<p>We study the stability of weighted <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> Bergman kernels on a sequence of domains in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and show that, under certain conditions on the domains and weights, the weighted <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> Bergman kernels converge compactly. For <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we provide a link between the stability of multiplier ideal sheaves and the stability of weighted Bergman kernels with weights of the form <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$e^{-\varphi _j}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>φ</mi> <mi>j</mi> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\{\varphi _j\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>φ</mi> <mi>j</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a sequence of plurisubharmonic functions. Our main result shows that on a bounded hyperconvex domain, such weighted Bergman kernels converge compactly whenever the associated multiplier ideal sheaves satisfy a stability condition, without requiring the weights <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\{\varphi _j\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>φ</mi> <mi>j</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> to be monotone.</p>

Stability of Multiplier Ideal Sheaves and Weighted $L^p$ Bergman kernels

期刊论文
发表日期：

2026年5月11日

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Liyou Zhang,
Ziyi Zhang

摘要

We study the stability of weighted $L^p$ $L^{p}$ Bergman kernels on a sequence of domains in $\mathbb {C}^n$ $C^{n}$ and show that, under certain conditions on the domains and weights, the weighted $L^p$ $L^{p}$ Bergman kernels converge compactly. For $p=2$ $p = 2$ , we provide a link between the stability of multiplier ideal sheaves and the stability of weighted Bergman kernels with weights of the form $e^{-\varphi _j}$ $e^{- φ_{j}}$ , where $\{\varphi _j\}$ ${φ_{j}}$ is a sequence of plurisubharmonic functions. Our main result shows that on a bounded hyperconvex domain, such weighted Bergman kernels converge compactly whenever the associated multiplier ideal sheaves satisfy a stability condition, without requiring the weights $\{\varphi _j\}$ ${φ_{j}}$ to be monotone.

Stability of Multiplier Ideal Sheaves and Weighted \(L^p\) Bergman kernels

摘要