6 problems

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…

Torus conjecture. The complex $X_{A_{\mathcal{X}}}$ is homotopic to a real torus $\mathbb{T}$.

Maximal-length fullness conjecture. Every strong exceptional collection of line bundles of maximal length on a Fano toric Deligne–Mumford stack is a full strong exceptional collect…

The dimension-three H-triviality conjecture. There are infinitely many $\mathrm{H}$-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists…

Let $\Delta$ be a convex lattice polytope in $\bR^n$ containing the origin in its interior. Let $X$ be the toric Fano stack associated with $\Delta$, let $W:(\bC^*)^n\to\bC$ be a L…

Let $\Delta$ be a convex lattice polytope in $\bR^n$ containing the origin in its interior. Let $X$ be the associated toric Fano stack, let $\bT$ be the $n$-dimensional torus actin…