21 problems

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…

Two smooth varieties $X$ and $X'$ are K-equivalent if there exist birational morphisms … from a smooth variety $Y$ such that … Let their quantum cohomology rings be considered over…

Let $A$ be the contraction algebra of a rational flopping curve $C$, namely a Jacobi algebra of the type defined in the cited work. For every $d\in\mathbb{N}$, let…

Let $X_1$ and $X_2$ be smooth varieties, and let $\mu:X_1\longrightarrow X_2$ be a birational map resolved by morphisms … such that $f_1^*K_{X_1}$ and $f_2^*K_{X_2}$ are linearly e…

Geometric Realisation Conjecture. Every $f$ whose Jacobi algebra satisfies

Donovan–Wemyss conjecture. The base singularities are isomorphic if and only if the contraction algebras are isomorphic:

Let $\mu:X_1\longrightarrow X_2$ be a birational map of smooth varieties, resolved by morphisms … such that $f_1^*K_{X_1}$ and $f_2^*K_{X_2}$ are linearly equivalent; this is a K-e…

Realisation Problem. If $f\in\mathbb{C}\langle\!\langle x,y\rangle\!\rangle$ satisfies $\operatorname{GKdim}\EuScript{J}\mathrm{ac}(f)\leq 1$, then $\EuScript{J}\mathrm{ac}(f)$ is…

The Classification Conjecture. The complete local cDV singularities are isomorphic if and only if their contraction algebras have equivalent bounded derived categories:

Let $({\widehat{Y}},{\widehat{f}},R)$ and $({\widehat{Y}}^{\prime},{\widehat{f}}^{\prime},R^{\prime})$ be two three-dimensional simple formal flopping contractions, with associated…

Quasi-homogeneity conjecture. The threefold $X$ is quasi-homogeneous if and only if the potential $\Phi$ is quasi-homogeneous. The source introduces this as a conjectural relation…

Polarized Donovan–Wemyss conjecture. The spaces $X$ and $X'$ are formally isomorphic at their singular points if and only if $\Lambda(Y,\Phi)$ and $\Lambda(Y',\Psi)$ are isomorphic…

Let $Y_{-C}\xleftarrow{\,l_{0,-C}\,}Y_0\xrightarrow{\,l_{0,C}\,}Y_C$ be the Schubert flop diagram for a one-dimensional face $C$ in the remaining case described in the paper. The m…

Let $\operatorname{Gr}(r,N)$ be the Grassmannian appearing in the stratified Atiyah flop, and suppose that $2r=N-1$ or $2r=N$. Expected tilting-type equivalence. There exists a til…

Let $f$ be a flop between nonsingular projective $3$-folds, and let … denote the corresponding flop identities for the Donaldson–Thomas invariants and their contributions from the…

Let $K$ be the subgroup of $\uppi_{1}(\mathbb{C}^{n}\backslash \mathcal{H}_{\mathbb{C}})$ generated by the $J$-twists, where $J$ ranges over all subsets of the flopping curves, and…

Let $C_i$ be a crepant curve on a $3$-fold $X$, and suppose that its universal sheaf is perfect and that $X$ has at worst Cohen–Macaulay rational singularities. Mutation theorem fo…

Let $X\to X_{\mathrm{con}}$ be a flopping contraction of $3$-folds, with $X$ having at worst Cohen–Macaulay rational singularities. Let the universal sheaf of the noncommutative de…

Suppose that $Y\to Y_{\mathrm{con}}$ and $Z\to Z_{\mathrm{con}}$ are flopping contractions of a single curve in smooth projective $3$-folds, to points $p$ and $q$ respectively. Ass…

Let $X$ and $X'$ be birational isomorphic varieties. Suppose that, for a resolution … of the birational isomorphism $X\stackrel{fl}{\dashrightarrow}X'$, one has…

Let $W$ be a rank $2N$ vector bundle equipped with a fibrewise non-degenerate symmetric bilinear form, and let $\mathbb{IG}(N,2N)^-$ and $\mathbb{IG}(N,2N)^+$ be the two components…