In the dictionary, lease means:
A lease is a legal agreement by which the owner of a building, a piece of land, or something such as a car allows someone else to use it for a period of time in return for money.
Therefore, I could not get the logic/implication/meaning of the following sentence, when Stark offered to put her name on the next clean self-sustained tower, but Pepper, his girlfriend, said :
“On the lease".
Does this mean, she's asking for (joking) the ownership of the next tower? Or she asks Stark to rent her the next tower? (But why? Stark does not need to earn the rent money from her.)
What's I am missing here??
Here is the sentence in the context, and the conversation just happened after Stark had lit up Stark Tower for the first time with Arc Reactor powering it:
In the penthouse apartment on top of the new Stark Tower, Pepper hit a switch. The tower began to light up, first from street level, and then hundreds of feet to the pinnacle, illuminating the New York skyline—and the sky itself.
“How does it look?” Pepper asked. He could see her grinning with excitement in the heads-up display inside the helmet of his Iron Man suit.
“Like Christmas, but with more me,” Tony said.
“We’ve got to go wider on the public awareness campaign,” she said, thinking like the CEO she was. The best decision Tony had ever made was handing the reins of Stark Industries over to her. “You need to do some press. I’m in DC tomorrow working on the zoning for the next three buildings. . . .”
“Pepper, you’re killing me,” Tony said as he arced past the Empire State Building. “The moment, remember? Enjoy the moment.”
“Get in here and I will,” she said.
Pepper stood watching a holographic display of Stark Tower and the Arc Reactor powering it. “Levels are holding steady, I think,” she said.
“Of course they are. I was directly involved, which brings me to my next question. How does it feel to be a genius?”
“Well, I really wouldn’t know, now would I?” Pepper teased him.
“What do you mean? All this came from you.”
“No, all this came from that.” Pepper tapped the miniaturized Arc Reactor in Tony’s chest.
“Give yourself some credit, please. Stark Tower is your baby. Give yourself . . . twelve percent of the credit.”
Tony had meant it as a joke, but he could see she hadn’t taken it that way. “An argument could be made for fifteen,” he said, trying to get a laugh out of her.
It wasn’t working. “Twelve percent,” she said. “My baby.”
Now he decided to tease her a little more. “Well, I did do all the heavy lifting. Literally, I lifted the heavy things. I’m going to pay for that comment about percentages, in some subtle way later, aren’t I?”
“It’s not going to be that subtle.”
“I’ll tell you what. The next building is going to say Potts on the tower.”
“On the lease,” she corrected him.
The Avengers I