Deformation formulas for parameterized hypersurfaces
type: Item
instance of: scholarly article
title: Deformation formulas for parameterized hypersurfaces
author: Brian Hepler
publication date: 2024-07-05T00:00:00Z
review text: This paper studies deformations of non-normal hypersurfaces \(X=V(f)\) that are obtained by deforming the normalisation map \(Y\to X\). The main case is where the normalisation \(Y\) is smooth or a rational homology manifold. When \(\mathbb Q_X^\bullet\) is perverse, the kernel \(\boldsymbol N_X^\bullet\) of the natural surjection of perverse sheaves from \(\mathbb Q_X^\bullet\) to the intersection cohomology complex \(\boldsymbol I_X^\bullet\) (with constant \(\mathbb Q\)-coefficients) is a perverse sheaf, called the comparison complex. Then \(\boldsymbol N_X^\bullet\) has stalk cohomology concentrated in degree \(-n+1\) if and only if \(Y\) is a rational homology manifold. Under the assumption of isolated instability the author expresses the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison complex.\N\NThe resulting formula gives for \(n=1\) Milnor's formula \(\mu=2\delta-r+1\), where the singularity degree \(\delta\) is interpreted as the number of double points in a general 1-parameter deformation. For the surface case one obtains that \(\lambda^0_{\boldsymbol N_{V(f_0)}^\bullet,\boldsymbol z}(\boldsymbol 0)- \lambda^1_{\boldsymbol N_{V(f_0)}^\bullet,\boldsymbol z}(\boldsymbol 0)= C-T-\delta-\chi(\mathbb L_{\Sigma f,\boldsymbol0})\), where \(T\) is the number of triple points, \(C\) the number of cross caps (\(D_\infty\) singularities), \(\delta\) the number of \(A_1\) singularities and \(\mathbb L_{\Sigma f,\boldsymbol0}\) the complex link of \(\Sigma f\) at \(\boldsymbol 0\).
reviewed by: Jan Stevens
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 32B10
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S60
Mathematics Subject Classification ID: 14B07
zbMATH DE Number: 7877209
zbMATH Keywords: Milnor fiber
zbMATH Keywords: perverse sheaf
zbMATH Keywords: intersection cohomology
zbMATH Keywords: non-isolated singularities
MaRDI profile type: Publication
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published in: Annales de l’institut Fourier
DOI: 10.5802/AIF.3613
description: scientific article; zbMATH DE number 7877209