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Register a new userOn minimal kernels and Levi currents on weakly complete complex manifolds
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A complex manifold $X$ is emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $phi$. The minimal kernels $Sigma_X^k, k in [0,infty]$ (the loci where are all $mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $Sigma_X^k$s, and give sufficient conditions for points in $Sigma_X^k$ to be in the support of some Levi current. When $X$ is a surface and $phi$ can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini,we prove the existence of a Levi current precisely supported on $Sigma_X^infty$, and give a classification of Levi currents on $X$. In particular,unless $X$ is a modification of a Stein space, every point in $X$ is in the support of some Levi current.
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Applying Lies theory, we show that any $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in the class $mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at any point $p in M$ to be the $v$-axis in coordinates $(z, zeta, w = u + i, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $gamma$ is the $v$-axis so that $M = {u=F(z,zeta,overline{z},overline{zeta},v)}$ has Poincare-Moser reduced equation: begin{align} u & = zoverline{z} + tfrac{1}{2},overline{z}^2zeta + tfrac{1}{2},z^2overline{zeta} + zoverline{z}zetaoverline{zeta} + tfrac{1}{2},overline{z}^2zetazetaoverline{zeta} + tfrac{1}{2},z^2overline{zeta}zetaoverline{zeta} + zoverline{z}zetaoverline{zeta}zetaoverline{zeta} & + 2{rm Re} { z^3overline{zeta}^2 F_{3,0,0,2}(v) + zetaoverline{zeta} ( 3,{z}^2overline{z}overline{zeta} F_{3,0,0,2}(v) ) } & + 2{rm Re} { z^5overline{zeta} F_{5,0,0,1}(v) + z^4overline{zeta}^2 F_{4,0,0,2}(v) + z^3overline{z}^2overline{zeta} F_{3,0,2,1}(v) + z^3overline{z}overline{zeta}^2 F_{3,0,1,2}(v) + z^3{overline{zeta}}^3 F_{3,0,0,3}(v) } & + z^3overline{z}^3 {rm O}_{z,overline{z}}(1) + 2{rm Re} ( overline{z}^3zeta {rm O}_{z,zeta,overline{z}}(3) ) + zetaoverline{zeta}, {rm O}_{z,zeta,overline{z},overline{zeta}}(5). end{align} The values at the origin of Pocchiolas two primary invariants are: [ W_0 = 4overline{F_{3,0,0,2}(0)}, quadquad J_0 = 20, F_{5,0,0,1}(0). ] The proofs are detailed, accessible to non-experts. The computer-generated aspects (upcoming) have been reduced to a minimum.
Define a subset of the complex plane to be a Rolles domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolles domain to be minimal if no proper subset is a Rolles domain. In this paper, we investigate minimal Rolles domains.
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 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.
We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Amp`ere operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials.
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