André Weil and the Hodge Conjecture

The text recounts André Weil’s longstanding skepticism of the Hodge conjecture and his influential role in shaping Hodge theory’s modern language. Weil reorganized Kähler-Hodge techniques into a clearer exposition, but he distrusted the conjecture’s plausibility. Early optimism came from Lefschetz’s (1,1)-theorem (the p=1 case) and the hope that the Hodge ring might be generated by degree‑2 classes, but Mumford’s construction of “exceptional” Hodge classes in a CM-type abelian fourfold showed that (2,2) classes need not be products of divisors. Tate informed Weil, who recognized a whole family of such examples: abelian varieties with an extra imaginary quadratic symmetry split complex directions into two equal groups, and the product over that field produces a two‑dimensional rational subspace of (n,n)-classes - now called Weil classes - coming from symmetry rather than known algebraic cycles.

Weil therefore pursued a counterexample to the conjecture, arguing that these automatically produced Weil classes might not correspond to algebraic cycles; his attempts failed, and the conjecture remained open. Later work inspired by Weil led Gross to adapt the ideas and, after interaction with Deligne, culminated in Deligne’s theorem that Hodge cycles on abelian varieties are absolutely Hodge. Commentators noted that abelian varieties, once thought an easier case, still resisted algebraicity of Weil classes for decades, reinforcing Weil’s insistence that a decisive counterexample would be valuable. The piece also cites Weil’s blunt plea that geometers would be served by settling the question with a counterexample.