ALBANY – Tuesday’s national elections held mixed messages for the political future of Mayor Giuliani.

On one hand, Mr. Giuliani can claim a share of credit for the Republican victories, having crisscrossed the country on behalf of Mr. Bush and GOP candidates for Congress. And the effort seems to have boosted his standing in the party: A poll released yesterday found that Mr. Giuliani is the front-runner for the Republican nomination in 2008.

On the other hand, separate exit polls found that much of Mr. Bush’s support came from voters who disagree sharply with Mr. Giuliani’s liberal positions on social issues, including support for legalized abortion and gay rights.

A 22% plurality of voters identified “moral values” as the most important issue in the election, and four out of five of those voters preferred Mr. Bush, according to the nationwide survey of 13,660 voters by the major TV networks and the Associated Press.

These figures foreshadow the challenge facing Mr. Giuliani, Senator Mc-Cain, and other moderate Republicans contemplating running for the White House in four years. They are some of the best known and most popular figures in their party, with broad appeal among independents and Democrats, but they may prove unacceptable to social conservatives.

“The nation appreciates Rudy Giuliani for his tremendous efforts in the wake of 9/11, but as a national party leader I don’t believe he could advance,” the vice president for public policy of Focus on the Family, Tom Minnery, told The New York Sun. “He parts company with cultural conservatives on the greatest evil of the day, which is abortion. And his position in favor of gay marriage puts him at odds with one of the major movements that carried the day on Tuesday. …As his positions on these issues get known, the further he’ll sink down the list of preferred candidates.”

“If Rudy Giuliani has a conversion of thought and becomes pro-life and an advocate of traditional marriage, then he has a chance of being president,” said a conservative legal activist, Mathew Staver of Liberty Counsel. “Absent a conversion of thought, he will only be able to be a cheerleader, but he certainly can’t be commander in chief.”

Other observers said Mr. Giuliani’s extraordinary popularity, and his strong positions on national security, might well persuade social conservatives to look past their differences with him.

“It’s no secret that Rudy Giuliani is to the left of where the party is on most if not all social issues,” said the president of the Club for Growth, Stephen Moore. “That is going to be a handicap for him if he decides to run for president in 2008, but I don’t think it’s crippling. … Giuliani is just such a star, it puts him almost in his own category.”

Mr. Giuliani “did himself a world of good” by working so hard for Mr. Bush this year. “Being a good soldier, which is clearly what Rudy was, helps him with cultural conservatives,” Mr. Moore said. “If I were Rudy Giuliani’s political adviser, I would have told him to do exactly what he did.”

The misgivings of social conservatives did not prevent Mr. Giuliani from emerging as the leading contender for the 2008 GOP nomination in poll on Election Day by McLaughlin & Associates. He was the choice of 30% of voters who intend to participate in Republican primaries, compared to 17.8% for Mr. McCain, 5.4% for Governor Bush of Florida,2.4% for Senate Majority Leader Frist, and 2% for Governor Pataki.

Senator Clinton was the leading candidate among Democratic primary voters at 27.3%, compared to 16% for Senator Kerry, 6.7% for Senator Edwards, 3.6% for Vice President Gore, and 3.5% for Howard Dean.

Pollster James McLaughlin said it was unclear whether the Republicans polled were aware of Mr. Giuliani’s stands on abortion and gay rights. “Trust me, the Iowa caucus-goers will find out real quick,” he said.

Mr. McLaughlin added, however, that his telephone survey of 1,000 people, with a margin of error of plus or minus 3.1%, did not detect the same emphasis on social issues among voters as the networks’ exit polling.

Only 12% of voters said moral issues were most important, including 5% who cited abortion and 1.6% who cited gay marriage, according to the McLaughlin poll. The top issues in this poll were terrorism, the economy, and the war in Iraq.

“He’s positioned himself to be a national figure, spending lots of time on the road, meeting average Republicans, as a way to reduce the angst that some may have,” said a Democratic political consultant, Hank Sheinkopf. “His traveling the country, using his experience on 9/11, standing with the president, is an attempt to reduce those problems, to make the negatives into positives. You’ve got to take the whole package, is what he’s trying to tell people.”

The socially conservative wing of the Republican Party “is a problem for [his] future aspirations, no question,” said a political science professor at Cooper Union, Fred Siegel. “The question is, is there anything that can modify that?…I can imagine Giuliani beginning to modify some of his positions, especially on partial-birth abortion.”

However, Mr. Giuliani resisted doing just that in 2000, when he was considering a run for the U.S. Senate against Hillary Clinton. The state Conservative Party chairman, Michael Long, said Mr. Giuliani could have the party endorsement if he would come out against late-term abortions, and Mr. Giuliani refused.

“He would have to temper his views if he was going to be a viable candidate as we go down the road,” Mr. Long said. “He’d have to find some common ground with social conservatives.”

Gay marriage was not a pressing issue during Mr. Giuliani’s term as mayor, but he did sponsor legislation extending spousal benefits to the domestic partners of city employees. And when he was forced out of Gracie Mansion during his divorce, he lived for a while in the apartment of two gay friends.

While these gestures went over well with residents of New York City, they would probably be used against him in a Republican presidential primary.

“He’s got a tough row to hoe, as do all of us who support moderate Republicans and want them to advance within the party,” said the director of the National Abortion and Reproductive Rights Action League of New York, Kelli Conlin. “There’d be nothing better for us than to have a situation where both [presidential] candidates are pro-choice. It’s hard for me to believe that people on the religious right would accept that.”