Bohner, a former journalist and TV personality who was once briefly married to author Michael Lewis, took up with the married Schmidt last year while she was living and working in south Florida. Since then, Schmidt "paid her an ungodly amount of money" to relocate her business, Kate Bohner Productions, to Los Angeles; she set up shop on Pacific Avenue in Venice, Calif. He's been spending the occasional weekend with her, flying down from the Bay Area on Thursday and leaving Saturday, staying at the Beverly Hills Hotel. (Schmidt saw her twice in October and at least once in November.)

But besides paying for the move and buying Bohner a new set of diamond earrings on every trip, Schmidt, whose net worth has been estimated at $6 billion, hasn't been forthcoming with cash. He attended a meeting at Bohner's firm to get pitched on funding a new $1 million documentary show. It sounds like he didn't bite, because Bohner's said to be losing her job.

Will Schmidt set Bohner up with a job at Google, as he did with a previous girlfriend, PR exec Marcy Simon? Bohner has told people she's already been working for Google — for free. She's been writing executive summaries about Google's push into the cell-phone market, work she estimates she could have billed at $250,000.

So why is Schmidt being so cheap? It's possible he's nervous about bad publicity that could come from leaving a money trail. He's said to have an arrangement with his wife Wendy, but she may not appreciate having the name of Schmidt's girlfriend make the headlines. So why doesn't he buddy up with some other media mogul and get them to fund Bohner's company on the sly? Ah, there's the rub: Schmidt has been so aggressively unlikeable in his dealings with other media CEOs that he doesn't have any chits to cash in.

The foundation's two main programs are the 11th Hour Project, an organization which publishes links to information it deems "scientific" about global warming, and Greenhound LLC, a bus operator on Nantucket Island. Schmidt is the founder of the 11th Hour Project, and a longtime summer resident of Nantucket, where she is also an investor in downtown real estate.

Both superficially good causes. But if Eric wanted to give Wendy, who has a master's in journalism from Berkeley, a job writing environmental press releases, why didn't he just hire her at Google, as he did with ex-girlfriend Marcy Simon? And if the Schmidts want to boost the value of their Nantucket real estate with bus service, why don't they just pay for it themselves, rather than with the help of a tax-exempt charitable foundation?

Eric Schmidt complains about the lack of investigative journalism today. This seems like a good place to start. Compared to Bill and Melinda Gates, whose charity reaches around the globe, the Schmidts don't just come across as small-hearted. They look downright unimaginative.

Well, Google shareholders can rest easy. Simon figured out that despite the massive diamond ring he gave her, Schmidt wasn't serious about promises he made Simon about splitting with Wendy. (According to a Simon confidante, Schmidt claimed he was "legally separated" from his wife. Can anyone confirm that?) "When he's with Wendy, he's completely with her," says an insider. And when he's not with her, he's with someone like Kate Bohner. Impressive that he manages to have so much free time while ostensibly running Google.