59 problems

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…

Let $X$ be the coadjoint variety of a simple algebraic group $\mathrm G$ over an algebraically closed field of characteristic zero. Let $\mathrm T(\mathrm G)$ be the Dynkin diagram…

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…

King's conjecture. Every smooth toric variety possesses a full strong exceptional collection of line bundles.

Let $X$ be a smooth projective surface. A full exceptional collection in $\mathrm{D}(X)$ is an exceptional collection that generates the derived category; in the line-bundle case,…

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

Let $V$ be a $2n$-dimensional symplectic vector space, let ${\mathsf{IGr}}(k,V)$ be the isotropic Grassmannian, and let ${\mathcal{A}}_t$ be the admissible subcategories defined by…

Transitivity conjecture. The group spanned by mutations of exceptional bases and the isometries of $K_0(\mathbb{P}_n)$ acts transitively on the set of exceptional bases of…

Let $X$ be a smooth complete toric variety. Suppose that $X$ is an $F$-fibration over $Y$, where $Y$ and $F$ are smooth toric varieties, and both $Y$ and $F$ have a full strongly e…

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…

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

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…

Let $X$ be a Fano variety of complex dimension $D$ with odd-vanishing cohomology, and let $F_0(\textbf{v})$ be a power series with convergence domain $\Omega$. Let…

Let $\mathbf G/\mathbf B$ be the generalized flag variety, let $W$ be its Weyl group with longest element $w_0$ and length function $\ell$, and let $\mathcal X_v$ be the objects in…

Let ${\mathsf{OGr}}(k,V)$ be an orthogonal Grassmannian, and let the collections in the source be the semiorthogonal collections constructed in Theorems,, and. The orthogonal Grass…

Let $V$ be a vector space of dimension $2n+1$ equipped with a skew-symmetric form of maximal rank $2n$, and let odd symplectic flag varieties be defined analogously to the odd isot…

Let ${\mathsf{Gr}}(k,V)$ be a Grassmannian, let $\mathrm{Y}^{\mathrm{mu}}_{k,n-k}$ denote the lexicographically minimal upper-triangular representatives of the cyclic orbits of You…

Let $\mathbf{G}$ be a simple algebraic group over an algebraically closed field $\mathsf{k}$ of characteristic $0$, and let $\mathbf{P}\subset \mathbf{G}$ be a maximal parabolic su…

Full exceptional collection conjecture. There is a full exceptional collection in

Reflection-vector conjecture. (a) If the quantum cohomology of $X$ is convergent and semisimple, then $\Psi_q(\mathcal{E})$ is a reflection vector for every exceptional object…

Zonotope generation conjecture. Under these assumptions,

Let $X$ be an adjoint variety that is not quasi-minuscule, let $Y\subset X$ be a general hyperplane section, let $c_1(X)$ and $c_1(Y)$ denote their first Chern indices, and let…

Torus and rationality conjecture. There exists a subcomplex $Y \subseteq X_A$ homotopic to $\mathbb{T}^n$. Furthermore, $\mathcal{X}$ is rational, with local systems on $Y$ corresp…

Let $X$ be the coadjoint variety of a simple algebraic group $\mathrm{G}$ over an algebraically closed field of characteristic zero. Let $\operatorname{Aut}(X)$ denote its automorp…