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Hodge theory of twisted derived categories and the period-index problem

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arxiv 2212.10638 v1 pith:6RSG4WJG submitted 2022-12-20 math.AG

James Hotchkiss This is my paper · ORCID
classification math.AG
keywords hodgecategoriesderivedperiod-indextheorytwistedclassesconjecture
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We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary smooth proper varieties and in families. As applications, we construct Hodge classes whose algebraicity would imply period-index bounds; construct new counterexamples to the integral Hodge conjecture on Severi-Brauer varieties; and prove the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces.

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Cited by 2 Pith papers

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  1. The period-index conjecture is false

    math.AG 2026-08 accept novelty 8.0 of 10

    The period-index conjecture is disproved: for every d≥3 there is a variety with a 2-torsion Brauer class of index 2^{d-1}, exceeding the conjectural bound.

  2. The hyperk\"ahler period-index conjecture is false

    math.AG 2026-08 accept novelty 7.0 of 10

    Huybrechts's hyperkähler period-index conjecture is false in dimension 4: on some K3^[2] and Kum2 fourfolds, period-2 classes have index divisible by 8, and on some K3^[2] fourfolds a period-5 class has index divisibl...

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