Constant families of $t$-structures on derived categories of coherent sheaves

We generalize the construction (due to D. Abramovich and the author) of a “constant” $t$-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a $t$-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with $t$-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced $t$-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of L. Alonso Tarrío, A. Jeremнas López, M.-J. Souto Salorio. As an application of the “constant” $t$-structures techniques we prove that every bounded nondegenerate $t$-structure on $D(X)$ with Noetherian heart is invariant under the action of a connected group of autoequivalences of $D(X)$. Also, we show that if $X$ is smooth then the only local $t$-structures on $D(X)$, i.e., those for which there exist compatible $t$-structures on $D(U)$ for all open $U\subset X$, are the perverse $t$-structures considered by R. Bezrukavnikov.