70 problems

Let $X$) be a compact Kähler manifold of dimension $n$, and let $[\omega_0]$ be any Kähler class on $X$. For a smooth $(1,1)$-form $T$, write $c_1(X)$ for the first Chern class of…

Let $f(z)\in\mathbb{C}[z_0,\dots,z_{d+1}]$ be a generic homogeneous polynomial of degree $d+2$, where $d\geq1$, and let … be the resulting maximally degenerate one-parameter family…

Let $M$ be a compact $4$-manifold equipped with an almost complex structure $J$ and a taming symplectic form $\Omega$. Let $\sigma$ be a smooth volume form on $M$ satisfying … If…

On a hyperhermitian manifold $(M,I,J,K,g)$, let $4\Omega=\omega_J-i\omega_K4$, where $4\omega_J4$ and $4\omega_K4$ are the fundamental forms associated with $J$ and $K$, and let…

Zero mass conjecture. The residual Monge–Ampère mass of $u$ at the origin must vanish whenever $u$ has an isolated singularity and zero Lelong number there. This conjecture concern…

Let $Y_{[3]}^\Gamma$ be a resolved threefold with exceptional divisor $\mathcal{ED}$, and let the Ricci-flat metric and the Kähler metric on $\mathcal{ED}$ be those considered in t…

Rotational symmetry conjecture. Every such solution $u$ on $\mathbb{R}^n$ is rotationally symmetric.

Fekete-array Monge–Ampère conjecture. The measures $\mu_n$ converge weak- to $\mu_K$.

Uniqueness conjecture. A Hermitian Einstein–Weyl structure on a compact complex manifold is unique, up to a holomorphic automorphism.

Affine Calabi conjecture. For every pair of polarizations $(\omega,\nu)$, there is a unique choice for $\phi^\vee\circ\phi^{-1}$ such that the local potentials solve the real Monge…

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 $u$ be a solution of the Monge–Ampère equation in a setting where the associated function $u^2$ may fail to be smooth, and let $g$ denote the Riemannian metric used in the smoo…

Let $Y$ be the smooth part of the limiting base of a maximal degeneration of a Calabi–Yau family, equipped with its Monge–Ampère metric and integral affine structure, and let…

A special exhaustion function on a complex manifold is an exhaustion satisfying a Monge–Ampère equation of the type described in the surrounding context. Burns–Diederich conjecture…

Let $X$ be the underlying projective variety, let $X_\mathrm{na}$ be its non-Archimedean analytification, and let $\mathcal{M}^1_\mathrm{na}$ be the space of finite-energy non-Arch…

Even $L_p$-Minkowski uniqueness conjecture. For $p\in[0,1)$, the Monge–Ampère equation

Let $X\to\mathbb{D}^\times$ be a projective meromorphic degeneration with relatively ample meromorphic line bundle $L$, and let $\phi_t\in\operatorname{CPSH}(L_t)$ be the continuou…

Schneider's conjecture. For $f_1,\ldots,f_n\in\operatorname{Conv}(\Omega)$, the mixed Hessian measure satisfies

Monge–Ampère origin conjecture. The family

Let $\Omega\subset\mathbb C^n$ be a bounded hyperconvex domain, and consider the eigenvalue problem for the complex Monge–Ampère operator on $\Omega$. Lions further conjectured tha…

Let $\mathcal{C}_{\theta}$ be the spherical cap, let $\alpha>\frac{1}{n+2}$ and let $\theta\in(0,\pi)$. Suppose that $h$ is a positive, strictly convex function on…

Zero mass conjecture. The zero mass conjecture claims that $\nu_n(u)=0$ when $\nu_1(u)=0$.

Z-critical analogy conjecture. The Z-critical equation is the correct analogue of Bridgeland stability on the PDE side.

Let $u\in {\mathcal P}(B_1)$ be a plurisubharmonic function whose alternating part has finite frequencies, meaning that near the origin, in a proper complex Hopf-coordinate, it can…

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…