<p>For plurisubharmonic functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> lying in the Cegrell class of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {B}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {B}^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> respectively such that the Lelong number of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> at the origin vanishes, we show that the mass of the origin with respect to the measure <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(dd^c\max \{\varphi (z), \psi (Az)\})^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>d</mi> <mi>c</mi> </msup> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> is zero for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A\in \text{ Hom }(\mathbb {C}^n,\mathbb {C}^m)=\mathbb {C}^{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mspace width="0.333333em" /> <mtext>Hom</mtext> <mspace width="0.333333em" /> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>m</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mi mathvariant="italic">nm</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> outside a pluripolar set. We establish a new approach and introduce a new concept coined the <i>log truncated threshold</i> of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> at 0 which reflects a singular property of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> via a log function near the origin (denoted by <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$lt(\varphi ,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>), and derive an optimal estimate of the residual Monge–Ampère mass of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> at 0 in terms of its higher order Lelong numbers <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\nu _j(\varphi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ν</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> at 0 for <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$1\le j\le n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, in the case that <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$lt(\varphi ,0)<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. These results unify and imply the well-known results about Guedj–Rashkovskii’s zero mass conjecture.</p>

On Guedj–Rashkovskii’s zero mass conjecture

期刊论文
发表日期：

2026年4月22日

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Fusheng Deng,
Yinji Li,
Qunhuan Liu,
Zhiwei Wang,
Xiangyu Zhou

摘要

For plurisubharmonic functions $\varphi $ $φ$ and $\psi $ $ψ$ lying in the Cegrell class of $\mathbb {B}^n$ $B^{n}$ and $\mathbb {B}^m$ $B^{m}$ respectively such that the Lelong number of $\varphi $ $φ$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max \{\varphi (z), \psi (Az)\})^n$ ${(d d^{c} max {φ (z), ψ (A z)})}^{n}$ on $\mathbb {C}^n$ $C^{n}$ is zero for $A\in \text{ Hom }(\mathbb {C}^n,\mathbb {C}^m)=\mathbb {C}^{nm}$ $A \in Hom (C^{n}, C^{m}) = C^{nm}$ outside a pluripolar set. We establish a new approach and introduce a new concept coined the log truncated threshold of $\varphi $ $φ$ at 0 which reflects a singular property of $\varphi $ $φ$ via a log function near the origin (denoted by $lt(\varphi ,0)$ $l t (φ, 0)$ ), and derive an optimal estimate of the residual Monge–Ampère mass of $\varphi $ $φ$ at 0 in terms of its higher order Lelong numbers $\nu _j(\varphi )$ $ν_{j} (φ)$ at 0 for $1\le j\le n-1$ $1 \leq j \leq n - 1$ , in the case that $lt(\varphi ,0)<\infty $ $l t (φ, 0) < \infty$ . These results unify and imply the well-known results about Guedj–Rashkovskii’s zero mass conjecture.