10 problems

Let $X$ be a compact Kähler manifold and let $lpha$ be a nef and big class on $X$. Assume that $D=E_{nK}(lpha)$ is a simple normal-crossings divisor, let $s_D$ be a defining sectio…

Let $X$ be a compact Kähler manifold and let $lpha$ be a nef and big class on $X$. Let $T=\theta+dd^c\boldsymbol{varphi}$ be an adapted closed positive $(1,1)$-current with minimal…

Let $\mathfrak{p}_\zeta:\mathbb{P}\to\mathbb{M}$ be the universal bundle with connection $\mathbb{A}$, and let $p_\zeta:P_\zeta\to M_\zeta$ denote its restriction to a fibre. The f…

Let $M^{\bullet,\bullet}=\bigwedge^{\bullet,\bullet}V^*\otimes\operatorname{End}(R)$, let $\alpha\in\mathcal{N}_0\cap\mu^{-1}(\zeta)$, and let…

Let $\overline{E}$ be a holomorphic vector bundle on $\overline{X}$ such that $\overline{E}|_D$ is $c_1(N_D)$-polystable and $\overline{E}$ is $(c_1(D),[\omega_0])$-stable. Let…

Yau's conjecture. There is a solution of the full Hull–Strominger system on $X$.

Let $\Sigma$ be a compact Riemann surface, and consider the dimensionally reduced twisted Kapustin–Witten equations on $\Sigma\times\mathbb R^+_y$ for a connection $A$, a…

Let $B$ be a ball in $\mathbb C^n$ and let $\mathcal E$ be a reflexive sheaf over $B$ with an isolated singularity at $0$. For an admissible Hermitian–Yang–Mills connection $A$ on…

A reflexive sheaf on a ball with an isolated singularity at the origin may have an algebraic tangent cone, meaning a torsion-free sheaf on the associated projective space. Tangent-…

Kobayashi's conjecture. These three conditions should be equivalent in general, independently of whether $X$ is algebraic: