18 problems

Let $G$ be a connected reductive algebraic group, let $\operatorname{Gr}=G(\mathcal K)/G(\mathcal O)$ be its affine Grassmannian, and let $\lambda_1^\vee,\ldots,\lambda_k^\vee$ be…

Weak del Pezzo characterization. A smooth projective surface has a $2$-tilting bundle if and only if it is weak del Pezzo.

Let $Q$, ${\mathbf{d}}$, and $\theta$ satisfy the assumptions of the theorem on partial tilting bundles, and let $\mathcal{U}$ be the universal representation on…

Let $X$ be the variety underlying the symplectic resolution ${\mathfrak{X}}$, let $s$ be a generic slope, and let $\{{\mathcal{E}}^K_{{\mathfrak{X}},s}(p)\}_{p\in X^H}$ be the $K$-…

Tseu's equiconstitution conjecture. For all $n$, the tilting generator $\mathcal{T}_n$ is equiconstituted to $\mathcal{Z}$ up to tensor product with a line bundle.

Equiconstitution conjecture. The tilting generator $\mathcal{Z}$ is equiconstituted with any Toda–Uehara tilting generator.

Let $X$ be a nice symplectic resolution. Consider an invariant basis for the equivariant K-theory of $X$ under a bar involution. Hikita's conjecture. Such an invariant basis should…

Let $X$ be a globally $F$-split variety. Write $D_X^{(e)}$ for the sheaf of $O_X^{p^e}$-linear endomorphisms of $\mathcal O_X$, so that…

Let $X$ be a smooth projective variety with nontrivial first-order deformations, meaning $h^1(T_X)>0$, where $T_X$ is the tangent sheaf. The absolute Frobenius pushforward is denot…

Quiver flag variety reconstruction conjecture. The morphism

General Grassmannian reconstruction conjecture. For any $1\leq r<n$, the morphism

Let $\operatorname{Gr}(r,N)$ be the Grassmannian appearing in the stratified Atiyah flop, and suppose that $2r=N-1$ or $2r=N$. Expected tilting-type equivalence. There exists a til…

Let $S$ be a smooth projective algebraic surface over $k$. A tilting bundle on $S$ is a vector bundle that is a tilting object in the derived category of coherent sheaves on $S$. T…

Twisted-homogeneous-variety tilting conjecture. The category $D^b\operatorname{Coh}(X)$ admits a tilting object.

Surface tilting-bundle conjecture. Smooth projective algebraic surfaces have a tilting bundle if and only if they are rational.

Tilting-bundle rationality conjecture. $X$ has a tilting bundle if and only if $X$ is rational.

Let $X$ be a smooth projective surface. A tilting bundle on $X$ is a vector bundle $\cE$ such that $\cExt^i(\cE,\cE)=0$ for $i>0$, and the zero sheaf is the only sheaf $\cF$ satisf…

Let $X$ be a smooth toric Fano variety. A sequence of basepoint-free line bundles is given by … Assume that this sequence has bound quiver of sections $(Q,\varrho)$, and let…