On t-structures adjacent and orthogonal to weight structures

We study t-structures (on triangulated categories) that are closely related to weight structures. A t-structure couple t= (Cₓ ₀, Cₓ ₀) is said to be adjacent to a weight structure w= (Cₖ ₀, Cₖ ₀) if Cₓ ₀=Cₖ ₀. For a category C that satisfies the Brown representability property we prove that t that is adjacent to w exists if and only if w is smashing (that is, "respects C-coproducts"). The heart Ht of this t is the category of those functors Hw^op Ab that respect products (here Hw is the heart of w) ; the result has important applications. We prove several more statements on constructing t-structures starting from weight structures; we look for a strictly orthogonal t-structure t on some C' (where C, C' are triangulated subcategories of a common D) such that C'ₓ ₀ (resp. C'ₓ ₀) is characterized by the vanishing of morphisms from Cₖ ₁ (resp. Cₖ -₁). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi) coherent sheaves on a scheme X that is projective over an affine noetherian one. We also study hearts of orthogonal t-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal t-structures.