In particular, Katzarkov wanted to break the mirror image’s curve count into pieces, then use the mirror symmetry program to show that there was a corresponding way to break up the four-fold’s Hodge structure. He could then work with these pieces of the Hodge structure, rather than the whole thing, to show that four-folds can’t be parameterized. If any one of the pieces couldn’t be mapped to a simple 4D space, he’d have his proof.
But this line of reasoning depended on the assumption that Kontsevich’s mirror symmetry program was true for four-folds. “It was clear that it should be true, but I didn’t have the technical ability to see how to do it,” Katzarkov said.
He knew someone who did have that ability, though: Kontsevich himself.
But his friend wasn’t interested.
Digging In
For years, Katzarkov tried to convince Kontsevich to apply his research on mirror symmetry to the classification of polynomials — to no avail. Kontsevich wanted to focus on the whole program, not this particular problem. Then in 2018, the pair, along with Tony Pantev of the University of Pennsylvania, worked on another problem that involved breaking Hodge structures and curve counts into pieces. It convinced Kontsevich to hear Katzarkov out.
Katzarkov walked him through his idea again. Immediately, Kontsevich discovered an alternative path that Katzarkov had long sought but never found: a way to draw inspiration from mirror symmetry without actually relying on it. “After you’ve spent years thinking about this, you see it happening in seconds,” Katzarkov said. “That’s a spectacular moment.”
Tony Pantev studies the structure of manifolds by holding them in front of a mathematical mirror. Felice Macera
Kontsevich argued that it should be possible to use the four-fold’s own curve counts — rather than those of its mirror image — to break up the Hodge structure. They just had to figure out how to relate the two in a way that gave them the pieces they needed. Then they’d be able to focus on each piece (or “atom,” as they called it) of the Hodge structure separately.
This was the plan Kontsevich laid out for his audience at the 2019 conference in Moscow. To some mathematicians, it sounded as though a rigorous proof was just around the corner. Mathematicians are a conservative bunch and often wait for absolute certainty to present new ideas. But Kontsevich has always been a little bolder. “He’s very open with his ideas, and very forward-thinking,” said Daniel Pomerleano, a mathematician at the University of Massachusetts, Boston, who studies mirror symmetry.
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