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Polynomial Optimization, Certificates of Positivity, and Christoffel Function
We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function...
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Degenerate J-flow on compact Kähler manifolds
In this note, we study a degenerate twisted J-flow on compact Kähler manifolds. We show that it exists for all time, it is unique and converges to a...
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The local properties of some subclasses of plurisubharmonic functions
The main aim of this note is first to give the definition of the local classes. It also aims at studying the local property of some subclasses of...
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The twisted conical Kähler-Ricci solitons on Fano manifolds
In this paper, we show the relation between the existence of twisted conical Kähler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound...
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Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for m -subharmonic functions with finite ( p , m )-en...
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Transfinite Diameter with Generalized Polynomial Degree
We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial...
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Special metrics in Kähler geometry
In this note we walk through some of the developments and contributions made in the last 50 years to the study of the complex Monge–Ampère equation...
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On a Family of Quasimetric Spaces in Generalized Potential Theory
We construct a family of quasimetric spaces in generalized potential theory containing m -subharmonic functions with finite ( p , m )-energy. These...
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Monge—Ampère Type Equations on Almost Hermitian Manifolds
In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C ∞ a priori estimates under the existence...
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The probabilistic vs the quantization approach to Kähler–Einstein geometry
In the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic manifold X —involving random point processes on X —a key...
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Weak solutions to the complex m-Hessian type equations for arbitrary nonnegative Radon measures on open subsets of \({\mathbb {C}}^n\)
In the paper, we prove the existence of weak solutions of the complex m -Hessian type equation for arbitrary nonnegative Radon measures on an open...
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Tian’s Properness Conjectures:An Introduction to Kähler Geometry
This manuscript served as lecture notes for a minicourse in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to...
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Kähler–Einstein Metrics on Toric Manifolds and G-manifolds
This is an expository paper. In the first part, we discuss variant approaches in the study of Kähler–Einstein metrics on toric Fano manifolds. In the...
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