Cycles, derived categories, and rationality

Abstract

Our main goal is to give a sense of recent developments in the (stable) rationality problem from the point of view of unramified cohomology and 0-cycles as well as derived categories and semiorthogonal decompositions, and how these perspectives intertwine and reflect each other. In particular, in the case of algebraic surfaces, we explain the relationship between Bloch's conjecture, Chow-theoretic decompositions of the diagonal, categorical representability, and the existence of phantom subcategories of the derived category.

54 pages, comments welcome!

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• Derived categories view on rationality problems
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• Categorical representability and intermediate Jacobians of Fano threefolds

Cited by in corpus (5)

• Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives
• Rational points on symmetric powers and categorical representability
• Categorical dimension of birational automorphisms and filtrations of Cremona groups
• Rationality of inner twisted flags of type $A_n$
• A note on semiorthogonal decompositions for Fano fibrations