“The Apparition” sounds like the title of a horror movie, and this is not a case where the United States distributors of this French film have goosed up the original language title, which was, yes, “L’apparition.” There are several points in the movie during which the viewer can see the story line veer into genre territory, as when some of the characters, a disparate group convened for an investigation, discuss the possibility of working with an exorcist.

But the movie, directed by Xavier Giannoli, in fact aims for tragedy (which it nearly achieves) and enigmatic spirituality (and here’s where there’s a problem). Vincent Lindon plays Jacques, a journalist whose best friend and colleague is killed, practically right next to him, in the Middle East. At home nursing a blown-out ear, and PTSD, he is summoned by a Vatican representative. A young woman in rural France, Anna (Galatéa Bellugi) has seen a vision of the Virgin Mary, and is being celebrated by locals — and now, tourists on pilgrimages — as a potential new Bernadette of Lourdes. The church wants an investigation, to which the local priest sheltering Anna has strident objections.

Mr. Lindon, who carries his powerful masculinity with canny reserve, is superb as a man inquiring into a faith he had previously thought had nothing to do with him. But Ms. Bellugi is a real find; she inhabits her character, who, even as she hides her secrets, is so genuinely beatific that you can hear it in her breathing. Which makes it even more of a shame that the movie, which for two hours is an absorbing, detailed procedural, becomes so willfully diffuse in its final 20 minutes.