Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost K\"ahler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal K\"ahler metric. This result rests on the Yau--Tian--Donaldson correspondence in terms of the moment polytope $\Delta$ for $\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal K\"ahler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal K\"ahler metric in every K\"ahler class.
In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$...
For a primitive symplectic variety $X$ admitting a holomorphic Lagrangian fibration $f: X \to B$, we classify the possible Mumford--Tate groups of the very general fiber. We extract as a consequence that the discriminant of $f$ has codimension one in $B$. In the special case when $X$ is a hyper-K\"ahler manifold which...
An old conjecture in non-K\"ahler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant $c$, then the metric must be K\"ahler when $c\neq 0$ and be Chern flat when $c=0$. The conjecture is known to be true in dimension two by the work of Balas--Gauducho...
Let $Z$ be a compact K\"ahler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie alge...
Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton s...
LeBrun's K\"ahler realization $g_m$ of the Taub--NUT metric on $\mathbb{C}^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $\alpha g_m$ admits a K\"ahler immersion into a finite- or infinite-dimensional complex projective space when $m>\alpha/2$, and conjectured that the same hold...
Shao-Yuan Huang· 0 citations
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