Stock markets on edge over Delta variant, Fed policy

* European stocks higher

* German inflation slows

* COVID-19 curbs in Australia, parts of Asia

* Oil prices recover

* Graphic: World FX rates https://tmsnrt.rs/2RBWI5E

LONDON, June 29 (Reuters) - Global shares inched back from record highs on Tuesday, with concerns about new coronavirus outbreaks in Asia undercutting an economic recovery, while investors remained on edge over the United States' exit from accommodative policy.

European stocks, as measured by the pan-European STOXX 600 index, were up 0.6%, helped by a jump in industrial, financial and mining stocks, sectors set to benefit from economic improvements.

Optimism around a steady recovery has put Europe's benchmark on course for its fifth straight month of gains.

Germany's DAX added 1%, grinding higher after data showed an easing of Germany's annual consumer price inflation in June. The reading of 2.1% was still above the European Central Bank's target of close to but below 2%.

Investors have been keeping an eye on inflation figures and what they may mean for continued central bank stimulus.

MSCI's broadest index of Asia-Pacific shares outside Japan sank 0.6% as recent positive momentum stalled as some countries re-imposed lockdowns to contain the spread of the Delta variant of the virus.

Japan's Nikkei fell 0.8%, while in Australia the ASX/200 index closed down 0.1% as increasing COVID-19 curbs across the country dented sentiment. The South Korean market closed 0.5% lower.

Chinese stocks lost 0.92% as investors booked profits after a rally on the back of the country's strong rebound from the impact of the COVID-19 pandemic.

U.S. stock futures, the S&P 500 e-minis, were 0.1% lower, while MSCI's all country world index, which tracks shares across 50 countries, was 0.1% weaker, off record highs scaled the day before.

Fears over the spread of the highly infectious Delta variant are denting sentiment at a time markets are on edge after the Fed shocked traders with a hawkish tilt earlier this month.

Indonesia is grappling with record-high cases, while Malaysia is set to extend a lockdown and Thailand has announced new restrictions.

"The outlook for policy in general and the U.S. specifically, both fiscal and monetary, is the more relevant factor in the market's mind right now rather than the spread of the Delta variant," said James Athey, investment director at Aberdeen Standard Investments.

"That may well prove to be naive or complacent."

In the U.S., a closely-watched jobs report for June will be released on Friday. It could sway the Fed's policy outlook and bring forward expectations for interest rate increases.

"Inflation is already much higher than the Fed was anticipating, so it is really the pace of improvement in the labour market that stands head and shoulders above every other indicator in terms of when the Fed will feel comfortable signalling the start of tapering," said Ray Attrill, Head of FX Strategy at National Australia Bank in Sydney.

News of a possible bipartisan U.S. infrastructure spending agreement over the weekend helped boost risk appetite on Monday.

On Wall Street, the Nasdaq and S&P 500 had gained 0.98% and 0.23%, respectively, on Monday to hit all-time highs, fuelled by tech stocks as investors bet on a robust earnings season.

Yields for benchmark 10-year U.S. Treasuries edged higher, but below levels of recent days, at 1.4934%. Last week, it notched its largest weekly gain since March.

Germany's 10-year bond yield edged higher at -0.177%, within sight of a recent one-month high.

In currency markets, the U.S. dollar was on course for its biggest single day gain in seven trading sessions. Against a basket of its rivals, the greenback rose 0.2% to 92.06.

Both the dollar and yen have benefited from some safe-haven demand, driven by concerns over the spread of the highly contagious Delta variant first identified in India.

The euro declined 0.2% to $1.1900, edging back toward the 2-1/2-month low of $1.8470 touched on June 18.

The British pound slipped back toward a two-month low, weakening 0.2% to $1.3861.

Concerns over renewed COVID-19 lockdowns across parts of Australia hampered the country's dollar, which fell 0.3% to $0.75580.

Oil prices recovered after earlier dipping on angst over the virus spread translating into slower fuel demand growth.

Brent crude was up 0.3% at $74.92 a barrel, while U.S. light crude added 0.3% to $73.12 per barrel.

Spot gold was 0.6% down at $1,767.60 per ounce.

(Reporting by Tom Arnold in London and Paulina Duran in Sydney; Editing by Shri Navaratnam, Emelia Sithole-Matarise and Alexander Smith)