ATLANTA -

A recent poll shows slim margins between Georgia's leading candidates running for U.S. Senate and governor.

An Atlanta Journal-Constitution poll conducted by a New York survey and research firm last week shows Michelle Nunn, a Democrat running for U.S. Senate, ahead of Republican opponents in a general election ballot test. However, Nunn is 1 point ahead of David Perdue, who may face a Republican runoff in July.

The poll also showed Gov. Nathan Deal several points ahead of Jason Carter -- a Democratic state senator from Atlanta, and grandson of former President Jimmy Carter.

Democrats haven't gotten much attention yet from Republicans who are competing against each other before the May 20 primary.  Early voting continues through Friday.

More than 1,000 voters were surveyed, and the poll has a 4 percent margin of error.