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Register a new userComplex Hessian equations with prescribed singularity on compact Kahler manifolds
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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.
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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 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.
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed Kahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which generalize the results of Hessian equations and Hessian quotient equations.
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 $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure on $Omega$. We study the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. First we give a sufficient condition on the measure $mu$ in terms of its domination by the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in the proofs is to establish a new capacity estimate showing that the $m$-Hessian measure of a continuous $m$-subharmonic function on $Omega$ with zero boundary values is dominated by an explicit function of the $m$-Hessian capacity with respect to $Omega$, involving the modulus of continuity of $varphi$. Another important ingredient is a new weak stability estimate on the Hessian measure of a continuous $m$-subharmonic function.
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