Some of Ms. Abrams’s most prominent Democratic supporters across the country, including some potential presidential candidates, claimed that the election had been “stolen” because Mr. Kemp’s office had purged hundreds of thousands of voters from the rolls, had placed tens of thousands of registrations on hold, and had overseen a problem-plagued Election Day process that critics said amounted to voter suppression.

Mr. Kemp is not named as a defendant in the suit; instead, it names his interim successor as secretary of state, Robyn A. Crittenden, and the members of the state election board. Even so, the suit repeatedly criticizes Mr. Kemp.

In Georgia, the secretary of state is responsible for overseeing the election process, and Mr. Kemp, a fervent supporter of President Trump, has been denounced for years for pursuing policies that his critics describe as voter suppression tactics. He ignored repeated calls from Democrats for him to step down as secretary of state before the election, and not act as both a referee and a contestant in the race.

The 66-page complaint filed in the United States District Court for the Northern District of Georgia reiterates a familiar list of issues that Ms. Abrams’s team and others had raised before and after Election Day. It argues that the state had imposed “discriminatory voting barriers reminiscent of the Jim Crow era” in recent years, with Republicans in control of the state legislature and the governor’s office.

Among the many specific requests made in the suit, the plaintiffs ask the court to force Georgia to stop purges of inactive voters from the rolls; to reinstate voters already removed by those purges; to ban the touchscreen voting machines the state uses, which leave no paper trail and which the plaintiffs say are not secure; and to ensure that voters “can vote without unreasonable delay or hardship during any election,” a reference to long lines seen at some polling places and the closing of others.

Notably, the complaint also asks that Georgia once again be placed under a federal “preclearance” regime, in which any changes in Georgia voting rules would require approval from a federal judge, to ensure that they did not impinge on the voting rights of minority voters.