The euro zone is reeling. Efforts to write consistent rules for global finance remain a tangle.

The Japanese have moved to hold down the value of their currency, leaving less room to criticize China for doing the same. The United States’ struggle with its economy and public debt has hobbled its authority on the world stage.

As the leaders of the Group of 20 opened their summit in this French resort Thursday, they were facing the limits of the G-20’s role as a sort of coordinating committee for the world economy. This year, the meeting of great powers has been overshadowed by outside events, jolted by surprises — Greece’s on-again, off-again referendum on an emergency bailout considered vital for the world economy — and buffeted by changing winds in small nations.

Greece’s referendum and possible exit from the euro zone dominated G-20 talks, with world leaders focused on how to prevent Europe’s problems from wrecking the international financial system and undermining the global economy.

The leaders explored possible responses, including new lending to troubled economies by the International Monetary Fund and direct loans to Europe from developing countries. Up for discussion again was how to persuade China to export less and import more, which could help fuel economic activity in places such as the United States.

But none of the ideas represented a “silver bullet,” said an official familiar with the talks.

Unlike two years ago, when G-20 leaders agreed on a massive global stimulus program to address the recession, the discussions here promised little dramatic impact. Government coffers are too bare in many countries to pursue such options again, and leaders are divided over the best strategies.

European efforts to persuade China and other major emerging economies to help finance Europe’s bailout fund have fallen short. U.S. officials, meanwhile, have insisted that the countries sharing the euro currency have enough money to pay for their own rescue program.

“We certainly believe that Europe has the capacity to step forward and to provide the kind of resources that can work through this period of time,” said U.S. deputy national security adviser Ben Rhodes.

President Obama, who held separate meetings with French President Nicolas Sarkozy and German Chancellor Angela Merkel, pushed for Europe’s leaders to move forward as quickly as possible with the emergency plan they approved last week to address the Greek debt crisis and the continent’s spreading financial problems. Greek Prime Minister George Papandreou’s surprise announcement Monday that he would subject the bailout plan to a vote of his populace threw the rescue program into disarray, and the uncertainty did not ebb even after he reversed direction Thursday, saying he was scrapping the referendum.

“The steps needed to be taken are clear irrespective of the political personalities or situation at any given moment,” Rhodes said. “What needs to be done as relates to Greece and the stabilization of the euro zone was outlined last week.”

Events overtake summit

At the height of the 2009 financial crisis, the G-20 agreed on a massive government spending plan that briefly had China, Europe, the United States and others pulling in the same direction. The stimulus agreement “provided a very strong sense that indeed it was the committee that ran the world. . . . After that, every single G-20 meeting has been less impactful than the prior one,” said Moises Naim, a senior associate at the Carnegie Endowment for International Peace.

In recent days, a series of developments has conspired to knock the G-20 summit off course.

World leaders and investors have raised grave concerns about the financial health of Italy, warning that the government in Rome could be next if Greece defaults. G-20 leaders had hoped to receive details of a broad reform package aimed at shoring up Italy’s finances. But Prime Minister Silvio Berlusconi failed to deliver. As investors continued to turn against Italy, the interest rates it must pay to borrow money ticked up again, moving closer to the level considered unsustainable. And, unlike Greece, Italy is considered too large to be bailed out by other European countries if it faces default.

A recent Swiss decision to hold down the value of its highly prized currency, meanwhile, also showed the limits of international cooperation. The Swiss move, along with the general instability in the euro area, has prompted investors to move their money into the Japanese yen. That has posed a burden for Japan by increasing the value of the yen and making Japanese exports more expensive abroad, crimping economic activity as the nation tries to rebound from the devastating March 11 earthquake and tsunami.

As a result, the Bank of Japan moved heavily into currency markets on the eve of the summit to push the price of the yen lower. This represented the type of unilateral economic moves that the G-20 is supposed to help prevent.

The move by Japan, a major economic power, also makes it more difficult for the group to issue any statement critical of how China manages its exchange rate. U.S. officials have been trying to enlist the G-20 in pressing China to let its currency, the renminbi, rise in value on world markets. This could lead China to import more goods from the United States and Europe, whose economies are flagging.

China’s currency policy

U.S. Treasury Secretary Timothy F. Geithner met with Chinese Vice Premier Wang Qishan on Thursday, and they discussed ways to aid Europe’s recovery, according to Lael Brainard, the U.S. Treasury undersecretary for international affairs.

Many U.S. lawmakers and presidential candidates have expressed concern about what they view as China’s efforts to manipulate the value of its currency, keeping it artificially low in relation to the dollar to make Chinese goods cheaper on world markets. Chinese officials have fought back, noting that their currency has actually been rising in value and saying that the United States is unfairly blaming China for an economic mess of its own making.

Brainard said that world leaders raised the issue during G-20 working sessions and that Chinese officials responded positively. She said she anticipated that China would agree to include formal notation related to “recognizing the role of greater exchange rate flexibility” in the “action plan” that emerges from the summit.

“I think China is recognizing the role of greater exchange rate flexibility in helping to shift to domestic demand,” Brainard told reporters Thursday evening. “It is one of the most powerful instruments that China has at its disposal in the near term to both counter inflationary pressure, which is a very high priority for them, and also to shift to domestic demand-led growth. It’s very much been part of the discussions here.”