Hi, thank you for your question. Just a bit more information required to fully assist you:
-Are you in England or Wales?
-How old is the child?
-Are/were you married to the father?
-Do you consider the UK as your main residence?
Legally, he would have been able to apply for a prohibited steps order to prevent you leaving as he is the father of the child - he does not necessarily require parental responsibility for this and just needs to demonstrate there is a relationship between them and a threat of you leaving. Given that you have now left, and if there is no plan for you to return to the UK, he is entitled to apply to the International Child Abduction and Contact Unit, either to return the child to the UK or to have contact.
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He will initially need to apply for permission from the court to seek an application and demonstrate his relationship with his daughter - this can be in the form of a statement, together with exhibits such as photographs or correspondence between you regarding her. Once permission is granted he can proceed with the prohibited steps order application.
In relation to rights - he has no automatic rights, but your child has a right to a relationship with him and this can only be reasonably be restricted if there are welfare or child protection concerns when she is with him or in his care. By you leaving the UK you have prevented the child from having contact and the court will consider this when deciding what order, if any, should be made.
Shared custody is usually when both parents are able to provide the child with day to day care needs equally and it is in the best interest of the child for this to happen - otherwise, yes one parent will be considered the main carer whilst the other will have contact arrangements.
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