Criminal Law Questions? Ask a Criminal Lawyer.
Sorry...no. A person can file charges. The DA believes the person unless the story is outrageous or suspicious. If the charges are then dropped, the state is IMMUNE from suit for having brought the charges unless perhaps there was police misconduct of some kind. Plus there are many reasons why charges are dropped. You say the complainant lied. Assuming that is true, that may or may not be the reason the charges were dropped. Unfortunately our system is not set up for defendants to sue the state every time charges are dismissed or there is an acquittal.
If the statements were coerced, unfortunately that is a matter for the victim to complain about only. That said, once a person files charges, the case becomes THE PEOPLE V XXXXX XXXXX. The complainant then becomes just a witness in that prosecution and cannot just drop charges. The DA can decide the original story was the truth and can compel the complainant to testify.