* SSEC +0.7%, CSI300 +1.1% * Investors turn to STAR50, ChiNext after Didi drama * State Council says to step up supervision of overseas-listed firms SHANGHAI, July 7 (Reuters) - China shares closed higher on Wednesday, helped by robust strength in Shenzhen’s start-up board ChiNext and Shanghai’s tech-focused STAR Market, after Beijing stepped up supervision of Chinese firms listed overseas. ** At the close, the Shanghai Composite index was up 0.66% at 3,553.72 and the blue-chip CSI300 index was up 1.13%. ** Leading the gains, the sub-index tracking new-energy vehicle makers and suppliers jumped 5.77%, the rare earth sub-index surged 5.01% and the healthcare sub-index rose 2.54%. ** The financial sector sub-index was lower by 0.32%, the consumer staples sector up 0.31%, the real estate index down 1.23% and the healthcare sub-index up 2.54%. ** The smaller Shenzhen index ended up 1.68%, the ChiNext Composite index was higher by 3.572% and STAR50 index was up 2.43% ** Battery maker EVE Energy Co gained 13% while Ningbo Ronbay New Energy Technology, a material supplier for lithium batteries, surged 20%. ** China will upgrade law enforcement on illegal activities in the securities market such as financial fraud, insider trading and market manipulation, its cabinet said on Tuesday. ** The country will also step up supervision of Chinese firms listed offshore, days after Beijing launched a cybersecurity investigation into ride-hailing giant Didi Global Inc on the heels of its U.S. stock market listing. ** Around the region, MSCI’s Asia ex-Japan stock index was weaker by 0.36%, while Japan’s Nikkei index closed down 0.96%. ** At 07:13, the yuan was quoted at 6.4684 per U.S. dollar, 0.16% firmer than the previous close of 6.4788. (Reporting by Shanghai Newsroom; Editing by Shailesh Kuber) Our Standards: The Thomson Reuters Trust Principles.