140 problems

Bridgeland stability conjecture. The category $\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(Y)$ admits a Bridgeland stability condition.

Contractibility conjecture. The space

Bayer–Macrì–Toda generalized Bogomolov–Gieseker conjecture. For any $Z_{\alpha,\beta}^{2}$-semistable object $E\in\mathcal{B}_{H}^{\beta}$ satisfying

Let $f_n$ be the chain-type invertible polynomial, let $S_n$ and $L_{f_n}$ be the associated ring and grading group, and let $\omega_1,\dots,\omega_n$ be positive rational numbers…

Let $\mathcal{T}$ be a triangulated category, and let $\operatorname{Stab}(\mathcal{T})$ denote its stability manifold. Folklore conjecture. If $\operatorname{Stab}(\mathcal{T})$ i…

Joyce's conjecture. For every $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $alphain N(X)$, there is an invariant $J^{alpha}(sigma)inLambda$ such that $J^{alpha}(sigma)$ does not de…

Joyce's conjecture. The holomorphic volume form $\Omega$ defines a stability condition in the Fukaya category of $X$.

Bayer–Macrì–Toda's conjecture. If $E$ is $\sigma_{s,q}$-semistable for a certain choice of $(s,q)\in\Delta$, then

Bridgeland–Joyce's conjecture. This correspondence can be understood as an isomorphism between some quotients of the spaces of stability conditions. Roughly, the space of stability…

Let a complex structure determine a Fukaya category, and let special Lagrangians be the relevant geometric objects in that category. Stability conjecture. Complex structures are st…

Let $X\subset\mathbb P^5$ be a smooth cubic fourfold, let $\mathcal A_X$ be its K3 category, and let $\operatorname{Aut}_{\rm s}(\mathcal A_X)$ denote the group of symplectic autoe…

Let $X$ be a smooth projective K3 surface, let $D^b\operatorname{Coh}(X)$ be its bounded derived category of coherent sheaves, and let the cohomological action define the morphism…

Let $(X,\alpha)$ be a projective twisted K3 surface, let ${\rm D}^{\rm b}(X,\alpha)$ be its bounded derived category of twisted coherent sheaves, let ${\mathcal P}_0(X,\alpha)$ be…

Let $X=\mathcal O_{\mathbb P^2}(-3)$, and let $\operatorname{Stab}^{0}_{n}(X)$ be the normalized connected submanifold of stability conditions defined by $Z(\mathcal O_x)=i$. Let…

Polishchuk's transitivity conjecture. The group $\operatorname{Aut}(D^b_{\operatorname{coh}}(C))$ should act transitively on the set of all spherical objects on $C$.

Let $X$ be a nonsingular projective variety, let $\sigma\in\operatorname{Stab}(X)$ be a stability condition, and let $\alpha\in\mathcal{N}(X)$ be a numerical equivalence class. Mod…

Let a Calabi–Yau category be a category equipped with a trace map … inducing a functorial duality … for all objects $A,B$. Consider two stability conditions on this category with t…

Let $M$ be a Calabi–Yau threefold, and let $D(\operatorname{Coh} M)$ be the derived category of coherent sheaves on $M$. A Pi-stable object is an object of this derived category sa…

Pi-stability monodromy conjecture. Pi-stability leads to such an autoequivalence ${\cal F}_{\cal P}$ for every closed loop ${\cal P}$. This conjecture asserts that transporting Pi-…

Let $Q$ be a quiver whose underlying unoriented graph is a disjoint union of Dynkin diagrams of type $A$, $D$, or $E$. A representation of $Q$ is called stable with respect to a we…

Let $X$ be a smooth proper variety, or more generally let $\mathcal{C}$ be a smooth proper category, and let $\sigma$ be a stability condition on the relevant derived category. A m…

Let $\mathcal{E}=(E_1,E_2,Phi)$ be a holomorphic triple of vortex type, with central charge $Z_{\mathcal{X}}(\mathcal{E})$ satisfying … and let…

Given an artinian abelian category $\mathcal A$ and a homomorphism $X \colon K_0(\mathcal A) \to \mathbb R$ positive on the class of every non-zero object, let the iterated HKKP fi…

Let $R$ be a ring and let $\mathcal{D}$ be an $R$-linear, smooth and proper category. A continuous path $\sigma_\bullet:[0,+\infty)\rightarrow\operatorname{Stab}(\mathcal{D}/R)$ of…

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