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Register a new userExistence of a complete holomorphic vector field via the Kahler-Einstein metric
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In this paper, we study the existence of a complete holomorphic vector fields on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kahler-Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kahler-Einstein metric whose differential has a constant length. Then we will construct a complete holomorphic vector field from the gradient vector field of the potential function.
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We prove that for a bounded domain in $mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set.
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $mathcal{C}^2$-boundary in $mathbb{C}^n$ into the unit ball of $mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.
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.
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.
We will show a rigidity of a Kahler potential of the Poincare metric with a constant length differential.
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