14 problems

Let $(X_t,\omega_{CY,t})$ be a polarised degeneration family of $n$-dimensional Calabi–Yau manifolds near the large complex structure limit, and let $\mu_t$ denote the normalised C…

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…

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…

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…