57 problems

For every compact Hermitian manifold $(M^n,J,g)$, if the Chern holomorphic sectional curvature of $g$ is identically equal to a constant $c\in\mathbb{R}$, then $g$ is Kähler when…

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 $X$ 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…

Fino–Vezzoni flow conjecture. The pluriclosed flow exists for all time and converges smoothly to a Kähler metric. In particular, $M$ is Kähler.

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\{\beta\}\in H^{1,1}(X,\mathbb{R})$ be a real $(1,1)$-class with smooth representative $\beta$. The…

Gauduchon's conjecture. There exists a Gauduchon metric $\omega$ on $M$ such that

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…

SKT degeneration conjecture. If $X$ admits an SKT metric, then its Frölicher spectral sequence degenerates at the second page $E_2$.

Strongly Gauduchon conjecture. The manifold $M$ does not admit any strongly Gauduchon metric. The claim extends the known obstruction for compact non-Kähler BKL manifolds; the sour…

Let $(M^n,g)$ be a compact Hermitian manifold. A Hermitian manifold is Bismut Kähler-like (BKL) if its Bismut curvature has the Kähler symmetries … for all type $(1,0)$ tangent vec…

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…

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

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…

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 $n\geq2$. Let $\alpha$ and $\beta$ be fixed real numbers with $\beta\neq0$, and let $\mathcal{C}_{\alpha,\beta}$ denote the mix…

Chen–Zheng conjecture. If the holomorphic sectional curvature of $\nabla^b$ is a non-zero constant, then $g$ must be Kähler, hence a complex space form.

Hermitian space-form conjecture. If the Chern connection of $M$ has constant holomorphic sectional curvature, then $M$ must be either Kähler, hence a complex space form, or Chern f…

Let $M$ be a compact complex manifold. A Hermitian-symplectic metric on $M$ is a Hermitian metric whose Kähler form is the $(1,1)$-part of a closed $2$-form. Equivalently, there is…

Streets–Tian–Fino–Vezzoni intersection conjecture. If $M$ admits a Hermitian-symplectic metric and a balanced metric, then it must admit a Kähler metric.

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$. For a real number $t$, let $\nabla^{(t)}=(1-t)\nabla+t\nabla^b$ be the $t$-Gauduchon connection, let…

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$, and let $H^r$ denote the Riemannian holomorphic sectional curvature, with $R^r$ the curvature of the R…

Let $M$ be a Hermitian manifold with a Hermitian metric $g$. Assume that $g$ has positive or quasi-positive real bisectional curvature. Yang–Zheng's conjecture. Then $M$ is project…