Foliations with persistent singularities (Q1996067)

Let \(X\) be a smooth projective variety over an algebraically closed field \(k\) of characteristic zero, and let \(\mathcal{F}\) be a foliation of codimension \(q\) on \(X\), with normal sheaf \(N_\mathcal{F}\). If \(S\) is a scheme over \(k\), and \(0 \in S\) a closed point, then following [\textit{T. Suwa}, in: Singularity theory. Proceedings of the symposium, Trieste, Italy, August 19-- September 6, 1991. Singapore: World Scientific. 817--865 (1995; Zbl 0944.32022)] the authors define an unfolding of \(\mathcal{F}\) over \(S\) as a foliation \(\tilde{\mathcal{F}}\) on \(X\times S\) such that \(\tilde{\mathcal{F}}_{|X \times \{0\}} = \mathcal{F}\). When \(S = \text{Spec}(k[x]/(x^2))\), any unfolding of \(\mathcal{F}\) is called an infinitesimal first order unfolding. A singularity \(p \in X\) of \(\mathcal{F}\) is called persistent if, for every infinitesimal first order unfolding \(\tilde{\mathcal{F}}\) of \(\mathcal{F}\), the point \((p,0)\) is a singularity of \(\tilde{\mathcal{F}}\). In this paper, the authors define a scheme structure for the persistent singularities of \(\mathcal{F}\), in such a way that they are defined by an ideal sheaf \(\mathcal{I}\). On the other hand, a singularity \(p \in \mathcal{F}\) is called a Kupka singularity, if \(d\omega(p) \neq 0\), where \(\omega \in H^0(X,\Omega^q_X \otimes \det(N_\mathcal{F}))\) is a twisted \(q\)-form defining \(\mathcal{F}\). In this paper, the authors give a more refined definition for the Kupka scheme, which is defined by an ideal sheaf \(\mathcal{K}\). They show that in case \(q = 1\), if the ideal sheaf \(\mathcal{J}\) associated to the singular locus of \(\mathcal{F}\) is a sheaf of radical ideals, \(c_1(N_\mathcal{F}^*) \neq 0\) and \(H^1(X,N_\mathcal{F}^*) = 0\), then \(\mathcal{F}\) has Kupka singularities. Furthermore, they relate Kupka singularities and persistent singularities in the following way: when \(q = 1\), they prove that \(\mathcal{J} \subset \mathcal{I} \subset \mathcal{K}\). This result generalizes the one obtained in [\textit{C. Massri} et al., Asian J. Math. 22, No. 6, 1025--1046 (2018; Zbl 1408.32032)] for foliations on \(\mathbb{P}^n\). Moreover, these inclusions remain true for foliations of higher dimension if the normal bundle of \(\mathcal{F}\) is locally free; however, in general this is not known to be true. Finally, the authors show that the inexistence of persistent singularities for a foliation \(\mathcal{F}\), satisfying \(\text{Ext}_{\mathcal{O}_X}^1(N_\mathcal{F}^*,\text{Sym}^2(N_\mathcal{F}^*)) = 0\), implies the existence of a connection on \(N_\mathcal{F}^*\). In particular, in this case, all Chern classes of \(N_\mathcal{F}^*\) vanish.