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

In the present paper, we show that given a compact Kahler manifold $(X,omega)$ with a Kahler metric $omega$, and a complex submanifold $Vsubset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any quasi-plurisubharmonic function $varphi$ on $V$ such that $omega|_V+sqrt{-1}partialbarpartialvarphigeq varepsilonomega|_V$ with $varepsilon>0$ can be extended to a quasi-plurisubharmonic function $Phi$ on $X$, such that $omega+sqrt{-1}partialbarpartial Phigeq varepsilonomega$ 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 Kahler manifolds which are not necessarily projective.

On the restriction formula

Let $varphi$ be a quasi-psh function on a complex manifold $X$ and let $Ssubset X$ be a complex submanifold. Then the multiplier ideal sheaves $mathcal{I}(varphi|_S)subsetmathcal{I}(varphi)|_{S}$ and the complex singularity exponents $c_{x}left(varphi|_{S}right)leqslant c_{x}(varphi)$ 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.

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.

On the extensions of Kahler currents on compact K{a}hler manifolds

Let $(X,omega)$ be a compact K{a}hler manifold with a K{a}hler form $omega$ of complex dimension $n$, and $Vsubset X$ 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 $varphi$ be a strictly $omega|_V$-psh function on $V$. In this paper, we prove that there is a strictly $omega$-psh function $Phi$ on $X$, such that $Phi|_V=varphi$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of K{a}hler currents. We also discuss possible extensions of Kahler currents in a big class.

CR eigenvalue estimate and Kohn-Rossi cohomology

Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $Box_b$ be the $overline{partial}_b$-Laplacian. Eigenvalue estimate of $Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $lambda$ of $Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $min mathbb{Z}_+$ and $q=0,1,cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m rightarrow +infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $ min mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.

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 Sius $partialoverline{partial}$-formula and Berndtssons eigenvalue estimate of $overline{partial}$-Laplacian and combining Bonaveros technique, we obtain the following result: given a holomorphic pseudo-effective line bundle $(L, h_L)$ on a compact Hermitian manifold $(X,omega)$, if $h_L$ is a singular metric with algebraic singularities, then $dim H^{q}(X,L^kotimes Eotimes mathcal{I}(h_L^{k}))leq Ck^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.

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

In this note, we reveal that our solution of Demaillys strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll{a}r and Jonsson-Mustatu{a} implies the truth of twist