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India saw a record single-day jump in new cases for the seventh straight day. It reported 6,535 new infections Tuesday, raising its total to 145,380, including 4,167 deaths.
Real are now second on 56 points after 27 games, two behind Barca who beat Real Sociedad 1-0 on Saturday.
Real's victory means they lead the standings on 56 points after 26 games, with Barca slipping to second spot on 55, while Zinedine Zidane's side also crucially have the better head-to-head record in the event both sides finish level on points.
Barca were beaten 1-0 by Bilbao at San Mames, where Inaki Williams headed in a dramatic 93rd-minute winner, after La Real had seen off Madrid 4-3 in a seven-goal thriller at the Santiago Bernabeu.
Zinedine Zidane's Real side demonstrated their new-found grit to re-take the lead four minutes later.
PSG pulled off an unlikely turnaround as Mbappe capitalised on a defensive mix-up to pull one back in the 81st minute before Pablo Sarabia, once of Real, drove into the top corner second later.
The Brazilian became the second-youngest player to score a hat-trick in the Champions League after Real great Raul in his team's 6-0 win.
The Kochi-based Blasters wear a formidable look, at least on paper, and are determined to come up with an improved show this time around.
Real, European champions 13 times and three years in a row from 2016-18, failed to muster a shot on target and suffered their first defeat of the season.
Peru were the superior team in the second half and could have scored more but for a series of excellent saves by Bolivian goalkeeper Carlos Lampe.