21 problems

Let $X$ be an integral quasi-excellent scheme. A desingularization means a proper birational morphism from a regular scheme to $X$. Grothendieck's conjecture. Every integral quasi-…

Let $L \in \text{\bbold Q}[z,E]$ have leading and trailing coefficients $a_d(z)$ and $a_0(d)$. Let $N_1$ be an integer larger than any integer root of $a_d(z-d)a_0(z)$, and let…

Kiem–Li's conjecture. The intermediate variety $\mathbf{K}_\sigma$ is isomorphic to the moduli space of Hecke cycles:

Let $K_{\sigma}$ be the intermediate variety in the factorization of the morphism from the Kirwan desingularization $K$ to the desingularization $S$. Narasimhan–Ramanan desingulari…

Let $K$ be the Kirwan desingularization and let $S$ be the desingularization by parabolic bundles described in the paper. Suppose that the morphism $K\to S$ factors through an inte…

Let $N_0$ be the compact singular special Lagrangian $3$-fold from Condition … . Suppose $\chi=0$, and that $(M,J_0,\omega_0,\Omega_0)$ extends to a smooth family…

Radial-profile conjecture. After a suitable scaling, the limit of the maximizers should be a radially decreasing function with respect to some point.

Let $X$ be an algebraic or analytic variety, and let $X^{\mathrm{nc}}$ denote its normal-crossings locus. A smooth blowing-up is a blowing-up with smooth centre, and it is admissib…

Let $X$ be an algebraic or analytic variety. A smooth blowing-up is a blowing-up with smooth centre, and it is admissible when it satisfies the local normal-crossings and Hilbert–S…

Let $(\overline{X},\varphi_{CS})$ be a nearly $G_2$ conifold with a singularity at $x_0$ modelled on the $G_2$ cone $(C=C(L),\varphi_C)$. An obstruction conjecture for nearly $G_2$…

For sufficiently large $n\in\mathbb{N}$, consider a complete, embedded, noncompact self-shrinker $\Phi_g\subset\mathbb{R}^3$ for mean curvature flow, where $g=2n-1$ denotes its gen…

Let $L_\infty$ be the limiting linear map appearing in the desingularization and obstruction setup, where the theorem was originally stated under the assumption…

Let $(M_o^4,\mathbf{g}_o)$ be a Ricci-flat orbifold, and suppose it can be desingularized in the Gromov–Hausdorff sense by Einstein metrics $(M^4,\mathbf{g}_n)_{n\in\mathbb{N}^*}$…

The singularity is modeled on $\mathbb{R}^4/\mathbb{Z}_2$, whose minimal resolution has the topology $T^*\mathbb{S}^2$ of the Eguchi–Hanson metric. Unique-topology conjecture. This…

Let $(\psi_t)_{t\in(-T,T)}$ be a $1$-parameter family of closed, tamed, definite $4$-forms on $Y$. Let $\hat P$ be an unobstructed singular associative submanifold in $(Y,\psi_0)$…

Intrinsic-flat desingularization conjecture. The desingularizations constructed in the paper should give rise to a $d_{\mathcal F}$-Cauchy sequence of smooth closed manifolds with…

Strict Bott-Samelson desingularization conjecture. If $w\ngeq v$ for every $v\in\Gamma_n$, then $X^w$ can be desingularized using a sequence of strict Bott-Samelson resolutions.

Bistable Grassmannian conjecture. The bistable Grassmannian equals the whole Grassmannian:

Let $M$ be a representation of a Dynkin quiver and let ${\bf e}$ be a dimension vector. For a generic subrepresentation type $[N]$ of $M$ in dimension ${\bf e}$, write…

Let $(X,B)$ be a $B$-variety, and let a $B$-strong desingularization mean a desingularization compatible with the $B$-scheme structure and functorial in the relevant exact regular…

A $B$-scheme is a scheme $X$ equipped with a boundary $B$; here $(X,B)$ is assumed to be qe and generically reduced, and the scheme has characteristic zero. Write…