven, with quirky offhandedness, that the doctor must be punished for his father’s death. He must kill one of his children—his choice of which. If he fails to do so, all his family will fall ill and die. First they will suffer paralysis of the legs, then be unable to eat, and finally blood will flow from their eyes—followed by death. Martin also suggests that Steven fill in for his dead father by taking up with his widow.

Steven dismisses the threats as the ravings of a troubled kid until Bob suddenly loses the use of his legs, and shortly thereafter refuses food of any sort. Kim soon follows. As Steven and Anna exhaust all medical options they debate what to do about Martin. Can he be persuaded, by whatever means, to retract what seems to be a curse? Does he even have the power to do so? If not, what can they do to save their children—or choose between them? And how will the children weigh in about their fate?

You should not expect Lanthimos to provide rational answers to any of the questions you might have about this scenario (although the way in which both Kim and Bib dicker over what might have to them has a streak of mordant logic.) Martin is clearly as much a stand-in for Artemis as Steven is for Agamemnon (and Anna, presumably for Clytemnestra, which might make one wonder whether a sequel might not be in order), but how his “curse” arose, and where it gets its power, are matters left utterly obscure. Nor is the style of the film in any way naturalistic. The cool detachment of Lanthimos’ approach owes a great deal to Kubrick, and to the Michael Haneke of “Caché” and “Funny Games.” That extends to the performances, with are deliberately studied and affected, with line deliveries that can seem almost impossibly stilted (though that characterization applies less to Cassidy and to Alicia Silverstone, who enjoys a strong single-scene cameo as Martin’s needy mother).

It would be a mistake, however, to think of “The Killing of a Sacred Deer” as mere imitation or homage. Though certainly reminiscent of the films of those other masters, it still speaks with Lanthimos’ voice, the one that made “Alps” and “The Lobster” so distinctive. His vision is abetted by Jade Healy’s antiseptic production design, Thimios Bakatakis’ elegant cinematography (complete with Kubrickian tracking shots) and Yorgos Mavropsaridis’ editing, which maintains the somber tone the story demands even when things explode. And explode they do in the last act, in which threats escalate and blood is spilled, especially after a hunting rifle is introduced. Still, a mood of grim necessity prevails to the very end.

With an eclectic music score that ranges from Bach and Schubert to Gyorgy Ligeti, this film is no likelier a mainstream crowd-pleaser than Lanthimos’ previous ones were. But it’s an engrossing modern variant of the doom-laden tragedies of Aeschylus and Euripides, exerting a hypnotic effect even as it makes you intensely uncomfortable.