72 problems

Spectral-network stability-condition conjecture. The collection $\mathcal{P}$ forms a pre-slicing of the standard $\mathbb{V}_{\mathcal{S}}$-collection, and $(Z,\mathcal{P})$ is a…

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…

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…

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…

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 $Q$ 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…

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

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…

Halpern-Leistner's spanning conjecture. For any Fano variety $X$, there exist $\tau\in\mathrm{H}^2(X)$ and a sector $\mathscr{S}\subset\mathbf{C}^*$ such that, for every $\tau$-adm…

Let $Q$ be a finite connected acyclic quiver. Write $\operatorname{Stab}(\mathcal{D}^b(Q))$ for the space of stability conditions on $\mathcal{D}^b(Q)$, and let…

Let $(X,H)$ be a polarised smooth projective threefold. Let $\operatorname{CH}_1(X)_{\mathbb R}$ denote the real Chow group of $1$-cycles, and let $\nu_{b,w}$, $\Delta_H$,…

Let $(M^n,\omega,\Omega)$ be a possibly non-compact Calabi–Yau manifold, with Kähler form $\omega$ and holomorphic volume form $\Omega$, suitably convex at infinity. Let…

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

Bayer–Macrì–Toda stability conjecture. The pair $(\mathcal{A}_{H}^{\alpha,\beta},Z^{3}_{\alpha,\beta})$ is a stability condition on $D^{b}(X)$. This conjecture is equivalent to the…

Manifold-with-corners conjecture. There is an open neighborhood $U$ of $\sigma$ such that this map is a homeomorphism from $U$ onto an open subset of $\mathcal{A}^{\mathbb{R}}_n$.

Boundedness conjecture. If $\mathcal{C}$ is smooth and proper over $k$, then every stability condition on $\mathcal{C}$ admits a mass-Hom bound.

Manifold-with-corners conjecture. The map $\ell_{(-)}(E_1,\ldots,E_n)$ induces a homeomorphism from an open neighborhood of $\sigma$ onto an open subset of…

K(π,1) conjecture. The space $\operatorname{Stab}^\dagger_{\mathcal C}(\mathcal D)$ is contractible; consequently, both $\Upsilon_{\operatorname{reg}}/\mathbb W$ and…

Faithfulness conjecture. The action of the Artin–Tits group $\mathbb B$ on the triangulated category $\mathcal D$ is faithful.

Let $A=(a_1,a_2,a_3)$ satisfy $\chi_A>0$. Let ${\mathcal X}_A$ be the associated parameter space, $\widehat{W}_A$ its extended Weyl group, ${\mathcal D}_A$ the associated triangula…

Contractibility conjecture. The principal connected component is the union of the geometric and algebraic stability conditions and is contractible:

Twisted stability-condition conjecture. There exists a bounded t-structure on $\mathrm{D}_{\mathrm{perf}}(X,\alpha)$ with heart $\mathcal{A}_{\omega,D}$ such that…

Let $\mathcal{MC}(m)^{\Gamma}_{\ast,s}$ be the open subspace of framed cubic differentials whose zeros are distinct. Let $\mathcal D$ be the associated $CY_3$ category, let…

Let $Y\to\mathbb{P}^3$ be a quartic double solid whose ramification divisor contains no lines, and let $\cX/B$ be the family of prime Fano threefolds of genus $6$ constructed in Th…