We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is of finite type over a field, then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

Two major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.