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Xiangyu Zhou

Publications and source records attributed to Xiangyu Zhou.

At least 73 records · Page 4Linked to original sources
arXiv2021

On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

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.

math.CV↗
arXiv2021

On the restriction formula

Let $φ$ be a quasi-psh function on a complex manifold $X$ and let $S\subset X$ be a complex submanifold. Then the multiplier ideal sheaves $\mathcal{I}(φ|_S)\subset\mathcal{I}(φ)|_{S}$ and the complex singularity exponents $c_{x}\left(φ|_{S}\right)\leqslant c_{x}(φ)$ by Ohsawa-Takegoshi $L^{2}$ extension theorem. An interesting question is to know whether it is possible to get equalities in the above formulas. In the present article, we show that the answer is positive when $S$ is chosen outside a measure zero set in a suitable projective space.

math.CV↗
arXiv2020

A generalization of Nadel vanishing theorem

In this paper we first prove a version of $L^{2}$ existence theorem for line bundles equipped a singular Hermitian metrics. Aa an application, we establish a vanishing theorem which generalizes the classical Nadel vanishing theorem.

math.CV↗
arXiv2020

On the extensions of Kähler currents on compact Kähler manifolds

Let $(X,ω)$ be a compact Kähler manifold with a Kähler form $ω$ of complex dimension $n$, and $V\subset X$ is a compact complex submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in $X$ as a zero section of a holomorphic vector bundle or rank $n-k$ over $V$. Let $φ$ be a strictly $ω|_V$-psh function on $V$. In this paper, we prove that there is a strictly $ω$-psh function $Φ$ on $X$, such that $Φ|_V=φ$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of Kähler currents. We also discuss possible extensions of Kähler currents in a big class.

math.CV↗
arXiv2020

Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$

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.

math.CV↗
arXiv2019

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.

math.CV↗
arXiv2019

Asymptotic estimate of cohomology groups valued in pseudo-effective line bundles

In this paper, we study questions of Demailly and Matsumura on the asymptotic behavior of dimensions of cohomology groups for high tensor powers of (nef) pseudo-effective line bundles over non-necessarily projective algebraic manifolds. By generalizing Siu's $\partial\overline{\partial}$-formula and Berndtsson's eigenvalue estimate of $\overline{\partial}$-Laplacian and combining Bonavero's technique, we obtain the following result: given a holomorphic pseudo-effective line bundle $(L, h_L)$ on a compact Hermitian manifold $(X,ω)$, if $h_L$ is a singular metric with algebraic singularities, then $\dim H^{q}(X,L^k\otimes E\otimes \mathcal{I}(h_L^{k}))\leq Ck^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.

math.CV↗
arXiv2019

Linear invariants of complex manifolds and their plurisubharmonic variations

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0 2, p\neq 2,4,\cdots$, provided that the $p$-Bergman kernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we show that, for all $p>0$, the $p$-Bergman kernel on a strongly pseudoconvex domain with $\mathcal C^2$ boundary or a simply connected homogeneous regular domain is exhaustive. These results shows that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative $m$-pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact Kähler manifolds is positively curved, with respect to the canonical singular Finsler metric.

math.CV↗
arXiv2018

New characterizations of plurisubharmonic functions and positivity of direct image sheaves

We give new characterizations of plurisubharmonic functions and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics. As applications, we present a different method to prove plurisubharmonic variation of generalized Bergman kernel metrics and Griffiths positivity of the direct images of twisted relative canonical bundles associated to holomorphic families of Kähler manifolds.

math.CV↗
arXiv2017

Strong openness of multiplier ideal sheaves and optimal $L^{2}$ extension

In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.

math.CV↗
arXiv2015

Multiplier ideal sheaves, jumping numbers, and the restriction formula

In the present article, we establish an equality condition in the restriction formula on jumping numbers by giving a sharp lower bound of the dimension of the support of a related coherent sheaf. As applications, we obtain equality conditions in the restriction formula on complex singularity exponents by giving the dimension, the regularity and the transversality of the support, and we also obtain some sharp equality conditions in the fundamental subadditivity property on complex singularity exponents. We also obtain two sharp relations on jumping numbers.

math.CV↗
arXiv2015

Characterization of multiplier ideal sheaves with weights of Lelong number one

In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda.

math.CV↗
arXiv2014

Effectiveness of Demailly's strong openness conjecture and related problems

In this article, stimulated by the effectiveness in Berndtsson's solution of the openness conjecture and continuing our solution of Demailly's strong openness conjecture, we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result. We explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier, which is implicitly contained in our paper arXiv:1401.7158. We also obtain optimal effectiveness of the conjectures of Demailly-Kollár and Jonsson-Mustată respectively.

math.CV↗