On the product of a complex manifold X by a complex curve S considered as a parameter space, we show a Riemann–Hilbert correspondence between regular holonomic relative scriptD‐modules (respectively, complexes) on the one hand and relative perverse complexes (respectively, S‐double-struckC‐constructible complexes) on the other hand.

We give a classification theorem for a relevant class of t-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of t-structures whose hearts have at most nfixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the t-tree, a new technique which generalizesthe filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by Happel, Reiten and Smalø. The last section provides applications to classical n-tilting objects, examples of t-trees for modules over a path algebra, and new developments on compatible t-structures

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Suppose that A is an abelian category whose derived category D(A) has Hom sets and arbitrary (small) coproducts, let T be a (not necessarily classical) (n-)tilting object of A and let H be the heart of the associated t-structure on D(A). We show that there is a triangulated equivalence of unbounded derived categories D(H) ∼ = −→ D(A) which is compatible with the inclusion functor H ֒→ D(A). The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has Hom sets and arbitrary products.