Rational homotopy and intersection-formality of complex algebraic varieties
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 \(X_{reg}\) and \(L\). Then the \textit{perverse algebraic model} \(I{\mathcal A}_{\overline{\bullet}}(X)\) for \(X\) is the pull-back of the evaluation map \(\delta_1 : \xi_{\leq p}{\mathcal A}_{pl}(L)(t, dt) \to {\mathcal A}_{pl}(L)\) at \(t =1\) along the map \(\iota^*\). We have an isomorphism \(H^*(I{\mathcal A}_{\overline{\bullet}}(X)) \cong IH_{\overline{\bullet}}^*(X; {\mathbb Q})\) in \({\mathcal P}\text{CDGA}\). A complex projective variety \(X\) of dimension \(n\) with isolated singularities is called \textit{intersection-formal} if there exists a string of quasi-isomorphisms between \(I{\mathcal A}_{\overline{\bullet}}(X)\) and its homology, namely \(IH_{\overline{\bullet}}(X; {\mathbb Q})\) in \({\mathcal P}\text{CDGA}\). We call \(X\) GM-\textit{intersection-formal} if there is a string of quasi-isomorphisms between \(I{\mathcal A}_{\overline{\bullet}}(X)\) and its homology in \({\mathcal P}_{2n}\text{CDGA}\). The assertion of the main theorem is as follows. With the same notation as above, assume further that there is a resolution of singularities \(f : \widetilde{X} \to X\) of \(X\) such that \(D= f^{-1}(\Sigma)\) is smooth, and that the link \(L_\sigma\) of \(\sigma\) in \(X\) for all \(\sigma \in \Sigma\) is \((n-2)\)-connected. Then \(X\) is GM-intersection-formal. Furthermore, if \(\Sigma =\{\sigma \}\) is given by a single point then \(X\) is intersection-formal. The strategy for the poof is the use of mixed Hodge structures on the intersection homology. More precisely, the increasing filtration given by the structure induces a spectral sequence called the \textit{perverse weight spectral sequence} converging to \(IE_{2. \overline{p}}^{r,s}(X) \cong Gr_s^WIH_{\overline{p}}^{r+s}(X; {\mathbb Q})\). Under the assumption of the theorem, we see that \(I{\mathcal A}_{\overline{\bullet}}(X)\) is connected to \(IE_{1. \overline{p}}(X)\) with quasi-isomorphisms and that \(IE_{2. \overline{\bullet}}(X) \cong IH_{\overline{\bullet}}(X; {\mathbb Q})\). Moreover, one has a perverse cdga \(M_{\overline{\bullet}}\) from which there exist quasi-isomorphisms to \(IE_{1. \overline{\bullet}}(X)\) and \(IE_{2. \overline{\bullet}}(X)\), respectively. The cdga \(M_{\overline{\bullet}}\) is defined step by step for perversities. It is also proved that isolated surface singularities are GM-intersection-formal by using the perverse weight spectral sequence.
scientific article; zbMATH DE number 6879860
scientific article