Orlov's perfect-dg-category phantom conjecture

A phantom is a nonzero admissible subcategory whose Grothendieck group vanishes. Let R\mathcal{R} be a smooth finite-dimensional dg-algebra, and let P ⁣erf ⁣--R\mathcal{P}\!\mathit{erf}\!\operatorname{--}\mathcal{R} denote the dg-category of perfect dg-modules over R\mathcal{R}. Orlov's conjecture. There are no phantoms of the form

P ⁣erf ⁣--R.\mathcal{P}\!\mathit{erf}\!\operatorname{--}\mathcal{R}.

The phantom arising from the non-full maximal-length exceptional collection on the ten-point blow-up of PC2\mathbb P^2_\mathbb C disproves this conjecture.

Sources & referencesView supporting material

Primary source

Johannes Krah, “A Phantom on a Rational Surface”, arXiv:2304.01269 (2023).

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