This paper is a generalization of [D. Abramovich and A. Polishchuk [J. Reine Angew. Math. 590, 89–130 (2006; Zbl 1093.14026)], where a “constant” \(t\)-structure on the bounded derived category \(D(X \times S)\) of coherent sheaves on \(X \times S\) was constructed starting with a \(t\)-structure on the category \(D(S)\), for \(X\) and \(S\) smooth and quasiprojective. In this paper, smoothness and quasiprojectivity assumptions are not needed for such a construction. By the way, the paper is not just straightforward generalization of the previous one. In particular the author introduces the induction of a \(t\)-structure by a “nice” functor, which turns out to be a very useful tool in the construction.
Recall that tilting a noetherian \(t\)-structure does not always give a noetherian \(t\)-structure. The author then defines close to noetherian \(t\)-structures and shows how to get such a \(t\)-structure extending a close to noetherian pre-aisle. This allows him to perform the construction of constant \(t\)-structures in this case. An important issue for the construction of the constant \(t\)-structure is the induction of \(t\)-structures by “nice” functors. With this in mind, consider a finite Tor-dimension finite morphism \(f:X \to Y\), construct a \(t\)-structure on the unbounded derived category \(D_{qc}(X)\) of quasicoherent sheaf and push it forward to \(D_{qc}(Y)\) to get by restriction to the bounded derived category \(D(Y)\) a \(t\)-structure.
As application, the author shows that every bounded nondegenerate \(t\)-structure on \(D(X)\) with noetherian heart is invariant under the action of a connected group of autoequivalences of \(D(X)\). If we define local \(t\)-structures to be those for which there exist compatible \(t\)-structure on \(D(U)\) for any open \(U\) of \(X\), then the author shows that if \(X\) is smooth any such \(t\)-structure is a perverse \(t\)-structure as defined in R. Bezrukavnikov [Perverse coherent sheaves (after Deligne), preprint 2000, math.AG/0005152].