In his obit essay on Nadezhda Mandelstam (in "Less Than One" collection), Joseph Brodsky writes, "It's an abominable fallacy that suffering makes for greater art. Suffering blinds, deafens, ruins, often kills. Osip Mandelsam was a great poet before the revolution. So was Anna Ahkmatova, so was Marina Tsvetaeva. They would have become what they became even if none of the historical events the befell Russia in the century had taken place: because they were gifted. Basically, talent doesn't need history."
On the other hand, in the intro to this tranlsation of AA, Stanley Kunitz comments that Osip Mandelstam made a remark to the effect that "great poetry is often a response to total disaster."
The connection between genius and environment is an intriguing one, though I don't think they are mutually exclusive. Attempts to separate the two remind me of what Flannery O'Connor once said about identifying the theme of a short story: "People talk about the theme of story as if the theme were like the string that a sack of chicken feed is tied with. They think that if you can pick out the theme, the way you pick the right thread in the chicken-feed sack, you can rip the story open and feed the chickens."
Likewise, though teasing out the thread of history in a poem might enchance one's appreciation for it, I'm not sure it's essential, or even wise, for experiencing a writer's genius. (I had a student once who argued that Frost's poem "After Apple Picking" was about negative effects of too many immigrants filling America in the 20s, a bizarre interpretation that raised the disturbing specter of Frost as the Pat Buchanan of poets.) One can see evidence of both views woven together in AA's poetry.
On the one hand, you have a poem like "Voronezh," written in 1936 and dedicated to Osip M: "But in the room of the banished poet / Fear and the Muse stand watch by turn, / and the night is coming on, / which has no hope of dawn." Kunitz argues that in this poem, AA "prophesied the horrors of 1937-38, when Stalin gave his demented Commissar of Internal Affairs seemingly unlimited license to kill and imprison."
And on the other hand, you have a poem like "Lot's Wife" (my personal favorite) written in 1922-24, that ends with, "Who will grieve for this woman? Does she not seem / too insignificant for our concern? / Yet in my heart I never will deny her, / who suffered death because she chose to turn." I suppose one could read some form of political defiance in those words, but they give me goosebumps because they seem to suggest some deeply personal response to grief. Whether the grief if AA's own or that of a friend or relative who knows. Regardless, it's a great poem.
Having heard so much about AA over the years, I worried that she couldn't possibly match all the profuse acclaim, that in reading her for the first time, I would feel a bit deflated, disappointed. Ha, fat chance! She is even better than I had hoped. I won't draw any grand conclusions about the effects of history on her work except to say that it both stares down and transcends the times.