62 problems

A Chow motive is a motive in the category of Chow motives. Kimura–O'Sullivan conjecture. Every Chow motive is Kimura-finite. Kimura finiteness is a structural finiteness property o…

Blanc's lattice conjecture. If $\cC$ is smooth and proper over $\C$, then

Let $X$ be a quasi-projective scheme. Write ${\mathbf{D}}^b({\mathbf{Coh}}(X))$, ${\mathbf{D}}({\mathbf{Qcoh}}(X))$, and ${\mathbf{Perf}}(X)$ for its bounded derived category of co…

Let $n\geq 2$ and let $A$ be an $n$-Poisson algebra. Full shifted Poisson complex conjecture. The $L_{\infty}$-algebra structure of the full shifted Poisson complex…

Let $k$ be a field of characteristic $0$, and let ${\mathcal A}$ be a smooth proper dg category over $k$. For every integer $n\geq 1$, consider the complex … where…

Geometricity conjecture. The derived stack $\mathbb{R}\underline{Vect}_{n}(X)$ of vector bundles on $X$ is geometric.

Let $X$ and $Y$ be smooth projective varieties, and let $D(X)$ and $D(Y)$ denote their bounded derived categories of coherent sheaves. A functor between these categories is standar…

Let $M$ be a symplectic manifold, let ${\cal W}$ be its Weyl algebra bundle, and let $(\Omega(M, {\cal W}), \nabla, R)$ be the associated $Q$-algebra. A finitely generated projecti…

Realization conjecture. There exist an exact category $\mathcal{B}$ of finite homological dimension and a group $H$ acting on $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ such that…

Mass-Hom conjecture. The mass-Hom inequality holds for all stability conditions on any smooth and proper category.

Decategorification isomorphism conjecture. The injective algebra homomorphism above is always an isomorphism.

Let $\mathcal{A}$ and $\mathcal{B}$ be small dg categories, and consider a larger source bicategory whose Hom category for $(\mathcal{A},\mathcal{B})$ is the whole derived category…

Let $X$ be a smooth projective scheme over a field, and let ${\mathbf D}^b_{\operatorname{Coh}}(X)$ denote its bounded derived category of coherent sheaves. An enhancement of this…

Let $X$ and $Y$ be smooth and projective schemes over a field. An exact functor … is a functor between bounded derived categories of coherent sheaves. Bernstein–Lunts–Lunts conject…

Uniqueness conjecture. The category $D^b\operatorname{Coh}(X)$ should have a unique enhancement.

Rivera–Takeda–Wang's conjecture. For an appropriate choice of the lift of $\bbDelta(1)$, the secondary coproduct coincides with the loop coproduct; in particular, the appropriate c…

Let $\mathcal{C}$ be a $\mathbb{C}$-linear dg-category. Write $K_*^{\operatorname{top}}(\mathcal{C})$ for its topological $K$-theory groups and $HP_*(\mathcal{C})$ for its periodic…

Let $\mathcal T$ be a triangulated category with $\operatorname{FM} \mathcal T\operatorname{NE}\emptyset$. The proper Fourier–Mukai locus is…

Let $\mathcal{T}$ be a smooth proper dg category, and let $\operatorname{Sym}^n\mathcal{T}$ denote its $n$th symmetric power in the sense of Ganter–Kapranov. Write…

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

Let $C$ be a smooth proper dg-category. Categorical non-commutative Hodge structure conjecture. There is an entirely categorical way to produce a non-commutative Hodge structure fr…

Standard derived equivalence characterization. The following conditions should be equivalent:

Let $\mathcal{A}$ be a small dg category. Its Hodge classes are the subspace … where $\operatorname{Ch}^{\operatorname{top}}$ is the topological Chern character, $j$ is the map fro…

Let $\Lambda$ be a finite-dimensional $\mathbb{k}$-algebra, let $\Lambda_0=\Lambda/\operatorname{rad}(\Lambda)$ be its semisimple quotient, and assume that $\Lambda_0$ is separable…

Let $k$ be a perfect field, let $l$ be a prime invertible in $k$, and let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $k$. Let…