The full answer is too broad (it's a 50-state survey question). Here is a starter, though. In Washington, annulment may be sought if
(i) The marriage or domestic partnership should not have been
contracted because of age of one or both of the parties, lack of
required parental or court approval, a prior undissolved marriage of
one or both of the parties, a prior domestic partnership of one or
both parties that has not been terminated or dissolved, reasons of
consanguinity, or because a party lacked capacity to consent to the
marriage or domestic partnership, either because of mental incapacity
or because of the influence of alcohol or other incapacitating
substances, or because a party was induced to enter into the marriage
or domestic partnership by force or duress, or by fraud involving the
essentials of marriage or domestic partnership, and that the parties
have not ratified their marriage or domestic partnership by
voluntarily cohabiting after attaining the age of consent, or after
attaining capacity to consent, or after cessation of the force or
duress or discovery of the fraud, shall declare the marriage or
domestic partnership invalid as of the date it was purportedly
contracted
But then also
(ii) The marriage or domestic partnership should not have been
contracted because of any reason other than those above, shall upon
motion of a party, order any action which may be appropriate to
complete or to correct the record and enter a decree declaring such
marriage or domestic partnership to be valid for all purposes from the
date upon which it was purportedly contracted
So an annulment would have to fit into one of these latter unspecified reasons. Although material fraud is considered to be such a reason, the closest case (an attempt to annul based on fraud), the WA Supreme Court avoided deciding whether a particular instance of alleged fraud sufficed to invalidate a marriage, since in addition one party was incompetent and the marriage was not solemnized, as required by state law.
In Radochonski v. Radochonski (1998 Wash. App. LEXIS 765), the husband sought a
declaration of invalidity of marriage based on fraud in the essentials of the marriage (the allegation was that the wife entered into the marriage to get permanent residency). The petition was denied because "the alleged fraud does not go to the 'essentials' of marriage" and because he "cannot demonstrate reasonable reliance on any statements Barbara made as to her motive in marrying him". The court notes that there is only one case, Harding v. Harding, addressing what the essentials of marriage are:
where one of the parties to a marriage ceremony determines before the
ceremony that he or she will not engage in sexual intercourse with the
other after marriage, not disclosing such intention to the other, and
carries out such determination, the offending spouse commits a fraud
in the contract of marriage affecting an essential of the marital
relation, against which the injured party may be relieved by annulment
of the marriage.
The court said that fraud in an essential may be found (citing cases in other states)
where one spouse has misled another on an attribute that prevents
sexual relations between the parties such as impotence, venereal
disease, and drug abuse, the latter on the theory that narcotics cause
impotence. These attributes have gone to the essentials of marriage
because they affected the sexual relations that are at the heart of
the marriage
but no so in the case of
premarital chastity, false representations as to love and affection,
misrepresentation of affection, failure to disclose out-of-wedlock
children, fraudulent representation of pregnancy, and failure to end a
previous relationship.
So it is highly unlikely that fraud in the essentials of marriage would be found at least in Washington.
