“It’s OK,” coos Maggie Gyllenhaal in this brilliantly modulated psychodrama about broken dreams and conflicted desires. “Kindergarten teachers are allowed to call their students.’’

Are they really? Gyllenhaal’s Lisa Spinelli, a disenchanted fortysomething Staten Islander, is on the phone to a five-year-old. She thinks he’s a new Mozart; a child possessed (in every sense?) of a preternatural talent “so fragile and so rare” that modern society “does everything to crush’’. But is Lisa simply projecting her own aspirations and anxieties on to this young boy? And how far will she go to see her ambitions fulfilled?

Based on Nadav Lapid’s 2014 Israeli film Haganenet, this second feature by writer-director Sara Colangelo (who made Little Accidents) is a brilliantly understated study of obsession, a perfectly poised portrait of a woman on the verge. As precariously balanced as its anti-heroine, The Kindergarten Teacher walks a tightrope between the intriguing and the unnerving, gesturing gently toward disparate genres (from heartbreaking love story to supernaturally inflected horror) while keeping its feet firmly rooted in everyday life. At the heart of its dark spell is Gyllenhaal, who delivers her most complex and accomplished role since 2006’s Sherrybaby, and who serves as producer on what is clearly a passion project.

Colangelo opens her film with shots of Lisa sitting and sighing. Whether in an empty classroom or on a crowded ferry, she’s alone, her isolation accentuated by a repetitive single-note musical motif. At work, Lisa plays Saint-Saëns and talks about Van Gogh, enveloping her young charges in the art that once fired her own youthful soul. At night, she attends “continuing education” classes, where Gael García Bernal’s sexy poetry teacher Simon is unimpressed by her haikus. “He says I need to put more of myself into the work,” Lisa tells her husband. She might as well be talking to herself.

When young Jimmy (the remarkable Parker Sevak) starts absentmindedly reciting poems that pierce Lisa’s heart, she thinks she’s discovered a prodigy whom she must nurture and protect. Belief turns to obsession as Lisa – convinced that she is both Jimmy’s muse and saviour – seizes control of his life, interrupting afternoon naps for private bathroom talks, getting his nanny fired so she can spend more time with him, and even taking him to a late-night Bowery poetry club, against his overly busy father’s wishes.

As with We Need to Talk About Kevin – both book and film – there’s a suggestion that what Lisa sees in Jimmy is merely a reflection of herself, her quiet desperation.

Disappointed by her own teenage kids (one wants to join the marines, the other is glued to her mobile phone) and disconnected from her husband (a rare moment of intimacy is all too easily interrupted by a call from Jimmy), she has become a shadow searching for the light.

Somehow the poems that drip randomly from this child’s mumbling lips fill that void. “With so few elements you make something very complex,” says Simon when Lisa passes Jimmy’s poems off as her own, making her seem suddenly thrilling – desirable rather than “derivative’’. Meanwhile, friends and colleagues look askance at Lisa’s fixation with a child who is not her own.

While there’s nothing overtly sexual about Lisa’s attentions, the almost imperceptible crossing of boundaries strikes a chilling note. Is Jimmy being groomed? Or are his alleged talents so prodigious that normal rules no longer apply? There’s also the unresolved issue of just how “good” his poems really are (“Anna is beautiful, beautiful enough for me… ”).

In her director statement, Colangelo describes The Kindergarten Teacher as discussing “what role, if any, poetry has in modern American life”, and whether there is “space for beauty, meaning and human expression” in “a world of smartphones, video games, guns’’. Yet as Simon observes, with cynical amusement, computers are now writing poems that could pass the Turing test. Is Jimmy really just a random-word generator, chancing upon profundity?

At times, I was reminded of the eerie tone of Jonathan Glazer’s Birth, in which Nicole Kidman’s bereaved widow comes to believe that a young boy is the reincarnation of her lost love. Elsewhere, it was the simpler sadness of a midlife crisis – a recognition of all that has been lost – that struck a poignant chord.

Throughout, I was mesmerised by the ambiguity that this smart, sharp film maintains, entranced by the deceptive simplicity of its complexities, unsettled by questions that are left deliberately unanswered.