The Period-Index Conjecture
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Statement
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field of characteristic and any there is a -dimensional variety over carrying a Brauer class that violates it, for Hodge-theoretic reasons. For the construction needs no uncountability, so the conjecture fails already over .
Context
A named conjecture in the arithmetic of Brauer groups, standing in the literature on period-index bounds over function fields and over algebraically closed fields, and a reference point for Hodge-theoretic obstructions.
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The paper's AI disclosure is unusually specific about a partial success. Prompted to find a counterexample by the Hodge-theoretic strategy, ChatGPT produced an example that was flawed, but whose shape survived into the final solution: a quotient of the same form, with an abelian surface, a genus two curve and the group Z/4. The author refined that into the working construction.
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