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Register a new userOn the extensions of Kahler currents on compact K{a}hler manifolds
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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.
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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.
Given a complex manifold $X$, any Kahler class defines an affine bundle over $X$, and any Kahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact Kahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--SzH{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.
Let $(X,omega)$ be a compact Kahler manifold of dimension $n$ and fix $1leq mleq n$. We prove that the total mass of the complex Hessian measure of $omega$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.
In this paper we survey the recent developments of the Ricci flows on complete noncompact K{a}hler manifolds and their applications in geometry.
A special Kahler-Ricci potential on a Kahler manifold is any nonconstant $C^infty$ function $tau$ such that $J( ablatau)$ is a Killing vector field and, at every point with $dtau e 0$, all nonzero tangent vectors orthogonal to $ ablatau$ and $J( ablatau)$ are eigenvectors of both $ abla dtau$ and the Ricci tensor. For instance, this is always the case if $tau$ is a nonconstant $C^infty$ function on a Kahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $tilde g=g/tau^2$, defined wherever $tau e 0$, is Einstein. (When such $tau$ exists, $(M,g)$ may be called {it almost-everywhere conformally Einstein}.) We provide a complete classification of compact Kahler manifolds with special Kahler-Ricci potentials and use it to prove a structure theorem for compact Kahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.
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