610 problems

Kuznetsov's conjecture. The fourfold $X$ is rational if and only if … for an ordinary, untwisted projective K3 surface $S$.

Spinor-modification hyperbolic-equivalence conjecture. If $X'/S$ is a spinor modification of $X/S$, then $X'/S$ is hyperbolic equivalent to $X/S$.

For a smooth Fano variety $X$ of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts … The refined Gamma conjecture II also identifies categorical and quantum data.…

DK hypothesis. K-equivalence should imply an equivalence … More generally, if $p^*K_X\geq q^*K_Y$, one expects a fully faithful embedding from $D^b\operatorname{Coh}(Y)$ into…

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…

Conjecture. For every smooth quasi-projective variety $X$, … The lower bound $\operatorname{Rdim}D^b(X)\geq\dim X$ is known quite generally. The difficult direction is to construct…

Let $X$ and $Y$ be smooth projective varieties. Orlov's motivic conjecture predicts … where $h(X)_{\mathbb Q}$ denotes the rational Chow motive. A Fourier-Mukai equivalence produce…

For every effective action of a finite group $G$ on a smooth projective complex curve $X$, there should exist an ordering of the conjugacy classes $[g]$ of $G$ and a semiorthogonal…

For every dg algebra $A$ over a field $k$ such that $\dim_k H^*(A)<\infty$, the regular module $A$ cogenerates the unbounded derived category $D(A)$ of dg $A$-modules; equivalently…

For a general Calabi–Yau threefold $X$ in the Picard-rank-$1$ family of degree-$33$ threefolds in $G(2,6)$ constructed by Inoue–Ito–Miura, there exists a non-birational Calabi–Yau…

Let $\Sigma_g$ be a closed oriented surface of genus $g\ge 2$, let $\Lambda$ be the complex Novikov field, and let $\mathscr F_g$ be the split-closed, strictly unobstructed, two-pe…

Let $S$ be a base scheme and let $G$ be a proper, flat, finitely presented commutative group stack over $S$ with finite flat inertia. Define its dual by…

Let $X$ be a smooth projective variety. A stability condition means a Bridgeland stability condition on $\operatorname{D^b}(X)$, defined with respect to the numerical lattice…

Let $X$ be a projective K3 surface with a generic polarization $H$, and let $F$ be an $H$-polystable coherent sheaf on $X$. Kaledin–Lehn's formality conjecture. The differential gr…

Let $Q$ be a Dynkin quiver, let $D^b(Q)$ denote its bounded derived category, and let $\sigma$ be a stability condition in $\mathrm{Stab}(D^b(Q))$. A full $\sigma$-exceptional coll…

Hyper-Kähler motivic conjecture. If $\operatorname{D}^b(X)$ and $\operatorname{D}^b(Y)$ are exactly equivalent, then $\mathfrak{h}(X)$ and $\mathfrak{h}(Y)$ are isomorphic as Frobe…

Let $W$ be an invertible polynomial in $n$ variables, let $R=\mathbb{C}[x_1,\dots,x_n]/(W)$, and let $W^\ast$ be its transpose polynomial. Write $D^b_{\mathrm{sing}}(R)$ for the bo…

Let $X$ be a singular variety and let … be a resolution of rational singularities. Write $\textbf{D}^{b}(X)$ and $\textbf{D}^{b}(\widetilde{X})$ for their bounded derived categorie…

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

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

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…

Let $G/P$ be a homogeneous roof, and $\mathcal{Z}$ a homogeneous roof bundle of type $G/P$ with projective bundle structures … Given a general section…

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

Let $X$ be a variety admitting crepant resolutions, including commutative resolutions and non-commutative crepant resolutions. Van den Bergh's conjecture. All crepant resolutions o…

Let $X$ be the canonical line bundle of $\mathbb{P}^3$, with projection $\pi:X\to\mathbb{P}^3$, and let $H$ be the pullback of the hyperplane class. Let…