<p>In this paper, we consider the limit behavior of a sequence of deformed Hermitian–Yang–Mills metrics <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$F_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{\otimes m}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <mo>⊗</mo> <mi>m</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> where <i>L</i> is an ample line bundle over a Kähler surface <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$(X, \omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. If the cohomology class <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$c_1(L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> admits a solution of the J-equation, then we prove that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$F_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> will converge to it. Furthermore, we also consider a boundary case. In this case, we prove that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_445_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$F_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> will converge to a singular Kähler metric away from a finite number of curves with negative self-intersection on the surface.</p>

Limit Behavior of Deformed Hermitian–Yang–Mills Metrics on Kähler Surfaces

期刊论文
发表日期：

2025年5月14日

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Xiaoli Han,
Xishen Jin

摘要

In this paper, we consider the limit behavior of a sequence of deformed Hermitian–Yang–Mills metrics $F_m$ $F_{m}$ on $L^{\otimes m}$ $L^{\otimes m}$ where L is an ample line bundle over a Kähler surface $(X, \omega )$ $(X, ω)$ . If the cohomology class $c_1(L)$ $c_{1} (L)$ admits a solution of the J-equation, then we prove that $F_m$ $F_{m}$ will converge to it. Furthermore, we also consider a boundary case. In this case, we prove that $F_m$ $F_{m}$ will converge to a singular Kähler metric away from a finite number of curves with negative self-intersection on the surface.