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Hvad er rigtig? 'var født' eller 'er født'?

Great question.
Hmmm… honestly speaking, I have never thought about it, so I’m not sure about the answer.

It might have something to do with whether or not the person in question is still alive;
‘H.C.Andersen blev født d. 2. april 1805’ (dead) compared to 'han er født d. 1. januar 1970’ (alive)
However, this may not be correct when I say; ‘jeg blev født på en tirsdag’ (I’m obviously still alive)
When using ‘er’ it is possible it’s referring to the current status (alive - I am), while ‘blev’ refers to the incident (the birth) itself which happened in the past. Because H.C. Andersen is dead, we say ‘blev’. But when talking about ourselves, we say ‘er’ because we are still alive. However, when we are talking about fx the day we were born, it’s an incident in the past and therefore said in past tense. In this case, both the subject + the exact meaning affect which tense is right.
(btw, we usually say ‘blev’ instead of ‘var’ here because ‘getting born’ is a passive form)

I discussed it with my grandmother, and she said that it might also depend on the context;
If Person A says, ‘han er født d. 2. april’, and Person B corrects it, ‘nej, han blev født d. 1. januar’

Sorry that I can’t be of more help; it’s just one of those times where the language teases us.