388 problems

Does every compact complex structure on the smooth manifold $T^2\times S^4$ split biholomorphically as a product of complex manifolds? Equivalently, for every integrable complex st…

Let $(M,I,J,K,g_0)$ be a compact HKT manifold, let $\Omega_0$ be the HKT form of the initial metric, and let $T_{\max}$ denote the maximal existence time of the unnormalized HKT-Ri…

Let $X$ be a compact complex manifold. If $X$ admits a Hermitian metric $h$ whose Chern holomorphic sectional curvature satisfies $H_h(v)\le 0$ for every nonzero $v\in T^{1,0}_xX$…

Let $Y$ be a projective Calabi–Yau variety with only ordinary double points, and let $X\to Y$ be a crepant resolution or let $Y$ occur as the central fiber of a smoothing. Then the…

Let $\pi\colon X\to S$ be a smooth projective family, let $s_0\in S$ and write $X_{s_0}=\pi^{-1}(s_0)$. Given an effective line bundle $L$ on $X$ and a prescribed singular Hermitia…

For every smooth complex projective variety $X$ and every slope-stable ample holomorphic vector bundle $E\to X$, does there exist a smooth strongly pseudoconvex Finsler metric on…

Let $f\colon X\to \Delta$ be a proper Kähler family over a disk, with central fiber $X_0$ a simple-normal-crossing fiber, and suppose that the relative canonical bundle…

For every smooth projective complex manifold $X$, is $X$ rationally connected if and only if its holomorphic tangent bundle $T_X$ admits a smooth uniformly RC-positive Hermitian me…

Let $K$ be a complete non-archimedean field, let $X$ be a normal projective variety over $K$, and let $L$ be a line bundle on $X$. For every continuous metric $\varphi$ on…

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $\alpha,\beta$ be Kähler classes on $X$. Set $\mu=\frac{n\,\alpha^{n-1}\cdot\beta}{\alpha^n}$. The Datar–Mete…

Let $n\geq 2$, let $\mathbb{B}^n$ be the complex unit ball with its canonical Kähler–Einstein metric $g_{\mathbb{B}}$, and let $\Omega_1,\ldots,\Omega_m$ be irreducible bounded sym…

Every projectively induced Kähler–Einstein metric on a smooth compact toric manifold is, up to automorphisms, a product of Fubini–Study metrics on projective spaces with matched mu…

For fixed singular or big-cohomology data for which twisted Kähler–Einstein currents are defined, if $T_1$ and $T_2$ are two twisted Kähler–Einstein currents satisfying the same tw…

For every complete, noncompact Kähler manifold $(M^n,g)$ with positive holomorphic bisectional curvature, i.e. $R(X,\overline{X},Y,\overline{Y})>0$ for all points $p\in M$ and all…

For every smooth projective variety $X$ over $\mathbb{C}$, the tangent bundle $T_X$ is ample if and only if $X$ is isomorphic to projective space $\mathbb{P}^{\dim X}$, that is,…

Let $X$ be a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma, and let $E$ be a numerically flat holomorphic vector bundle of rank $2$ on $X$. Must…

Let $C\subset \mathbb{P}^{2}$ be a smooth cubic curve, let $p_{1},\ldots,p_{9}\in C$ be distinct points, and let…

Let $X$ be a connected smooth complex projective surface, let $L$ be a line bundle on $X$, and let $\sigma$ be a divisorial Bridgeland stability condition. If $L$, or the appropria…

For every integer $n\ge 1$, if a Reinhardt domain $\Omega\subset\mathbb{C}^n$ has smooth boundary and its Bergman metric is complete and Einstein, then $\Omega$ is biholomorphic to…

Let $(X,x)$ be an $n$-dimensional isolated log terminal singularity. If $\alpha_x^{\mathrm{an}}(X)$ and $\alpha_x^{\mathrm{alg}}(X)$ denote the two local alpha invariants introduce…

Let $X$ be a smooth projective complex variety and let $\mathcal{F}$ be a foliation on $X$. Assume that, for almost all primes $p$, the reduction of $\mathcal{F}$ modulo $p$ is $p$…

For every admissible lattice point $(\chi,c)$, does there exist a closed, simply connected, spin symplectic $4$-manifold $X$ such that $(\chi_h(X),c_1^2(X))=(\chi,c)$?

Let $Y$ be a compact Kähler manifold, let $p\in Y$, and let $\pi:M=\operatorname{Bl}_pY\to Y$ be the blow-up at $p$. Suppose that a Kähler–Ricci flow on $M$ with initial Kähler cla…

For every real parameter $t>1$, does there exist a real-analytic CR-embedding $F_t:M_t^3\hookrightarrow\mathbb{C}^3$ such that $F_t(M_t^3)$ is a compact real-analytic strongly pseu…

Let $X$ be a compact complex manifold of complex dimension $n$. If $X$ admits a balanced Hermitian metric with fundamental form $\omega_{B}$ satisfying $d(\omega_{B}^{n-1})=0$ and…