Most were crushed by trees uprooted in winds of up to 140 kilometres per hour, which also cut power across large swathes of southern India on Monday.
India's meteorological department said Vardah, which forced the evacuation of 18,000 people, was the worst cyclonic storm to hit the capital of Tamil Nadu state in more than two decades.
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"So far 10 people are confirmed dead. We will get reports from rural areas as the day progresses, but the worst is over," said Abhishek Shandiyal, spokesperson for the National Diaster Management Authority.
The cyclone was weakening as it moved inland, he added. Television footage from Chennai, where huge floods last year killed at least 250 people, showed cars overturned by the strong winds and heavy rain.
The city is a centre for auto manufacturing and IT outsourcing. It is also scheduled to host the fifth Test match between England and India starting Friday, now in doubt after the storm. Team India said no decision had yet been taken about moving the Test.
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Chennai international airport was shut for almost 12 hours but resumed operations on Tuesday, an airport official said. Train services have also partially resumed.
India's eastern coast and neighboring Bangladesh are routinely hit by severe storms between April and December that cause deaths and widespread property damage.
In 1999, more than 8,000 people were killed when a cyclone battered the eastern state of Orissa.
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