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Once won, and activated via passing into the US, what obligations do I have which are specific to the ownership of this visa category (as opposed to the regular tax and citizen obligations required by the virtue of living and working in the US)?
Specifically:
DV1 visa is an immigrant visa. Once you enter the US it expires immediately and is replaced with I-551 (green card). You will receive a physical green card within a couple of months of your arrival, until then - the stamped visa becomes your green card.
What you need to maintain is the green card (permanent residency status).
When you enter the US with this visa, you are issued with permanent resident status (green card).
The United States Citizen and Immigration Services website has a detailed page on maintaining permanent residence.
It states that you may lose permanent residence status if you are convicted of certain offences. It also states:
You may be found to have abandoned your permanent resident status if you:
Move to another country intending to live there permanently
Remain outside of the United States for more than 1 year without obtaining a reentry permit or returning resident visa. However, in determining whether your status has been abandoned, any length of absence from the United States may be considered, even if less than 1 year
Remain outside of the United States for more than 2 years after issuance of a reentry permit without obtaining a returning resident visa. However, in determining whether your status has been abandoned any length of absence from the United States may be considered, even if less than 1 year
Fail to file income tax returns while living outside of the United States for any period
Declare yourself a “nonimmigrant” on your tax returns
