It is rare for any major museum to play daring with a wild card of a show, still less during times of recession, but so it is with Per Kirkeby at Tate Modern. Ten galleries have been devoted to this unfamiliar painter. For although Kirkeby (born in 1938) is a household name in Denmark, and the nation's most acclaimed artist since Vilhelm Hammershøi and Asger Jorn, he can hardly be well known to many people over here, since he has never had a full-dress show in Britain before, despite a career lasting more than 40 years.
It might have been longer had Kirkeby not started out as an Arctic geologist, a fact that becomes more significant the deeper one looks into his work. And depth is always at issue. For the first thing to say is that Kirkeby is a paradoxical painter, a neo-expressionist whose enormous canvases of flaring colour and passionate gesture might appear to be purely abstract were it not that there is some kind of realist in him, too. Look into his surfaces and you see figures and forms tangled up in the paint.
Take the very funny picture that the curators have shrewdly chosen to open this show. It is called The World's Northernmost House. But where is this fabled place? The canvas is a maze of ice-cracking lines, rickety black slate and rock, drips, damp outcrops of claggy brown and grey, rising up to a threatening green sky (or so one perceives it). Is there a hint of roof, doorstep or track? It's not clear. There is no obvious vantage point - indeed, one could as easily be looking at a map as a landscape. But either way there is a powerful sense of absurdity: a room at the top of the freezing world.
What is discernible in Kirkeby's art is often foolish or extreme: a medieval knight, a horse so strangely angled it could tip out of the picture, sinister huts and doorless dwellings. One painting is named after the ship Nansen was forced to abandon in his attempt to reach the North Pole and certainly there is some sense of deadlock and of splintering horizons, though these seem equally bound up with the action of painting itself. Forge ahead, keep going, don't stop trying.
Now there is scarcely anything less enticing one could say about a painter's work than that it is concerned with painting itself, and that is not the case with Per Kirkeby. It is obvious that the mind's meanderings are as susceptible to expression for him as for any poet (and Kirkeby is also a poet), also that the infinite variety of the world inspires in him an infinite variety of representational methods.
He paints Pop on hard Masonite in the 60s. He glues on, tears off, splits the canvas into four screens in the 70s. By the 80s, images are overlaid and interleaved, paint is scraped and coagulated, thinned, dripped and squeezed in lush smears. He goes against the grain - huge stabbing strokes of black and white in Crystal, making a cataclysm out of delicate refraction - and he walks out into the landscape like some latter-day romantic. In several works, a calligraphy of lines spread across the canvas, rather like the branches of trees latticing a beautiful vista.
If Kirkeby is prolific and uncommonly various, he is also up to his eyes in art history, seeing the world, and his art, through that of others. You get a hint of what is to come in the second room where his commentaries on fellow painters fill a whole wall of bookshelves. I freely confess that I don't know quite what Kirkeby is doing in paintings that reprise other artists - Monet's poplars and water lilies, Soutine's flayed carcasses - other than to isolate and celebrate their motifs, though the overwhelming sense is of exploration.
Kirkeby wants to paint a version of The Flight into Egypt for modern times and his imagination turns to the heat, the route, the ankle-turning rocks of the terrain; the painting navigates an immense range of hazardous lines and hot colours. He wanders into the woods and the resulting pictures suggest elusive spaces, alternately airy and densely gnarled, the shadows breached by flashes of exultant colour.
Kirkeby's colour - radiant violet, cobalt, glowing ochre - is like a gift, a compensation for the complexity of his art. For he never offers any easy statements. None of his paintings is sewn up, resolved, and very often you feel more certain of the mood than the subject matter. His early work has been compared to that of contemporaries such as Sigmar Polke and Georg Baselitz, but in its primitive and irreducible pleasures seems more connected to Cy Twombly.