arXiv subjects
Publications and source records attributed to Jiafu Ning.
Let $(X,\omega)$ be a compact K\"ahler manifold and let $V\subset X$ be a closed complex submanifold. Coman-Guedj-Zeriahi proposed the problem: is every $\omega|_V$-plurisubharmonic function on $V$ the restriction of an $\omega$-plurisubharmonic function on $X$? In this paper, we solve this problem affirmatively, even for a compact Hermitian manifold.
In this paper, we present a converse to a version of Skoda's $L^2$ division theorem by investigating the solvability of $\bar{\partial}$ equations of a specific type.
We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.
Let $X$ be a compact Kähler manifold and $(L,h)\rightarrow X$ be a pseudoeffective line bundle, such that the curvature $iΘ_{L,h}\geq 0$ in the sense of currents. The main result of the present paper is that $H^n(X,\mathcal{O}(Ω^p_X\otimes L)\otimes \mathcal{I}(h))=0$ for $p\geq n-nd(L,h)+1$. This is a generalization of Bogomolov's vanishing theorem.
We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.
In the present paper, we show that given a compact Kähler manifold $(X,ω)$ with a Kähler metric $ω$, and a complex submanifold $V\subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any quasi-plurisubharmonic function $φ$ on $V$ such that $ω|_V+\sqrt{-1}\partial\bar\partialφ\geq \varepsilonω|_V$ with $\varepsilon>0$ can be extended to a quasi-plurisubharmonic function $Φ$ on $X$, such that $ω+\sqrt{-1}\partial\bar\partial Φ\geq \varepsilon'ω$ for some $\varepsilon'>0$. This is an improvement of results in \cite{WZ20}. Examples satisfying the assumption that there exists a holomorphic retraction structure contain product manifolds, thus contains many compact Kähler manifolds which are not necessarily projective.
We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension condition for some $p>0$. These results can be roughly viewed as converses of Hörmander's $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type extension theorems. As an application of the main result, we get a totally different method to Nakano positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic families of complex manifolds.
We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic functions.
In this article, we establish a support function of the weighted $L^2$ integrations on the superlevel sets of the weights with optimal asymptoticity near the positive infinity, which is an analogue of the truth of Demailly's strong openness conjecture.