Rational homotopy and intersection-formality of complex algebraic varieties
type: Item
instance of: scholarly article
title: Rational homotopy and intersection-formality of complex algebraic varieties
zbMATH Open document ID: 1396.55007
author: David Chataur
author: Joana Cirici
published in: Revista Matemática Complutense
publication date: 2018-06-01T00:00:00Z
review text: This paper studies the (GM-)intersection-formality of a complex projective variety \(X\) with only isolated singularities by applying the perverse algebraic model for a topological pseudomanifold which has been introduced by \textit{D. Chataur, M. Saralegi-Aranguren} and \textit{D. Tanré} in [Mem. Am. Math. Soc. 1214, v-viii, 112 p. (2018; Zbl 1418.55003)]. Let \(\Sigma\) be the singular locus of \(X\) and \(X_{reg}:=X -\Sigma\) its regular part. Then the \(k\)-th intersection cohomology group of \(X\) with perversity \(p\) is given by \[ IH_{\overline{p}}^k(X; R) \cong \begin{cases} H^k(X_{reg}; R) & \text{if \(k\leq p\)} \\ \text{Im}\;(H^k(X; R) \to H^k(X_{reg}; R)) & \text{if \(k= p+1\)}\\ H^k(X; R) & \text{if \(k> p+1\)}, \end{cases} \] where \(R\) is a commutative ring. By definition, \textit{a perverse commutative differential graded algebra} (perverse cdga for short) over \({\mathbb Q}\) is a commutative monoid in the category of functors from the poset \({\mathcal P}=({\mathbb Z}_{\geq 0}, \geq)\) to the category of cochain complexes of \({\mathbb Q}\)-vector spaces. Let \({\mathcal P}\text{CDGA}\) be the category of perverse cdga's over \({\mathbb Q}\). The category \({\mathcal P}_n\text{CDGA}\) of \(n\)-perverse cdga's is also defined by replacing \({\mathcal P}\) by the poset \({\mathcal P}=\{0, 1, 2,\dots, n-2 \}\). Let \(\Lambda (t, dt)\) be the free cdga over \({\mathbb Q}\) generated by \(t\) in degree \(0\) and \(dt\) in degree \(1\). For a cgda \(A\) and an integer \(p \geq 0\), we construct a truncation \(\xi_{\leq p}A(t, dt)\) of \(A\otimes \Lambda (t, dt)\) by perverse degree \(p\). For the projective variety \(X\) mentioned above, we chose a closed algebraic neighborhood \(T\) of the singular locus \(\Sigma\). The inclusion \(\iota : L:=\partial T \to X_{reg}\) induces a morphism \(\iota^* : {\mathcal A}_{pl}(X_{reg}) \to {\mathcal A}_{pl}(L)\) of cdga's over \({\mathbb Q}\) between the rational algebras of piecewise linear forms of \(X_{reg}\) …
reviewed by: Katsuhiko Kuribayashi
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55N33
Mathematics Subject Classification ID: 32S35
zbMATH DE Number: 6879860
zbMATH Keywords: intersection cohomology
zbMATH Keywords: isolated singularities
zbMATH Keywords: rational homotopy
zbMATH Keywords: mixed Hodge theory
zbMATH Keywords: formality
zbMATH Keywords: weight spectral sequence
MaRDI profile type: Publication
full work available at URL: s13163-018-0255-8
OpenAlex ID: W2794076368
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DOI: 10.1007/S13163-018-0255-8
description: scientific article; zbMATH DE number 6879860