36 problems

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 $\mathrm{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P^8$ be the Segre variety. Griffiths–Harris rigidity conjecture. The Segre variety can be characterized by a second-or…

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

Let $X$ and $X'$ be two connected indecomposable homogeneous projective varieties. For a homogeneous variety, define its minimum (respectively, maximum) semisimple stabilizer rank…

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 $X$ be a Fano contact manifold with $b_2(X)=1$. A Fano contact manifold conjecture asserts that $X$ is a homogeneous variety which is the unique closed orbit in the projectiviz…

Let $X$ be a Fano manifold, meaning a smooth projective variety whose anticanonical divisor $-K_X$ is ample. Its tangent bundle $T_X$ is nef if the tautological line bundle on the…

Let $X\subset\mathbb{P}^N$ be a maximal defective QEL-manifold, meaning that $X$ is not a hyperplane section of a non-degenerate manifold in $\mathbb{P}^{N+1}$, and suppose that…

Let $V$ be a Fano manifold of Picard number one, let ${\mathcal O}_V(1)$ be an ample generator of ${\rm Pic}(V)$, and suppose that $\dim V=2r+1$ is odd. Let…

Buch–Thomsen–Lauritzen–Mehta conjecture. Partial flag varieties that are not products of projective spaces do not satisfy Bott vanishing.

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 $X$ be a Fano contact manifold. Lebrun–Salamon conjecture. The manifold $X$ is an adjoint variety, so adjoint varieties are the only examples of Fano contact manifolds. This co…

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…

Coskun–Costa–Huizenga–Miró-Roig–Woolf's conjecture. The variety $Fl(k_{1},k_{2};\mathbb{C}^{n})$ does not admit an initialized irreducible homogeneous Ulrich bundle with respect to…

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

Full exceptional collection conjecture. The category $D^b\operatorname{Coh}(X)$ admits a full exceptional collection.

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 $Z\subsetneq \mathbb{P}^N$ be an irreducible and linearly non-degenerate projective variety. If the dual variety $\check{Z}\subset \check{\mathbb{P}}^N$ is a hypersurface with…

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…

Rational homogeneity conjecture. Then $X$ is a rational homogeneous variety.

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…