97 problems

Let $X$ be a Fano manifold whose big quantum cohomology is semisimple, and suppose that $\mathcal{D}^b_{\rm coh}(X)$ admits a full exceptional collection. Let $U\subset B_{ss}$ be…

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

A rational homogeneous variety is a variety of the form $G/P$, where $G$ is a connected reductive algebraic group and $P$ is a parabolic subgroup. Its bounded derived category is d…

Let $\mathrm{G}$ be a semisimple algebraic group and let $\mathrm{P}\subset \mathrm{G}$ be a parabolic subgroup. Let ${\mathbf D^{\mathrm{b}}}(\mathrm{G}/\mathrm{P})$ denote the bo…

Kuznetsov–Polishchuk's vector bundle conjecture. The exceptional collections constructed by Kuznetsov and Polishchuk consist of vector bundles.

Let $X$ be a rational surface. A standard augmentation is a toric system obtained from a toric system on a Hirzebruch surface by successive blow-ups and the corresponding augmentat…

Novaković's conjecture. $D^b(X)$ does not admit a full strongly exceptional collection.

Catanese's conjecture. The variety $G/P$ should possess a full strongly exceptional poset together with a bijection between the elements of the poset and the Schubert varieties of…

Let $G$ be a semisimple algebraic group and let $G/P$ be a projective homogeneous space of $G$. A collection of vector bundles on $G/P$ is exceptional if its derived endomorphism c…

Let $f$ be a weighted homogeneous polynomial associated with a regular weight system, and let $D^b_{\mathbb Z}(\mathcal A_f)$ be the triangulated category of graded matrix factoriz…

Let $\operatorname{IGrass}(n,V)$ be the Lagrangian Grassmannian with $\dim V=2n$, and let $\mathcal{R}$ be its tautological bundle. The defining exact sequence is … A relation betw…

Let $Q$ be the quadric under discussion, with spinor bundles $\Sigma^+$ and $\Sigma^-$. Consider the strong complete exceptional sequence … A braid-orbit conjecture for the quadric…

Let $X=G/P$ be a rational homogeneous variety, let $W^P$ denote the relevant set of Schubert representatives, and let $n=|W^P|$, the number of Schubert varieties in $X$. A strong c…

Let $\mathbb{P}^n$ be projective space, and let $D^b(\operatorname{Coh}\,\mathbb{P}^n)$ be its bounded derived category of coherent sheaves. An exceptional object is an object $E$…

Let $X$ be a projective variety. Its $(p,p)$-part of quantum cohomology is the component represented by cohomology classes of Hodge type $(p,p)$, and let $D^b_{\rm coh}(X)$ denote…

Strengthened Dubrovin conjecture. The space $\bigoplus_{p\in\mathbb{Z}} H^{p,p}(V)$, with quantum multiplication, is semisimple if and only if $V$ is Fano and…

Mutation-generation conjecture. All exceptional bundles and sheaves on $X$ can be obtained by mutations of the helix ${\cal R}$. In particular, there exists a mutated helix

Large-volume monodromy exceptional-helix conjecture. The large-volume monodromy action on ${\cal O}$ in the ambient variety $X$ produces an exceptional collection forming the found…

Let $X$ be a smooth projective variety with $\operatorname{rk} K_0(X)=\dim X+1$, and let ${\cal D}^b_{\operatorname{coh}}(X)$ be the bounded derived category of coherent sheaves on…

Exceptional-collection and Landau–Ginzburg correspondence conjecture. The maximal length of an exceptional collection on $X$ equals the number of fibers with ordinary double points…

Monotonicity and vanishing conjecture. The function $f(d)$ is strictly monotone decreasing for $d>1$, and

Homological mirror symmetry exceptional-collection prediction. The bounded derived category $\mathcal{D}^b(X)$ has a full exceptional collection consisting of $k$ objects.

Let $Q$ be a superquiver and let $\mathcal C$ be a hereditary category. An irreducible representation of $Q$ in $\mathcal C$ is a representation whose arrow morphisms do not lie in…

Let $G$ be an algebraic group, let $P\supset B$ be a parabolic subgroup, and let $\mathsf{k}[(G/P)(\mathbb{F}_q)]$ be the permutation representation over $\mathsf{k}$. Let…

Let $X$ be the canonical line bundle of $\mathbb{P}^3$, with projection $\pi:X\to\mathbb{P}^3$. Let $\beta\in\mathbb{R}$ and let $\mathcal{E}$ be an exceptional locally free sheaf…