605 problems

Noncommutative minimal model conjecture. For any smooth Fano variety $X$, there exist $\tau\in {\mathrm H}^\bullet(X)$, a sector $\mathscr{S}\subset\mathbf{C}^*$, $\epsilon>0$, and…

Let $G$ be a semisimple simply-laced complex group, let $\theta$ be a diagram automorphism of $G$, and let $C$ be a Calabi–Yau Riemann surface. For any…

Let $V$ be a $2n$-dimensional symplectic vector space, let ${\mathsf{IGr}}(k,V)$ be the isotropic Grassmannian, and let ${\mathcal{A}}_t$ be the admissible subcategories defined by…

Perry's noncommutative integral Hodge conjecture. The homomorphism

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $mathbbm{B}$ be the vector space of Betti vectors, and let $\mathcal{E}$ be a vector bundle on $mathbbm{P}^{n-1}$ with…

Let $X$ be the coadjoint variety of a simple algebraic group $\mathrm G$ over an algebraically closed field of characteristic zero. Let $\mathrm T(\mathrm G)$ be the Dynkin diagram…

Let $X$ be a Fano variety of index $m$ over an algebraically closed field $\boldsymbol{\Bbbk}$ of characteristic zero, and assume that the big quantum cohomology…

Trichotomy conjecture. For every vertex $i\neq 0$, there is a unique $k(i)\in\\{-1,0\\}$ such that

Chen–Xi's conjecture. Suppose two algebras $A$ and $B$ are derived equivalent. If $A$ is a Morita algebra and the dominant dimension of $B$ is at least two, then $B$ is also a Mori…

Let $X_1$ and $X_2$ be Gushel–Mukai varieties, and let $\mathsf{A}(X1)$ and $\mathsf{A}(X2)$ be their associated Lagrangian subspaces. Assume that neither subspace contains decompo…

Cao–Jiang conjecture. For any exceptional object $\mathcal{E}\in D(X)$, there exists an autoequivalence $\Phi\in\operatorname{Auteq}(D(X))$ such that $\Phi(\mathcal{E})$ is an exce…

Let $X$ be the symplectic manifold under consideration. For each Lagrangian $L_i\subset X$, let $\mathcal L_i$ be the corresponding object in the category of $\alpha_X$-twisted she…

Let $X\subset\mathbb{P}^4$ be a smooth quintic 3-fold, and assume the strong Bogomolov–Gieseker conjecture used to construct the heart $\mathcal{A}_G$ and central charge…

Let $X$ be a smooth complete toric variety. Suppose that $X$ is an $F$-fibration over $Y$, where $Y$ and $F$ are smooth toric varieties, and both $Y$ and $F$ have a full strongly e…

Let $\{\Delta_1',\ldots,\Delta_r'\}$ and $\{\Delta_1”,\ldots,\Delta_r”\}$ be the translated nef-partitions described in the preceding discussion, with corresponding mirror Calabi–Y…

Categorical McKay correspondence conjecture. There exists such a category and functor satisfying all of the following: $\mathcal{D}$ is 2-periodic, with $T^2=\operatorname{id}$; ev…

Let $X$ be a K3 surface. Let … be the homomorphism induced by the action of Fourier--Mukai autoequivalences on the Mukai lattice, and let…

Let $\mathcal{C}$ be a hereditary finitary category. A derived functor is called friendly when it sends every object into two adjacent cohomological degrees. Automorphism conjectur…

Calabi–Yau Drinfeld-double conjecture. The assignment

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. An exact functor $F:D^b(\mathcal{C}_1)\to D^b(\mathcal{C}_2)$ is friendly…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. Their reduced Drinfeld doubles are denoted by…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, let ${\mathcal M}_e^{perv}$ be the full subcategory of perverse objects in…

Let $X$ be a smooth projective variety and let $L$ be an ample line bundle on $X$. A Lefschetz decomposition of $\mathit{D}^b(X)$ is minimal when it is strict and minimal with resp…

Let $Y$ be a singular variety admitting crepant categorical resolutions, and call a categorical resolution indecomposable when it has no nontrivial decomposition in the relevant ca…

Let $Y$ be a singular variety, and let a categorical resolution of $Y$ correspond to a Lefschetz decomposition of $4D^b(\mathit{Z})4D$, where $\mathit{Z}$ is the exceptional diviso…