The controversy started after Abjijeet, bizarrely, called the murder of a techie in Chennai, a case of Love Jihad, wrongly suggesting that her attacker was a Muslim.
When the female journalist Swati Chaturvedi and other Twitter users pointed out that he should be arrested for unnecessarily stoking communal hatred, Abhijeet lashed out with choicest abuses calling her "an old woman" in his expletive-filled rant.
Chaturvedi said that the singer's tweets were "just obscenities and profanities" and she would file an FIR against him.
She said Mumbai Police has reached out to her and assured a strong action against Abhijeet.
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"I am going to file an FIR and I hope that police will take the required action. I very strongly feel that enough is enough. We as journalists and women have been targeted enough on Twitter. I am tired of these sexual innuendos we all face."
Later in the day, Chaturvedi, a Delhi resident, lodged a complaint at Vasant Vihar Police Station in South Delhi in the matter.
She had claimed in her complaint that the singer had used "foul" and "obscene language" in his tweets.
"These abusive messages drew condemnation from the public, and Mumbai Police, who I had tagged, contacted me (and) asked me to file a complaint. Request you to file an FIR and send this to Mumbai Police for further action," the complaint, addressed to station house officer of Vasant Vihar police station, said.