40 problems

Let $\uppi\colon \EuScript{X}\to\mathop{\mathrm{Spec}}\EuScript{R}$ be a flopping contraction, and let $\Lambda_{\mathrm{con}}(\uppi)$ denote its contraction algebra, the stable en…

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…

Calabi-Yau transition conjecture. Any two Calabi–Yau 3-folds can be connected to each other by a sequence of flops or transitions and their inverses.

Let $X$ and $X'$ be $K$-equivalent models with Kähler cones $\mathcal{K}_X$ and $\mathcal{K}_{X'}$. Let $\Phi^X_{ijk}$ and $\Phi^{X'}_{ijk}$ be their total quantum-cohomology serie…

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 $X$ and $Y$ be algebraic varieties related by a birational transformation called a flop. The bounded derived categories of coherent sheaves on $X$ and $Y$ are the categories un…

Let $X_{\zeta_{+}}$ be the geometric phase of the GLSM, let $D_{0}$ be the indicated divisor, and let $M_{0}$ and $\pi_{\zeta_{+}}$ denote the monodromy and projection functors use…

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 $X$ be the threefold considered in the paper, let $R$ be its base ring, and let $\operatorname{\mathbf{D}^{0}}X$ denote the relevant triangulated subcategory of…

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…

DK conjecture. If $X_+$ and $X_-$ are K-equivalent, then they are D-equivalent. This conjecture proposes a parallel between derived categories and birational geometry. It is attrib…

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

Let $X$ and $Y$ be smooth varieties related by flops. Orlov–Bondal's flop conjecture. The varieties $X$ and $Y$ are Fourier–Mukai partners. The source recalls this conjecture in th…

Let $X$ and $Y$ be smooth varieties related by a sequence of flops. Their quantum potentials are functions defined from the corresponding Gromov–Witten invariants, and the Novikov…

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:

Conjecture on the two flopped Kähler cones. These two neighbouring regions are the Kähler cones of $X'^{(1)}$ and $X”^{(1)}$.

Conjecture on the flopped Kähler cone. The cone on the other side of the boundary separating it from $\mathcal{K}(X)$ is the Kähler cone of $X'$.

Conjecture on the flopped nef cone. The cone adjacent to $\mathcal{K}(X)$ across the shared vertical boundary, shown as the green region, is the nef cone of $X'$.

Donovan–Wemyss conjecture. Any potential algebra of a rose quiver, and more generally any potential algebra arising from a quiver in the relevant wider list, can be realised as a c…

Strong-rationality conjecture. The BPS invariant $\mathrm{BPS}_\updelta$ depends only on $\operatorname{\mathrm{rk}} \updelta$. This is presented as a refined version of the strong…

Let a flop be presented by a wall-crossing with a derived $Q$-construction producing a kernel for the associated window functor. A Fourier–Mukai equivalence is an equivalence induc…

Let a $G$-model be an elliptic fibration with triple $(\mathbf{R},G,\mathfrak{g})$, and let $I(\mathfrak{g},\mathbf{R})$ be the hyperplane arrangement associated with the Lie algeb…

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…

Let $R$ be the hypersurface singularity underlying a simple flopping contraction, and let $\Gamma$ be the associated exact 3-Calabi–Yau algebra. Its canonical class $[w]$ defines q…