<p>We study polarised degenerations of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2024_1314_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> <EquationSource Format="TEX">$n$</EquationSource> </InlineEquation>-dimensional Calabi-Yau hypersurfaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2024_1314_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="174" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">{</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>…</mo> <msub> <mi>F</mi> <mi>m</mi> </msub> <mo>+</mo> <mi>t</mi> <mi>F</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$\{ F_{0}F_{1}\ldots F_{m}+tF=0\} $</EquationSource> </InlineEquation>, where the essential skeleton has dimension <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2024_1314_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$1\leq m\leq n-1$</EquationSource> </InlineEquation>. We will describe the limiting behaviour of the Calabi-Yau potential at the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2024_1314_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{0}$</EquationSource> </InlineEquation>-level, in terms of an optimal transport problem, and in terms of non-archimedean potential theory.</p>

Intermediate complex structure limit for Calabi-Yau metrics

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2025年2月26日

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Yang Li

摘要

We study polarised degenerations of $n$ $n$ -dimensional Calabi-Yau hypersurfaces ${F_{0} F_{1} \dots F_{m} + t F = 0}$ $\{ F_{0}F_{1}\ldots F_{m}+tF=0\} $ , where the essential skeleton has dimension $1 \leq m \leq n - 1$ $1\leq m\leq n-1$ . We will describe the limiting behaviour of the Calabi-Yau potential at the $C^{0}$ $C^{0}$ -level, in terms of an optimal transport problem, and in terms of non-archimedean potential theory.