In this work, a sharp K\"ahler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu--Wang--Zhang. For compact K\"ahler manifolds satisfying $\Ric(\omega)\geq\omega$, pinching the first $n^2+3$ nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index $n^2+1$ is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension $r$, then the multiplicity of the critical eigenvalue is at most $r+\lfloor r/2\rfloor^2$, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.
We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini...
C. Arezzo, A. Loi, G. Placini et al.· 1 citation· ⚡1
Let $(M,g,J)$ be a closed K\"ahler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler--Lagrange equations associated with the Beckner--Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity...
We develop an \(m\)-Hessian approach to the construction of finite-Monge--Amp\`ere weights on complete noncompact K\"ahler manifolds. Let \((M^n,g)\) be a complete noncompact K\"ahler manifold of complex dimension \(n\ge3\) with positive holomorphic bisectional curvature. The main new ingredient is a quantitative capac...
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...
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...
In this paper, we develop the weighted level set method for a K\"ahler manifold $(X^n,\omega)$ admitting an almost holomorphic map to a possibly singular base $Z$, which is not uniruled. As a key intermediate result, we prove that any blowup $\operatorname{Bl}_SX$ of $X$ along smooth submanifolds $S$ of $ \operatorname...
Ze-Hao Sha, Jian Wang· 0 citations
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