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Xiangsen Qin

Publications and source records attributed to Xiangsen Qin.

12 recordsLinked to original sources
arXiv2026

Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures

Motivated by Bedford's question, we study the possible cluster sets, in the vague topology, of Monge--Amp\`ere measures associated with bounded plurisubharmonic sequences $u_j\to\varphi$ locally in $L^1$, without assuming a common bound. On a bounded domain $\Omega\subset\mathbb C^n$ ($n\geq2$), and for a bounded maximal plurisubharmonic $\varphi$, the possible sets of all vague subsequential measure limits are precisely the vaguely closed sets of nonnegative Radon measures, including the empty set. This conclusion holds for every bounded plurisubharmonic $\varphi$ when $\Omega$ is hyperconvex, pseudoconvex and Runge, or a domain obtained by removing a relatively closed pluripolar set from a bounded hyperconvex domain. Under these hypotheses, every nonnegative Radon measure $\nu$ can also be realized by such a sequence with $(dd^cu_j)^n\to\nu$ vaguely. The main analytic result holds on every bounded domain: for any bounded plurisubharmonic $\varphi$ and nonnegative Radon measure $\mu$, we construct $u_j\leq\varphi$ converging to $\varphi$ in $L^p(\Omega)$ for every $1\leq p<\infty$, with $(dd^cu_j)^n\to(dd^c\varphi)^n+\mu$ vaguely. The approximants can be chosen smooth when $\varphi$ is continuous. On arbitrary pseudoconvex domains, prescribed measure limits can also be realized using locally bounded approximants. Under an additional localization condition, the approximants can be chosen globally bounded when the limit function is bounded.

math.CV↗
arXiv2026

Prescribed Lelong Numbers for One-Pole Green Functions on Complex Projective Space

Let $\omega_{\mathrm{FS}}$ be the normalized Fubini--Study form on $\mathbf P^n$, with $n\geq2$. We prove that the one-pole Lelong-number range in $DMA(\mathbf P^n,\omega_{\mathrm{FS}})$ is exactly $[0,1]$: for every $\lambda$ in this interval there is a Green function with a single pole, Monge--Amp\`ere measure equal to the Dirac mass at that pole, and Lelong number $\lambda$. This answers Question~9 in the survey of Dinew--Guedj--Zeriahi, where the problem is attributed to Coman and Guedj. The construction adapts Li and Xia's local zero-Lelong-number scheme through variable degrees and homogeneous finite stages, while also establishing the exact Lelong number and membership in the global $DMA$ class. We then study the relation between the one-pole range $\mathcal R_\alpha(x)$ and the Seshadri interval $[0,\varepsilon(\alpha,x)]$. An application of Koike's equivalence theorem gives a point on a degree-one del Pezzo surface where the Seshadri endpoint is not attained. Conversely, a finite-pullback criterion and an explicit finite morphism show that every ample rational class on a product of projective spaces realizes its full Seshadri interval at every point.

math.CV↗
arXiv2026

Failure of the Proposed Local Decay Formula for Local Arnold Multiplicities under Twisted K\"ahler--Ricci Flow

Let $\lambda(u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza--Guedj--Lu asked whether every maximal weak solution $\varphi_t$ of the twisted K\"ahler--Ricci flow satisfies $\lambda(\varphi_t,x)=\max\{\lambda(\varphi_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface $\mathbb F_e=\mathbb P_{\mathbb P^1} (\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-e))$, $e\ge2$. Let $S$ be its negative section and $F_1,\ldots,F_k$ be distinct fibres. If $a,b_i>0$, $\sum_i b_i>ea$, and the initial current is $a[S]+\sum_i b_i[F_i]$, then $\lambda(\varphi_t,x)=a-\min\{k/e,1\}t$ for $x\in S\setminus\bigcup_iF_i$ and $0<t<\min\{a,b_1,\ldots,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge2$.

math.DG↗
arXiv2025

Quantitative Carleman-type estimates for holomorphic sections over bounded domains

This paper establishes quantitative Carleman-type inequalities for holomorphic sections of Hermitian vector bundles over bounded domains in $\mathbb{C}^n$ with $n \geq 2$. We first prove a Sobolev-type inequality with explicit constants for the Laplace operator, which leads to quantitative Carleman-type estimates for holomorphic functions. These results are then extended to holomorphic sections of Hermitian vector bundles satisfying certain curvature restrictions, yielding quantitative versions where previously only non-quantitative forms were available. The proofs refine existing methods through careful constant tracking and by estimating the radius of the uniform sphere condition of the boundary through the Lipschitz constant of its outward unit normal vector.

math.CV↗
arXiv2025

Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact Kähler manifolds

The main goal of this paper is to generalize the Sobolev-type inequalities given by Guo-Phong-Song-Sturm and Guedj-Tô from the case of functions to the framework of twisted differential forms. To this end, we establish certain estimates of heat kernels for differential forms with values in holomorphic vector bundles over compact Kähler manifolds. As applications of these estimates, we also prove a vanishing theorem and give certain $L^{q,p}$-estimates for the $\bar\partial$-operator on twisted differential forms.

math.CV↗
arXiv2025

$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and apply them to derive a generalization of the Sobolev Inequality with Trace in $\mathbb{R}^n$. As applications to complex analysis, we get an integral form of Maximum Modulus Principle for holomorphic functions, and an improvement of Hörmander's $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains.

math.CV↗
arXiv2025

Dirichlet Green kernel estimates and Sobolev-type inequalities for twisted differential forms

We study the full-trace Dirichlet realization of the Dolbeault Laplacian on differential forms with values in a Hermitian holomorphic vector bundle over a relatively compact smooth domain in a K\"ahler manifold. We prove global Green kernel estimates that are uniform up to the boundary, including one- and two-boundary-factor bounds and estimates for the \(\bar\partial\)- and \(\bar\partial^*\)-derivatives. These estimates yield Sobolev-type inequalities with boundary terms. In real dimension two, the first-order bound has a logarithmic loss. In top antiholomorphic degree, we obtain quantitative \(L^r\)-to-\(L^k\) solvability for \(\bar\partial\) without pseudoconvexity. Analogous results hold for the twisted de Rham complex of a flat metric connection on a compact Riemannian manifold with smooth boundary.

math.AP↗
arXiv2025

$L^2$-estimates on flat vector bundles and Prékopa's theorem

In this paper, we will construct Hörmander's $L^2$-estimate of the operator $d$ on a flat vector bundle over a $p$-convex Riemannian manifold and discuss some geometric applications of it. In particular, we will generalize the classical Prékopa's theorem in convex analysis.

math.DG↗
arXiv2024

Uniform estimates of Green functions and Sobolev-type inequalities on real and complex manifolds

We prove certain $L^p$ Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds for the gradient operator $\nabla$, the Laplace operator $Δ$, and the operator $\bar\partial$. Integral representations for functions are key to get such inequalities. The proofs of the main results involves certain uniform estimates for the Green functions and their gradients on Riemannian manifolds, which are also established in the present work.

math.CV↗
arXiv2024

Curvature formulas and curvature strict positivity of direct image bundles

In this paper, we consider the curvature strict positivity of direct image bundles (vector bundles) associated to a strictly pseudoconvex family of bounded domains.The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded domains is strictly positive in the sense of Nakano even if the curvature of the original vector bundle is just Nakano positive. Based on our (I and my coauthors) previous results, this result further demonstrates that strictly pseudoconvex domains and pseudoconvex domains have very different geometric properties. To consider the curvature strict positivity, we will first construct a curvature formula for a direct image bundle, then the curvature strict positivity will be a simple consequence of it. As applications to convex analysis, we get a corresponding version of Prékopa's Theorem, i.e., we get the curvature strict positivity of a strictly convex family of bounded domains.In the last, we give a flatness criterion for the direct image bundles.onvex family of bounded domains.

math.CV↗
arXiv2022

Curvature strict positivity of direct image bundles associated to pseudoconvex families of domains

We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains. The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular domains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic. This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis. We investigate that the main result implies a remarkable result of Berndtsson which states that, for an ample vector bundle $E$ over a compact complex manifold $X$ and any $k\geq 0$, the bundle $S^kE\otimes\det E$ admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where $S^kE$ is the $k$-th symmetric product of $E$. The two main ingredients in the argument of the main theorems are Berndtsson's estimate of the lower bound of curvature of direct image bundles and Deng-Ning-Wang-Zhou's characterization of the curvature Nakano positivity of Hermitian vector bundles in terms of $L^2$-estimate of $\bar\partial$.

math.CV↗