Let \({\mathcal {T}}\) be a k-linear Hom-finite Krull-Schmidt 2-Calabi-Yau triangulated category with a cluster tilting object X and A be the endomorphism algebra \(\operatorname {End}_{{\mathcal {T}}}(X)\) . Adachi, Iyama and Reiten established bijections between: (a) 2-term silting complexes in \(\operatorname {K}^{\operatorname {b}}(\operatorname {proj}A)\) , (b) support \(\tau \) -tilting pairs in \(\operatorname {mod}A\) and (c) cluster tilting objects in \({\mathcal {T}}\) . In this paper, let \({\mathcal {C}}\) be a k-linear Hom-finite Krull-Schmidt 2n-Calabi-Yau \((n+2)\) -angulated category with an Oppermann-Thomas cluster tilting object T, \(\Lambda \) be the endomorphism algebra \(\operatorname {End}_{{\mathcal {C}}}(T)\) and \({\mathcal {D}}\) be the essential image of \(\operatorname {Hom}_{{\mathcal {C}}}(T,-)\) . We introduce \((n+1)\) -term (pre)silting complexes related to the n-cluster tilting subcategory \({\mathcal {D}}\) , which are called \({\mathcal {D}}\) - \((n+1)\) -term (pre)silting complexes and are generalizations of 2-term (pre)silting complexes. And we give bijections between: (a) \({\mathcal {D}}\) - \((n+1)\) -term presilting complexes in \(\operatorname {K}^{\operatorname {b}}(\operatorname {proj}\Lambda )\) , (b) \(\tau _n\) -rigid pairs in \({\mathcal {D}}\) and (c) n-rigid objects in \({\mathcal {C}}\) . Finally, we give an example in a 5-angulated cluster category to illustrate our main results.
