Ukrainian heritage is under threat - and so is the truth about Soviet-era Russia | Anna Reid

In the midst of the unfolding human tragedy is the appalling cultural loss the war may wreak. A rich mixture of influences - Polish, Russian, Viennese, Soviet, Jewish and even Ottoman, as well as its own folk traditions - informs the country's art, architecture, music and literature, producing a gorgeous potpourri that is, paradoxically, distinctly Ukrainian. And as with any underdog country, culture is particularly important for Ukrainians, having carried the national identity through long centuries of foreign rule.

The big city currently most at risk is Kharkiv, still in Ukrainian hands but under heavy shelling. Long a commercial centre, it has handsome 19th-century merchants' houses and warehousing (home, until now, of a thriving contemporary art scene), but is most remarkable for its constructivist government buildings, built during the 1920s when Kharkiv was the capital of the Ukrainian Soviet Socialist Republic. As the architectural historian Owen Hatherley says, "Only Dessau in the old East Germany, or maybe Tel Aviv, compares to it for buildings of that era."

So far the most prominent of them - a futurist vision of blocky concrete and flying walkways - is undamaged. But another - one of several reclad in Stalinist neo-classical style - lost every window and part of its roof when a missile landed nearby on 1 March. The city's landmark "Slovo" ("Word") building has already been directly hit three times. Shaped like a C - the first letter of the word "slovo" in Cyrillic - it was built to house prominent writers, scholars and artists; for Ukrainians, it represents the period's brief cultural flowering before Moscow cracked down again and sent most of its residents to the gulag.

Kharkiv's Fine Arts Museum has been damaged, too. Photos posted on social media by staff show blinds hanging in tatters, and parquet floors scattered with broken glass. Some pictures have been laid face-down on the floor; others still hang on the walls. Among other treasures, it houses 11 canvases by the great 19th-century painter Ilya Repin, who was born nearby but built his career in St Petersburg, Paris and Moscow. "The irony of the situation," the head of the museum's foreign art department told Reuters, "is that we are having to save Russian artists' works from their own people."

One of the most historic cities under siege is Chernihiv, a couple of hours' drive north of Kyiv on the main road to the border with Belarus. On a quick visit four weeks ago, it was as quiet and beautiful as ever, its clutch of churches white, green and gold against a blue sky and the sparkling snow. A local guide, now fled with her daughter and granddaughter to Lublin in Poland, tells me on the phone that so far as she knows the churches are still - "glory to God" - undamaged, but that the Soviet-era cinema on the main square has been destroyed, "and our lovely fountain too". She breaks down in tears talking about the kindness of the Poles.

In Kyiv and elsewhere, archive staff are working round the clock to scan documents and to move digitised material on to servers abroad. For some it is already too late. The building that houses the Chernihiv province's KGB records lost its roof to shelling a few days ago; it's not yet clear how much was lost.

The Budapest-based art historian Konstantin Akinsha is terrified for the fate of Odesa, where the Russian troopships lurking just offshore are expected to attempt a landing any day. The words tumble over themselves as he lists its collections. One of the best is at the Odesa art museum. Volunteers have been helping staff pack up paintings and take them to safety. They are fearful of looting or Russian confiscation, and so are understandably tight-lipped about where they are now. But since they have thousands of items to deal with, and many are evacuating (the museum's director took the hard decision to leave with her baby son), much remains in situ.

Akinsha also worries about Odesa's Museum of Western and Eastern Art, which has Frans Hals's St Matthew and his St Luke (their fellow evangelists hang in Moscow's Pushkin Museum and Los Angeles's Getty). And then there's the Archaeological Museum and the Literature Museum. "It has unbelievable manuscripts," he says. "I want to cry. It's absolutely scary."

Another concern is vandalism and looting by occupying troops; on 14 March, Russian soldiers broke into offices in a neo-baronial castle near the city of Zaporizhzhya, and deliberately smashed up office furniture and equipment. What Akinsha would like to see - apart from more and bigger anti-aircraft guns from the west - is strong anti-war public statements from prominent museum-world figures in Russia. "I'm not asking them to go to Red Square and set themselves on fire. But they could come out with something at least. They are completely silent. It's disgusting."

As we wait to see what Putin has in mind for ravishing Kyiv - a siege, artillery barrages, air-dropped chlorine bombs like in Syria? - the UK government announces that it is sending more and more sophisticated anti-aircraft guns to Ukraine. They will not be able to protect cities from short-range shelling, but should be able to prevent bombing from the air and by long-range ballistic missiles. It's late; we should have started arming the country months ago, when Putin started building up his forces on the border. But it's much better than never, and will help to save not only lives.