Whatever U.S. President Barack Obama was hoping to achieve at the Asia-Pacific summit in Singapore, it seems on the surface that he failed on two of the headline issues.

First, the summit of 21 nations, including Russia, Mexico and Australia as well as the United States and China, backed away from the commitment in the draft communique to halve greenhouse gas emissions by 2050. Instead, they finally agreed only to pursue “an ambitious program” at next month’s climate-change conference in Copenhagen, Denmark.

This is probably the final nail in the coffin for any hopes that Copenhagen would produce a binding international commitment to cut carbon emissions.

Second, Obama failed to secure any Chinese commitment to revalue its currency. This a major issue in the U.S. Congress, where critics of China maintain their currency is undervalued by about 30 percent, making Chinese exports to the United States much cheaper.

The summit’s draft communique, which had been approved by Chinese officials before the heads of government sat down to ratify it, pledged to support “market-oriented exchange rates.” At Chinese insistence, the final communique deleted that reference.

The main achievement of the summit was to make yet another pledge to complete the Doha Round of world trade talks next year, nine years after the process began. This is mixed news for Obama, since his chances of getting such a new free-trade deal through Congress are very slim.

It remains to be seen whether his offstage meeting with Russian President Dmitry Medvedev will succeed in producing a new treaty to cut U.S. and Russian strategic nuclear weapons. So perhaps the only visibly positive outcome for Obama was the purely theatrical and even cosmetic sound bite of claiming to be his country’s “first Pacific president.”

Even that must have rung a little hollow in Indonesia, the Asia-Pacific country where Obama spent part of his boyhood and where he is highly popular as something of a native son. But Indonesia, despite being the world’s most populous Muslim nation, is not on his travel schedule for this trip.

The trip focused on the main Asia-Pacific powers, staring with China, his next stop after Singapore. He had begun in Japan, where he sought to calm concern that the new Japanese government was backing away from that country’s traditionally firm alliance with the United States.

After China, Obama goes on to South Korea, where he was hoping for some progress on the complex problem of North Korea’s nuclear ambitions. But even in South Korea, where the U.S. relationship remains robust, there is a problem. Spooked by claims that free trade takes jobs from American workers, the U.S. Congress has yet to ratify a bilateral free-trade deal with South Korea.

But China remains the high point of the trip, and after the failure at APEC to get any Chinese concessions on its currency or on climate change, it will not be easy for Obama to turn the visit into the kind of success that resonates with the voters back home.

This is not for want of trying. The Obama administration has made a number of concessions to China that have yet to earn any reward. U.S. Treasury Secretary Timothy Geithner has backed away from his demonstrably true earlier claims that China manipulates its currency.

And Secretary of State Hillary Clinton has already signaled that the United States will not be making a big issue of China’s human-rights record. Indeed, State Department veterans have told this reporter that her deputy, Jim Steinberg, has issued standing instructions against U.S. diplomats meeting Chinese dissidents without his prior approval. And to the dismay of many human-rights activists, Obama pointedly did not meet the Dalai Lama on his last visit to Washington.

The Chinese, however, focus on two trade-restriction measures that the Obama administration has taken against Chinese exports of tires and steel tubes. President Hu Jintao spoke forcefully in Singapore of the need to resist “unreasonable” protectionist measures, and China’s top banking regulator Liu Mingkang last week warned that ultra-low U.S. interest rates were fueling speculation and creating bubbles that threaten the world’s hopes of economic recovery.

So the stage is set for some bumpy meetings in China, despite Obama’s pledge to deepen and intensify the relationship, and his claim to see China as “a strategic partner.” With China poised over the next two years to overtake Japan and become the world’s second-largest economy, the U.S.-China relationship is critical.

“The rise of a strong, prosperous China can be a source of strength for the community of nations,” Obama said in Tokyo.

But it can also be a major headache. Lee Kuan Yew, the veteran Singapore leader, sounded a warning recently when he declared: “The size of China makes it impossible for the rest of Asia, including Japan and India, to match it in weight and capacity in about 20 to 30 years. So we need America to strike a balance.”

The question for Obama is whether this balance is a role he seeks to play. It implies containing China and limiting its regional ambitions, and Obama said specifically in Tokyo that containment is not part of his policy. He has yet to say what his policy is, and the evidence from the APEC summit is that his concessions to Beijing have produced little result. And the U.S. trade deficit widened again last month.

In the long run, Obama’s presidency will be judged by the kind of relationship he can build with China almost as much as by his economic record. And his ability to haul the United States out of recession and into a far more robust recovery will also depend heavily on the economic relationship with China. Wherever he looks, Beijing remains the big issue.