The paper "The Minimal Exponent and k-Rationality for Local Complete Intersections" by Qianyu Chen, Bradley Dirks, and Mircea Mustaţă explores the relationship between the minimal exponent of a local complete intersection subvariety \( Z \) of a smooth complex variety \( X \) and the \( k \)-rationality of \( Z \). The main result states that \( Z \) has \( k \)-rational singularities if and only if the minimal exponent \( \widetilde{\alpha}(Z) \) is greater than \( k + r \), where \( r \) is the pure codimension of \( Z \) in \( X \). This condition is also characterized in terms of the Hodge filtration on the intersection complex Hodge module of \( Z \). The authors provide several corollaries and consequences of this main result. For instance, they show that if \( Z \) has \( k \)-rational singularities, then the Hodge filtration on the local cohomology sheaf \( \mathcal{H}_Z^p(\mathcal{O}_X) \) is generated at level \( \dim(X) - [\widetilde{\alpha}(Z)] - 1 \). Additionally, they prove that if \( Z \) has \( k \)-rational singularities and \( k \geq 1 \), then \( \mathcal{H}^k(\Omega^{d-k}_Z) \neq 0 \), where \( d \) is the dimension of \( Z \). The paper also discusses the connection between \( k \)-rationality and \( k \)-Du Bois singularities, extending previous results for hypersurfaces. It provides a characterization of \( k \)-rationality in terms of the minimal exponent and the Hodge filtration on the local cohomology sheaf. Furthermore, the authors show that if \( Z \) has \( k \)-rational singularities, then the morphism \( \psi_k: \Omega^k_Z \to \mathbf{R} \mathcal{H} o m_{\mathcal{O}_Z}(\Omega_{Z}^{d-k}, \omega_{Z}) \) is an isomorphism. The paper concludes with a detailed proof of the main theorem, using techniques from mixed Hodge modules, the Kashiwara-Malgrange \( V \)-filtration, and the theory of mixed Hodge modules. The results are significant for understanding the singularities of local complete intersections in smooth varieties.The paper "The Minimal Exponent and k-Rationality for Local Complete Intersections" by Qianyu Chen, Bradley Dirks, and Mircea Mustaţă explores the relationship between the minimal exponent of a local complete intersection subvariety \( Z \) of a smooth complex variety \( X \) and the \( k \)-rationality of \( Z \). The main result states that \( Z \) has \( k \)-rational singularities if and only if the minimal exponent \( \widetilde{\alpha}(Z) \) is greater than \( k + r \), where \( r \) is the pure codimension of \( Z \) in \( X \). This condition is also characterized in terms of the Hodge filtration on the intersection complex Hodge module of \( Z \). The authors provide several corollaries and consequences of this main result. For instance, they show that if \( Z \) has \( k \)-rational singularities, then the Hodge filtration on the local cohomology sheaf \( \mathcal{H}_Z^p(\mathcal{O}_X) \) is generated at level \( \dim(X) - [\widetilde{\alpha}(Z)] - 1 \). Additionally, they prove that if \( Z \) has \( k \)-rational singularities and \( k \geq 1 \), then \( \mathcal{H}^k(\Omega^{d-k}_Z) \neq 0 \), where \( d \) is the dimension of \( Z \). The paper also discusses the connection between \( k \)-rationality and \( k \)-Du Bois singularities, extending previous results for hypersurfaces. It provides a characterization of \( k \)-rationality in terms of the minimal exponent and the Hodge filtration on the local cohomology sheaf. Furthermore, the authors show that if \( Z \) has \( k \)-rational singularities, then the morphism \( \psi_k: \Omega^k_Z \to \mathbf{R} \mathcal{H} o m_{\mathcal{O}_Z}(\Omega_{Z}^{d-k}, \omega_{Z}) \) is an isomorphism. The paper concludes with a detailed proof of the main theorem, using techniques from mixed Hodge modules, the Kashiwara-Malgrange \( V \)-filtration, and the theory of mixed Hodge modules. The results are significant for understanding the singularities of local complete intersections in smooth varieties.