There is a huge list of what are referred to by immigration as "aggravated felonies." I believe domestic violence, even if only a misdemeanor, is among them.
Discuss immigration consequences with your criminal defense lawyer for sure but also consult an immigration law specialist for further edification.
Obviously, the stakes are personally extremely high for you.
Is this in NY? "High misdemeanor" is not a NY criminal term. NY has Class A and B misdemeanors.
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First, you should discuss this with your lawyer. If you do not have one, get one. You could be deported because it is domestic violence. There is also the chance that nothing will come of it until you try to reenter the country after leading. You may want to reconsider entering a plea and taking it to trial.
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Consult with an immigration lawyer.
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DWI is an unclassified misdemeanor in New York. Most domestic violence crimes are misdemeanor level offenses, except simple Harassment in the Second Degree, which is a violation level offense. However, that is not your main concern.
As my colleagues suggest, consult as quickly as possible with a qualified Immigration Law attorney. He or she will be able to tell you how the various offense you have been charged with are viewed by INS. As suggested above, many crimes of domestic violence are viewed very harshly, no matter how NY or any jurisdiction may classify these offenses.