22 problems

Let $R$ be a convex domain in $\mathbb R^k\times\mathbb R^l$, and let $V,W$ be a $(\sigma,\tau)$-type solution of the split Monge–Ampère equation in $R$. A $(\sigma,\tau)$-type sol…

Let $\mathcal{Y}^{p,q}$ be a Gauntlett--Martelli--Sparks--Waldram toric diagram and let $\mathcal{L}$ be a lattice segment in $\mathbb{R}^2$ such that … is a toric diagram. Irregul…

Let $X\to \mathbb{D}^*_t$ be a polarised meromorphic degeneration of $n$-dimensional Calabi–Yau manifolds, with relatively ample line bundle $L$, and let $X_K^{an}$ be the associat…

Compactification conjecture. For any $\kappa_1>\lambda$ and $\kappa_2>\mu$, $(X,I,\omega,\Omega)$ can be compactified complex analytically to a weak Fano manifold. Furthermore, the…

Let $l,m,k$ be parameters and let $\varphi:(\mathbb R_+)^l\times \mathbb R^{m}\to \mathbb R_{\geq 0}$ solve … in $(\mathbb R_+)^l\times \mathbb R^{m}$, with … on…

Let $(M,g,J)$ be a complete Calabi–Yau manifold with Euclidean volume growth, let $C$ be its unique asymptotic Calabi–Yau cone, and let $W$ be the central fibre of the associated…

Fix a polystable negative valuation $v$ on $X$. Let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics on $X$ with $v=v_\omega$, let $\mathcal C_v$ be the corresponding w…

Let $X$ be a normal affine variety, let $v$ be a fixed polystable negative valuation on $X$, and let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics $\omega$ on $X$ sa…

Let $X$ be a complete Calabi–Yau manifold in the setting of Theorem … should hold without the quadratic curvature decay assumption. This is the expected extension of the paper's ma…

Let $\mathbf{C}^3$ carry a Calabi–Yau metric whose tangent cone at infinity is $\mathbf{C} \times A_2$. Identify metrics that differ by biholomorphism and scaling. Székelyhidi's pa…

Tangent-cone conjecture. At the singular point $z\in Z$, the tangent cone to $(Z,d)$ is isometric to the blowdown of the zero section in $\mathcal{O}_{\mathbb{P}(V)}(-1)$, equipped…

Let $1\leqslant k<p-1$, and let $\overline{Y}$ be the partial resolution obtained by blowing up $k$ of the lines $x=z-\zeta^jw=0$. Its isolated singularity is modeled by … Let $d$…

Let $n>4$. A $\partial\bar\partial$-exact Calabi–Yau manifold is a Calabi–Yau manifold of complex dimension $n$ whose Calabi–Yau metric is $\partial\bar\partial$-exact. Suppose its…

Let $\omega$ be a Calabi–Yau metric on $\mathbf{C}^3$, and suppose its tangent cone at infinity is $\mathbf{C}\times A_2$. Metrics are considered up to scaling and isometry. One-pa…

Let $n\geq 3$ and let $f\in C^{\infty}(\mathbb C^n)$ be a real strictly plurisubharmonic function satisfying … The associated Kähler metric … is complete and has Euclidean volume g…

Let $(X,D)$ be a pair with curve singularity locus $C$, let $p\in C$, and let $g$ be a weak conical Kähler–Einstein metric on $(X,D)$ with prescribed cone angle parameters. The met…

Tian–Yau convergence conjecture. As $\beta\to0$, these Kähler–Einstein cone metrics converge to the complete Calabi–Yau metrics on $X\setminus D$ constructed by Tian and Yau.

Let $\mathbf{C}^n$ be complex affine space, and consider complete Calabi–Yau metrics on it with maximal volume growth. Tian's uniqueness conjecture. The flat metric is the unique C…

Tian's flatness conjecture. Every Calabi-Yau metric of Euclidean volume growth on $\mathbb{C}^{n}$ is flat.

Determinantal representation conjecture. The unique smooth normalized solution $\varphi$ may be represented as a limit in $L^{1}(X)$:

Zero-angle limit conjecture. As $\beta$ tends to zero, $(S,C,\omega_\beta)$ converges in an appropriate sense to a generalized Kähler–Einstein metric $\omega_\infty$ on…

Let $M$ be a Fano manifold and let $D$ be a smooth anti-canonical divisor. By the cited result, there is a complete Calabi–Yau metric on $M\setminus D$. For $0<\beta\leq 1$, let a…