18 problems

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…

Let $G$ be a simple simply connected complex algebraic group, let $P\subset G$ be a parabolic subgroup, and let $V^\vee\subset G^\vee/P^\vee$ be the top projected open Richardson s…

Let $X$ and $X'$ be connected indecomposable homogeneous projective varieties. For such a variety, write $\operatorname{minss\ rank} X$ and $\operatorname{maxss\ rank} X$ for its m…

Let $X=G/P$ and $X'=G'/P'$ be homogeneous projective varieties, where $G$ and $G'$ are connected simple algebraic groups and $P$ and $P'$ are parabolic subgroups. For a realization…

Positivity conjecture for the signed Euler characteristic. One has

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…

Let $X$ be an adjoint variety of type $A_n$, and let $Y\subset X$ be a general hyperplane section. Let ${\rm QH}(X)_{\rm can}$ and ${\rm QH}(Y)_{\rm can}$ denote the restrictions o…

Let $X$ be an adjoint or quasi-minuscule variety as above, and let $Y\subset X$ be a general linear section. Write ${\rm BQH}(Y)$ and ${\rm QH}(Y)$ for its small and big quantum co…

Let $X$ be an irreducible homogeneous quasi-affine variety with … and with $\mathbb{K}[X]^{\times}=\mathbb{K}^{\times}$. Here $\theta(X)$ denotes the transitivity degree of the aut…

Let $X$ be the coadjoint variety of a simple complex algebraic group $G$. Let $D_s(G)$ denote the short-roots subdiagram of the Dynkin diagram of $G$, which is the entire Dynkin di…

Let $X$ be a Fano manifold with a contact structure $D \subset T_X$, and let $L=T_X/D$ be the quotient line bundle. Assume that $\operatorname{Pic} X \cong \mathbb{Z}\cdot L$. Fano…

Let $G/P$ be a homogeneous roof, and $\mathcal{Z}$ a homogeneous roof bundle of type $G/P$ with projective bundle structures … Given a general section…

Exceptional collection conjecture. There is a full exceptional collection of vector bundles on $G/P$.

A quasihomogeneous variety is a variety with a group action having an open orbit, while a homogeneous variety has a transitive group action. Homogeneity conjecture. All quasihomoge…

Exceptional collection conjecture. There exists a full exceptional collection of vector bundles in $\mathsf{D}^{\mathrm{b}}(X)$.

Contact Fano homogeneity conjecture. A contact Fano manifold is homogeneous.

Let $X$ be a cominuscule variety, and let $d_{\rm \max}$ denote the minimal integer such that a curve of degree $d_{\rm \max}$ passes through any two points of $X$. For…

Let $X=\operatorname{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P(\mathbb C^3\otimes\mathbb C^3)$. Let $Z^4\subset\mathbb P V$ be a variety, and let $II_{Z,z}$ denote its sec…