André Weil and the Hodge Conjecture
5 days ago
- OpenAI may have constructed a counterexample to the Hodge conjecture.
- André Weil was an early doubter of the Hodge conjecture and attempted to find counterexamples.
- Weil recognized Hodge theory's importance but criticized its notation as a 'horrible salad of tensors'.
- Weil's book 'Introduction à l'étude des variétés kählériennes' modernized Hodge theory notation.
- Mumford found exceptional Hodge classes on a CM abelian fourfold that could not be products of divisors.
- Weil generalized Mumford's example into 'Weil classes' arising from symmetry, leading him to doubt the conjecture.
- Weil challenged supporters to find algebraic cycles for these classes while he sought to prove none exist.
- Despite efforts, Weil did not disprove the Hodge conjecture, which later became a Millennium Problem.
- Weil's approach influenced others: Deligne and Gross explored Hodge cycles on abelian varieties.
- Weil's skepticism remains relevant as mathematicians still cannot prove Weil classes are algebraic.