9 problems

Let $n$ be a nonnegative integer, set $d(n)=6+\max\\{3,n\\}$, and let $X$ be obtained by blowing up $\mathbf{F}_n$ at $d(n)$ general points. Let $\mathcal{C}_n$ be a subcategory of…

Let $A$ be a noncommutative projective-plane algebra, so that its noncommutative projective plane is represented by the category $\operatorname{qgr}(A)$. A phantom category is a no…

A phantom is a nonzero admissible subcategory whose Grothendieck group vanishes. Let $\mathcal{R}$ be a smooth finite-dimensional dg-algebra, and let…

Let $X$ be a Barlow surface, and let $\mathrm{D}(X)$ denote its derived category. Barlow surface phantom conjecture. The category $\mathrm{D}(X)$ has an exceptional collection of l…

Quadric-bundle phantom-category conjecture. The semiorthogonal decomposition of the derived category of such a degeneration contains a phantom category.

Sarkisov phantom-category conjecture. The semiorthogonal decomposition of $D^b(X)$ contains a phantom category.

Phantom-category conjecture. The derived categories of the Barlow surface and of the rational blow-down described in the source contain a phantom category as a semiorthogonal piece…

Let $M$ be a symplectic manifold homeomorphic to a Del Pezzo surface, and let its Fukaya category be considered. A phantom orthogonal component is an orthogonal component of the Fu…

Let $S$ be a Barlow surface equipped with a generic Kähler form. An orthogonal decomposition of its Fukaya category is a decomposition into mutually orthogonal full subcategories,…