18 problems

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…

Kuznetsov–Orlov no-phantom conjecture. There exists no phantom subcategory of $D^b\operatorname{Coh}(X)$ when $X$ is a smooth rational surface.

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 $n\ge 0$ be given, and let $X$ be a surface obtained by blowing up $\mathbf{F}_n$ at either $d$ general points with $d<6+\max\\{3,n\\}$, or any number of points in general posi…

Let $E$ be the anticanonical divisor of a noncommutative projective plane, and consider a blow-up at points on $E$. A phantom category is a nontrivial admissible full triangulated…

Let $S$ be a smooth projective surface admitting an effective canonical divisor or an effective anticanonical divisor. A phantom category is a nontrivial admissible full triangulat…

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…

Let $S$ be a del Pezzo surface, in particular let $S=\mathbb{P}^2$. A phantom category is a nontrivial admissible full triangulated subcategory of $D^b\operatorname{Coh}(S)$ with t…

A quasi-phantom category is a nontrivial admissible full triangulated subcategory of the bounded derived category of coherent sheaves on a smooth projective variety with finite Gro…

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

Let $\mathcal{T}=\langle E_1,\dots,E_n\rangle$ be a triangulated category generated by an exceptional collection. Kuznetsov's conjecture. Any exceptional collection of length $n$ i…

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…

Let $X$ be a smooth projective variety, let $\mathrm{D}(X)$ be its bounded derived category of coherent sheaves, and let $\mathcal{A}_{X}\subset \mathrm{D}(X)$ be an admissible sub…

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. Their derived categories 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,…