It really depends on the state. Each state has different laws for divorce and annulment. An annulment is sometimes granted where there were issues such as being forced or threatened into the marriage, having some sort of mental incapacitation at the time of marriage or being temporarily or permanently insane at the time, or marrying based on fraudulent statements or actions. I can't speak for all states, but I know that some states require that the annulment be requested by the "innocent" party.
Mr. Ruiz's answer is quite good. I'll add that the option of divorce may be more available to you than that of annulment, since divorce can be accomplished without all the mental incapacity or fraud or duress stuff that attaches to annulment. Divorce ends marriage just as effectively as does annulment. In annulment, you are asking the court to rule that the marriage never occurred at all. Saying something that actually happened never happened is tough for a judge to do, hence the heightened requirements. In divorce, you're simply saying that the two of you need to go your separate ways and asking the court to allow you to do so.
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Your question does not provide your state of residence. In California, and many states, the grounds for nullification of marriage are limited to:
(1) The spouses are related; (2) Either or both spouses are married to someone else; (3) One of the spouses was underage, under duress, or legally insane; or (4) the marriage was obtained by fraud. Homosexuality does not satisfy any of these categories.
Given that a petition for dissolution or for nullity of marriage is a public court document, you may find it more advisable to simply file for dissolution of marriage (divorce). This requires only that either or both spouses assert that there are "Irreconcilable differences" without any further explanation. There is no legal advantage for nullity vs. dissolution.
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