8 problems

Let $V$ be a smooth Fano manifold of complex dimension $n-1$, and let $I(V)\in\mathbb{N}$ be its Fano index. GMSY's Fano inequality conjecture. … with equality if and only if…

Let $\mathcal{X}$ be a toric Calabi–Yau $3$-fold with no factors of $\mathbb{C}$, let $\zeta_R$ be the Reeb symmetry, and let $\eta_h$ be the test symmetry associated with a genera…

Let $X(\Delta_{n-1})$ be the toric variety associated with a reflexive toric diagram $\Delta_{n-1}$, and let $\widetilde{X(\Delta_{n-1})}$ denote the relevant resolution. Let…

Sasaki–Einstein existence conjecture. For each integer $k\leq 8$ and $n\geq 2$, the Smale $5$-manifold

Let ${\cal X}$ be a reflexive toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, let $V(b_i^{*};Y)$ be the minimum volume, and let…

Let ${\cal X}$ be a toric Calabi–Yau $n$-fold with toric diagram $\Delta_{n-1}$, let $Y$ be its Sasaki–Einstein base, and let $V(b_i^{*};Y)$ denote the minimum volume. Let…

Sasaki–Einstein circle-bundle conjecture. If there exists a Kähler–Ricci soliton on $M$, then the $S^1$-bundle associated to $K_M$ admits a Sasaki–Einstein metric with a suitable c…

Let $Z$ be a smooth Fano manifold of complex dimension $n-1$, with Fano index $I(Z)mathrel{\in}\mathbb{Z}_{>0}$. Fano volume inequality conjecture. … Equality should hold if and on…