1,060 problems

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

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.…

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…

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…

Kontsevich's homological mirror symmetry conjecture. Under the mirror map $\Greekmath 0124 \longleftrightarrow\Omega^{\vee}$ and $\Omega\longleftrightarrow\Greekmath 0124 ^{\vee}$,…

Let $X$ be a smooth cubic fourfold. Write … where $\mathcal{A}_X$ is the Kuznetsov component. An associated K3 surface is a polarized K3 surface $S$ whose primitive degree-two Hodg…

Orlov's motivic conjecture. The rational Chow motives of $X$ and $Y$ are isomorphic.

Let $X$ be a smooth algebraic variety whose bounded derived category of coherent sheaves is equipped with a Frobenius structure, and let $D^b\operatorname{Coh}(X)$ denote that boun…

Mirror symmetry conjecture. On the A-side, there are three non-birational Calabi–Yau threefolds with isomorphic derived categories.

Let $X$ be a quasi-projective variety with an action of a finite group $G$, let $X/G$ have Gorenstein singularities, and let $Y$ be any crepant resolution of $X/G$. Denote by…

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

Let $X$ be a singular variety admitting crepant resolutions, and let $D^b(\tilde X)$ denote the bounded derived category of coherent sheaves on a crepant resolution $\tilde X$. Cre…

Kawamata's K-equivalence conjecture. If $\phi$ is K-equivalence, then

Bayer–Macrì–Toda conjecture. The heart obtained by the double-tilt construction on $\operatorname{Coh}(X)$, together with the central charge $Z^X_{a,b}$, defines a Bridgeland stabi…

Bondal–Polishchuk conjecture. This action should always be transitive, up to independent shifts of the $n$ exceptional objects.

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 $G$ be a finite subgroup of $\operatorname{SL}(n,\mathbb{C})$, and let $\tau:Y\to\mathbb{C}^n/G$ be a crepant resolution. Here $D(Y)$ denotes the derived category of coherent s…

Let $X$ be a smooth projective variety, and let $\operatorname{FM}\operatorname{Perf}X$ denote the set of its Fourier–Mukai partners. Kawamata's finiteness conjecture. … The source…

Orlov's conjecture. If $X$ possesses a full exceptional collection, then $X$ is rational.

O'Grady's conjecture. For every ${\mathcal{E}}\in D^b(S)$,

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

Căldăraru's HKR module-compatibility conjecture. The isomorphism $\mathrm{IK}$ is compatible with the module structures on differential forms over polyvector fields and on Hochschi…

Formality conjecture. The dg-algebra $A_X$ is formal: it is quasi-isomorphic to the graded algebra

Let $X$ be a normal algebraic variety with two crepant resolutions … Here $\operatorname{D}^{\operatorname{b}}(\operatorname{coh}Y_i)$ denotes the bounded derived category of coher…