101 problems

Let $X$ be a compact complex manifold. A balanced metric is a Hermitian metric whose fundamental form $omega$ satisfies $domega^{n-1}=0$, and a pluriclosed metric is a Hermitian me…

Let $(X^n,\omega)$ be a compact Hermitian manifold and let $\alpha$ be a closed real $(1,1)$-form whose Bott–Chern class $[\alpha]$ is nef. A class is big if it contains a current…

Gauduchon's conjecture for balanced metrics. One can always find a balanced metric in the prescribed Bott–Chern cohomology class whose first Chern–Ricci form coincides with $\Psi$.

Let $M$ be a compact Hermitian manifold with Hermitian metric $g$, and let its Chern or Levi-Civita connection have holomorphic sectional curvature equal to a constant $c$. Constan…

Let $(M,g)$ be a Hermitian gravitational instanton with either ALF or AF asymptotics. Aksteiner–Andersson toricity conjecture. Every such gravitational instanton is toric. The conj…

Chen–Nie conjecture. Let $(M^n,g)$ be a compact Hermitian manifold. Assume that the holomorphic section curvature of $D^r_s$ is a constant $c$. If $c\neq 0$, then $g$ must be Kähle…

Bismut space-form conjecture. If a compact Hermitian manifold $(M^n,g)$ has constant Bismut holomorphic sectional curvature $c$ and $c\neq 0$, then $g$ must be Kähler.

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. For a $(1,0)$-vector $X$, let $\mathcal{C}_{\alpha,\beta}(X)$ denote the mixed curvature, formed from the relevant Ch…

Let $(X^n,\omega)$ be a compact Hermitian manifold and let $\alpha$ be a closed real $(1,1)$-form whose Bott–Chern class $[\alpha]$ is nef. Fix points $x_1,\ldots,x_m\in X$ and pos…

Let $(M^n,g)$ be a compact Hermitian manifold with $n\geq 2$. Let $H^s$ denote the holomorphic sectional curvature of the Strominger connection. Strominger curvature conjecture. If…

Let $X$ be a Hopf surface, and call a Hermitian metric $\omega$ Levi-Civita Ricci-flat when its Levi-Civita Ricci curvature satisfies $\mathfrak{Ric}(\omega)=0$. Existence conjectu…

GIP conjecture. Any compact complex manifold with vanishing first Chern class admits a Hermitian metric and a Hermitian connection with totally skew-symmetric torsion and restricte…

A compact complex manifold $M$ of complex dimension $n$ has vanishing first Chern class, $c_1(M)=0$. A Hermitian connection is a connection preserving the Hermitian structure; it h…

Let $M$ be a compact complex manifold of real dimension $2n$ with vanishing first Chern class. A hermitian structure on $M$ is a Riemannian metric compatible with its complex struc…

Let $X$ be a compact complex manifold, let $K_X$ denote its canonical bundle, and let holomorphic sectional curvature refer to the curvature of a Hermitian metric on $X$. A Hermiti…

Let $X$ be a compact complex manifold. A Hermitian metric on $X$ is balanced if $dF^{n-1}=0$, and astheno-Kähler if $partial\bar{\partial}F^{n-2}=0$, where $F$ is its fundamental f…

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $(G/H,J)$ is either a Kähler metric or a multiple of the standard metric $g_o$.

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $G/H$ is either a Kähler metric or a multiple of the standard metric.

Let $X$ be a compact complex manifold of complex dimension greater than $2$. Among balanced, pluriclosed, and locally conformally Kähler (LCK) metrics, suppose that $X$ admits two…

Let $X$ be a compact holomorphic manifold. A Hermitian-symplectic metric is a Hermitian metric that is the $(1,1)$-component of a closed real $2$-form. Streets--Tian conjecture. If…

Let $(M,\omega)$ be a complete balanced Hermitian manifold of complex dimension $n$, and suppose that the equality case in the first-eigenvalue estimate of Theorem 1.1 holds, so th…

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. For $k\in\{1,2,3,4\}$ and $\beta\neq0$, define the kth-mixed curvature by … where $Ric^{(k)}$ is the kth Chern Ricci…

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. The holomorphic sectional curvature is $H(X)=R(X,\overline{X},X,\overline{X})/|X|^{4}$ for nonzero $(1,0)$-vectors…

Let $M$ be a compact complex manifold with a complex structure $J$. An LCK metric and a pluriclosed metric on $M$ are said to be compatible with the same complex structure when bot…

Let $X$ be a compact complex manifold endowed with a Hermitian metric $g$, and let $K$ denote its holomorphic sectional curvature for the Chern connection. Constant holomorphic sec…