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<p>Saito’s microlocalization construction has been used to great effect in understanding hypersurface singularities. In this paper, we introduce what we believe to be a suitable analogue of the microlocalization construction for local complete intersection subvarieties. As evidence, we relate our construction to Saito’s in the codimension one case. Moreover, we use this construction to study various natural questions concerning the minimal exponent of LCI subvarieties. We show that the minimal exponent agrees with the smallest Bernstein–Sato root, which was expected to be true. We also show that, in the isolated complete intersection singularities case, the minimal exponent agrees with the smallest non-zero spectral number. As applications of these results, we prove constructibility of the function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3169_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(x\mapsto \widetilde{\alpha }_x(Z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <msub> <mover accent="true"> <mi>α</mi> <mo stretchy="true">~</mo> </mover> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> along certain Whitney stratifications and we prove that the spectrum (hence, the minimal exponent) is constant in equisingular families of ICIS varieties.</p>

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Some applications of microlocalization for local complete intersection subvarieties

  • 期刊论文

  • 发表日期：

    2025年5月8日

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  • Bradley Dirks

摘要

Saito’s microlocalization construction has been used to great effect in understanding hypersurface singularities. In this paper, we introduce what we believe to be a suitable analogue of the microlocalization construction for local complete intersection subvarieties. As evidence, we relate our construction to Saito’s in the codimension one case. Moreover, we use this construction to study various natural questions concerning the minimal exponent of LCI subvarieties. We show that the minimal exponent agrees with the smallest Bernstein–Sato root, which was expected to be true. We also show that, in the isolated complete intersection singularities case, the minimal exponent agrees with the smallest non-zero spectral number. As applications of these results, we prove constructibility of the function \(x\mapsto \widetilde{\alpha }_x(Z)\) x ↦ α ~ x ( Z ) along certain Whitney stratifications and we prove that the spectrum (hence, the minimal exponent) is constant in equisingular families of ICIS varieties.