The Period-Index Conjecture is False
A foundational conjecture in algebraic geometry, predicting a sharp bound on the index of a Brauer class in terms of its period and transcendence degree, has been proven false. Through explicit geometric constructions leveraging Hodge theory, the paper demonstrates counterexamples in all dimensions three and higher, revealing that the conjectured relationship fails even over fields as familiar as the rational numbers. This result forces a fundamental reconsideration of how period and index relate in higher-dimensional arithmetic geometry.Script
For decades, mathematicians believed they understood the relationship between two fundamental invariants of division algebras: period and index. The period-index conjecture promised a precise bound connecting these quantities over function fields. That certainty has just shattered.
The conjecture claimed that for a variety of dimension d, the index must divide the period raised to the power d minus 1. For a threefold, that means index should divide period squared. But the paper constructs a concrete threefold with period 2 and index 8, violating the bound decisively.
The counterexample builds on a Dwork quartic K3 surface with a carefully chosen group action, paired with an elliptic curve carrying a compatible symmetry. Taking the quotient by this diagonal action produces a smooth projective threefold fibered by K3 surfaces, the stage for the contradiction.
The violation emerges through Hodge theory. A 2-torsion Brauer class arises from a cohomology class, and the Hodge-theoretic index machinery reveals a congruence obstruction. Certain integral Hodge classes fail to satisfy a quadratic equation modulo 2, forcing the index to jump beyond the conjectured bound.
The construction extends to all higher dimensions by taking products with projective space. For dimension d at least 4, the index grows as 2 to the power d while the period remains 2, systematically exceeding the conjectured bound. The mechanism relies on the period dividing the factorial of dimension minus 1.
The conjecture is now known to fail for all dimensions three and higher when the period divides the factorial of dimension minus 1. Whether it survives for other periods remains an open question, one you can explore further at EmergentMind.com, where you can dive deeper into cutting-edge research and create your own explanatory videos.
