Russia’s invasion of Ukraine has led arts organizations to reconsider who performs, forcing them to confront questions about free speech and policing political views.In Canada, an acclaimed 20-year-old Russian pianist’s concert was canceled amid concerns about his silence on the invasion of Ukraine. The music director of an orchestra in Toulouse, France — who is also the chief conductor at the Bolshoi Theater in Moscow — was instructed to clarify his position on the war before his next appearance. In New York, Anna Netrebko, one of opera’s biggest stars, saw her reign at the Metropolitan Opera end after she declined to denounce President Vladimir V. Putin of Russia.As global condemnation of Russia’s attack on Ukraine grows, cultural institutions have moved with surprising speed to put pressure on Russian artists to distance themselves from Mr. Putin, a collision of art and politics that is forcing organizations to confront questions about free speech and whether they should be policing artists’ views.Institutions are demanding that artists who have supported Mr. Putin in the past issue clear condemnations of the Russian president and his invasion as a prerequisite for performing. Others are checking their rosters and poring over social media posts to ensure Russian performers have not made contentious statements about the war. The Polish National Opera has gone so far as to drop a production of Mussorgsky’s “Boris Godunov,” one of the greatest Russian operas, to express “solidarity with the people of Ukraine.”The tensions pose a dilemma for cultural institutions and those who support them. Many have long tried to stay above the fray of current events, and have a deep belief in the role the arts can play in bridging divides. Now arts administrators, who have scant geopolitical expertise, find themselves in the midst of one of the most politically charged issues in recent decades, with little in the way of experience to draw on.“We’re facing a totally new situation,” Andreas Homoki, the artistic director of the Zurich Opera, said. “Politics was never on our mind like this before.”The new scrutiny of Russian artists threatens to upend decades of cultural exchange that endured even during the depths of the Cold War, when the Soviet Union and the West sent artists back and forth amid fears of nuclear war. The Russian maestro Valery Gergiev, who has long been close to Mr. Putin, was fired as chief conductor of the Munich Philharmonic and saw his international engagements dry up. The Hermitage Amsterdam, an art museum, broke ties with the Hermitage in St. Petersburg. The Bolshoi Ballet lost engagements in London and Madrid.Citing that Cold War tradition, the Cliburn — a foundation in Fort Worth named for the American pianist Van Cliburn, whose victory at the International Tchaikovsky Competition in Moscow in 1958 was seen as a sign that art could transcend political differences — announced that it would welcome 15 Russian-born pianists to audition next week for the 2022 Cliburn Competition, noting that they are not officials of their government.Jacques Marquis, the president and chief executive of the Cliburn, said the organization felt it was important to speak out as it watched Russian artists come under scrutiny. “We can help the world by standing our ground and focusing on the music and on the artists,” he said.The American pianist Van Cliburn’s victory at the International Tchaikovsky Competition in Moscow in 1958 was seen as a sign, at the height of the Cold War, that art could transcend political differences.The Van Cliburn FoundationEven as many institutions are eager to show support for Ukraine, and to distance themselves from artists who embrace Mr. Putin, they are uncomfortable with trying to vet the views of performers — and worry that Russian artists, who must often rely on the support of the state for their careers to thrive at home, could face reprisals if forced to publicly disavow the Kremlin.“You can’t just put everybody under general suspicion now,” said Alexander Neef, the director of the Paris Opera. “You can’t demand declarations of allegiance or condemnations of what’s going on.”The situation is tense and fast moving. Leaders of organizations are facing pressure from donors, board members and audiences, not to mention waves of anger on social media, where campaigns to cancel several Russian artists have rapidly gained traction.Institutions are also grappling with what to do about the Russians who are among their most important donors. On Wednesday the Guggenheim Museum announced that Vladimir O. Potanin, one of Russia’s richest men and a major benefactor, was stepping down as one of its trustees.Leila Getz, the founder and artistic director of a recital series in Vancouver, Canada, canceled an appearance by the Russian pianist Alexander Malofeev planned for August. Mr. Malofeev, 20, had not made any statements on the war, nor did he have any known ties to Mr. Putin. But Ms. Getz issued a statement saying she could not “in good conscience present a concert by any Russian artist at this moment in time unless they are prepared to speak out publicly against this war.”Soon she received dozens of messages. Some accused her of overstepping and demanded that Mr. Malofeev be allowed to perform.In an interview, Ms. Getz defended her decision, saying she was worried about the potential for protests. She said she had not asked Mr. Malofeev to condemn the war and that she was concerned for his safety.“The first things that came to my mind were, why would I want to bring a 20-year-old Russian pianist to Vancouver and have him faced with protests and people misbehaving inside the concert hall and hooting and screaming and hollering?” she said.Mr. Malofeev declined to comment. In a statement posted on Facebook, he said, “The truth is that every Russian will feel guilty for decades because of the terrible and bloody decision that none of us could influence and predict.”On Friday the Annapolis Symphony in Maryland announced that it would replace the Russian violinist Vadim Repin, who had been scheduled to play a Shostakovich concerto in upcoming concerts, “out of respect to Repin’s apolitical stance and concerns for the safety of himself and his family.”“We don’t want to put him in an uncomfortable, even impossible position,” the orchestra’s executive director, Edgar Herrera, said in a statement. In an interview, Mr. Herrera said that there had been threats to disrupt Mr. Repin’s performances and that the symphony was concerned that hosting a Russian artist could hurt its image and alienate donors.Deciding which artists are too close to Mr. Putin is not easy. Mr. Gergiev, the longtime general and artistic director of the Mariinsky Theater in St. Petersburg, has a relationship with Mr. Putin that goes back decades, and he has often supported the government’s policies. Mr. Gergiev led concerts in 2008 in South Ossetia, a breakaway region of Georgia that was aided by Russian troops, and at the Syrian site of Palmyra in 2016 after it was retaken by Syrian and Russian forces.Ms. Netrebko, the star soprano, issued a statement opposing the war in Ukraine but withdrew from performing after declining to distance herself from Mr. Putin, whom she has expressed support for in the past. The war brought renewed attention to a photograph from 2014 of her holding a flag used by Russian-backed separatists in Ukraine.The pianist Evgeny Kissin, who was born in Moscow, said he believed that “supporters of a criminal war waged by a dictator and a mass murderer should have no place on the concert stages of the civilized world.”Milan Bures for The New York TimesThe eminent pianist Evgeny Kissin, who was born in Moscow and is now based in Prague, said that while many artists in Russia needed to support Mr. Putin to some degree because their institutions relied on state aid, others went too far. He said he believed that “supporters of a criminal war waged by a dictator and a mass murderer should have no place on the concert stages of the civilized world.”He added that while he thought it was natural for Western institutions to ask Mr. Putin’s most prominent supporters to speak out against the war, he did not think it should be required of artists who had not been particularly political in the past.How the Ukraine War Is Affecting the Cultural WorldCard 1 of 5Anna Netrebko. More