arXiv subjects
Publications and source records attributed to Zhitong Mi.
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 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 present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain on weakly pseudoconvex Kähler manifolds. As applications, we give a necessary condition for the concavity degenerating to linearity, and a characterization for the holding of the equality in optimal jets $L^2$ extension problem on open Riemann surfaces.
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 note, we present an optimal $L^2$ extension theorem for holomorphic vector bundles with smooth hermitian metrics for continuous gain on weakly pseudoconvex Kähler manifolds, which is a unified version of the optimal $L^2$ extension theorems for holomorphic vector bundles with smooth hermitian metrics of Guan-Zhou and Zhou-Zhu.
In the present paper, we study the properties of singular Nakano positivity of singular hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex Kähler manifolds. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.
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, for singular hermitian metrics on holomorphic vector bundles, we consider minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions on weakly pseudoconvex Kähler manifolds related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal $L^2$ integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a strong openness property of the modules and a twisted version, an effectiveness result of the strong openness property of the modules, and an optimal support function related to the modules.
In this article, we consider minimal $L^2$ integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex Kähler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal $L^2$ integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to the modules, a strong openness property of the modules and a twisted version, an effectiveness result of the strong openness property of the modules.
In this article, we consider a modified version of minimal $L^2$ integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points, and obtain a concavity property of the modified version. As an application, we give a characterization for the concavity degenerating to linearity on open Riemann surfaces.
In this article, we consider the minimal $L^2$ integrals related to modules at boundary points on fibrations over open Riemann surfaces, and present a characterization for the concavity property of the minimal $L^2$ integrals degenerating to linearity.
In this note, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^{2}$ extension problem to hold.