arXiv subjects
Publications and source records attributed to Shijie Bao.
An algebraic approach is presented for the valuative interpolation problem, which recovers and generalizes prior characterizations known in the complex analytic setting by the authors. We use the asymptotic Samuel function to give the characterization of the existence of valuative interpolation. We also give a characterization of the existence in the infinite valuative interpolation problem.
The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular points of irreducible analytic subvarieties. By applying these tools, we establish the necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and weakly holomorphic functions at a singular point. As applications, we characterize the existence of valuative interpolations on the quotient ring of the ring of convergent power series in real variables. We also present separated necessary and sufficient conditions for the existence of valuative interpolations on the quotient ring of polynomial rings with complex coefficients and real coefficients. Furthermore, we show that the conditions become both necessary and sufficient under certain conditions on the zero set of the given polynomials.
In this article, using key tools including Zhou valuations, Tian functions and a convergence result for relative types, we establish necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and real analytic functions at the origin in $\mathbb{C}^{n}$ and $\mathbb{R}^{n}$, respectively. For the cases of polynomial rings with complex and real coefficients, we establish separate necessary conditions and sufficient conditions, which become both necessary and sufficient when the intersection of the zero sets of the given polynomials is the set of the origin in $\mathbb{C}^{n}$. Furthermore, we obtain a necessary and sufficient condition for a valuation to be of the form given by a relative type with respect to a tame maximal weight. We demonstrate a result of Boucksom--Favre--Jonsson on quasimonomial valuations also holds for quasimonomial Zhou valuations. Finally, we obtain a relationship between Zhou valuations and the differentiable points of Tian functions.
In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.
In the present paper, we generalize the notion of the $p$-Bergman kernel and the $ξ$-Bergman kernel to the $p$-Bergman kernel with respect to a functional $ξ$, and establish some properties of the $p$-Bergman kernel with respect to $ξ$. We also study the relations between the $L^p$ versions of higher order Bergman kernels and $ξ$-Bergman kernels, and as applications we give the reproofs and generalizations of some previous results of Blocki and Zwonek about higher order Bergman kernels.
In this note, we demonstrate the convergence of the Demailly approximation of a general (weakly) upper semi-continuous weight.
In this article, we prove that for any Zhou valuation $ν$, there exists a graded sequence of ideals $\mathfrak{a}_{\bullet}$ and a nonzero ideal $\mathfrak{q}$ such that $ν$ $\mathscr{A}-$computes the jumping number $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$, and that for the subadditive sequence $\mathfrak{b}^φ_{\bullet}$ related to a plurisubharmonic function $φ$, there exists a Zhou valuation which $\mathscr{A}-$computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^φ_{\bullet})$, where the ``$\mathscr{A}-$compute'' coincides with the ``compute'' in Jonsson-Mustaţă's Conjecture when the Zhou valuation $ν$ is quasimonomial. There are also some results obtained for Zhou valuations, including a characterization for a valuation being a Zhou valuation, and a denseness property of the cone of Zhou valuations.
In this paper, we establish the log-plurisubharmonicity of fiberwise $ξ$-Bergman kernels for a family of variant functionals, thereby addressing a question posed by Bo Berndtsson to the authors. As an application, we prove that for a plurisubharmonic function $ϕ$ and a locally finitely generated ideal sheaf $\mathscr{I}$ on the polydisc $Δ^{n+m}=Δ^n\timesΔ^m$, the set of points $w\inΔ^m$ for which $\mathcal{I}(ϕ|_{H_w})_{(o,w)}\subseteq (\mathscr{I}|_{H_w})_{(o,w)}$ is a pluripolar set, where $H_w$ represents the fiber over $w$, and $\mathcal{I}(ϕ|_{H_w})$ denotes the multiplier ideal sheaf of $ϕ|_{H_w}$ on $H_w$.
In this paper, we show the equivalence of the sharp effectiveness results of the strong openness property of multiplier ideal sheaves obtained in \cite{BG1} using $ξ-$Bergman kernels and in \cite{Guan19} using minimal $L^2$ integrals.
In this article, we obtain a class of tame maximal weights (Zhou weights). Using Tian functions (the function of jumping numbers with respect to the exponents of a holomorphic function or the multiples of a plurisubharmonic function) as a main tool, we establish an expression of relative types (Zhou numbers) to these tame maximal weights in integral form, which shows that the relative types satisfy tropical multiplicativity and tropical additivity. Thus, the relative types to Zhou weights are valuations (Zhou valuations) on the ring of germs of holomorphic functions. We use Tian functions and Zhou numbers to measure the singularities of plurisubharmonic functions, involving jumping numbers and multiplier ideal sheaves. Especially, the relative types to Zhou weights characterize the division relations of the ring of germs of holomorphic functions. Finally, we consider a global version of Zhou weights on domains in $\mathbb{C}^n$, which is a generalization of the pluricomplex Green functions, and we obtain some properties of them, including continuity and some approximation results.
In this article, we give some necessary conditions for the concavity property of minimal $L^2$ integrals degenerating to partial linearity, a charaterization for the concavity degenerating to partial linearity for open Riemann surfaces, and some relations between the concavity property for minimal $L^2$ integrals and the log-convexity for Bergman kernels.
For the purpose of proving the strong openness conjecture of multiplier ideal sheaves, Jonsson-Mustaţă posed an enhanced conjecture and proved the two-dimensional case, which says that: the Lebesgue measure of the set $\big\{c_o^F(ψ)ψ-\log|F|<\log r\big\}$ divided by $r^2$ has a uniform positive lower bound independent of $r$, for a plurisubharmonic function $ψ$ and a holomorphic function $F$ near the origin $o$. Jonsson-Mustaţă's conjecture was proved by Guan-Zhou depending on the truth of the strong openness conjecture. However, it is still a question whether one can prove Jonsson-Mustaţă's conjecture without using the strong openness property, and obtain a sharp effectiveness result for this conjecture. In this article, we use an $L^2$ method with the weight functions $ψ-\log|F|$ and firstly consider a module at at a boundary point of the sublevel sets of a plurisubharmonic function. By studying the minimal $L^{2}$ integrals on the sublevel sets of a plurisubharmonic function with respect to the module at the boundary point, we establish a concavity property of the minimal $L^{2}$ integrals. As applications, we obtain a sharp effectiveness result related to Jonsson-Mustaţă's conjecture, which completes the approach from the conjecture to the strong openness property. We also obtain a strong openness property of the module and a lower semi-continuity property with respect to the module.
In the present note, we introduce the $ξ-$complex singularity exponents, which come from the asymptotic property of $ξ-$Bergman kernels on sub-level sets of plurisubharmonic functions; give some relations (including a closedness property) among $ξ-$complex singularity exponents, complex singularity exponents, and jumping numbers; generalize some properties of complex singularity exponents (such as the restriction formula and subadditivity property) to $ξ-$complex singularity exponents.
In the present paper, we give the definition and properties of the multipoled global Zhou weights. Some approximation and convergence results of multipoled global Zhou weights are given. We also establish a semi-continuity result for the Zhou numbers.
In this article, we consider Bergman kernels related to modules at boundary points for singular hermitian metrics on holomorphic vector bundles, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted $L^2$ integrals on sublevel sets of plurisubharmonic functions, and reprove an effectiveness result of the strong openness property of the modules.
In this article, we present characterizations of the concavity property of minimal $L^2$ integrals with negligible weights degenerating to linearity on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem with negligible weights on the fibrations over open Riemann surfaces and the fibrations over products of open Riemann surfaces.
In this article, we consider Bergman kernels related to modules at boundary points on Stein manifolds, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted $L^2$ integrals on Stein manifolds, and reprove an effectiveness result of strong openness property of modules at boundary points on Stein manifolds.
In this article, we present characterizations of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of fibrations over products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets $L^2$ extension problem from fibers over products of analytic subsets to fibrations over products of open Riemann surfaces, which implies characterizations of the equality parts of Suita conjecture and extended Suita conjecture for fibrations over products of open Riemann surfaces.