Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWeight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
162
0
0.0
(
0
)
Authors
M.V. Bondarko
Ask ChatGPT about the research
No Arabic abstract
This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodskys categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial up to morphisms that are zero on cohomology. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conservative weakly exact functor $t$ from our $C$ to a certain weak category of complexes $K_w(Hw)$. For any (co)homological functor $H:Cto A$ for an abelian $A$ we construct a weight spectral sequence $T:H(X^i[j])implies H(X[i+j])$ where $(X^i)=t(X)$; it is canonical and functorial starting from $E_2$. This spectral sequences specializes to the usual (Delignes) weight spectral sequences for classical realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that $K_0(C)cong K_0(Hw)$ and $K_0(End C)cong K_0(End Hw)$. The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain $t$-structure which is adjacent to $w$ and vice versa. This is the case for the Voevodskys $DM^{eff}_-$ (one obtains certain new Chow weight and t-structures for it; the heart of the latter is dual to $Chow^{eff}$) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology.
rate research
Read More
We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding motives (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t-structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the homotopy hearts $Ht_{hom}^{eff}$ of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions $-_{-1}$. Respectively, we express the condition for an object of $Ht_{hom}^{eff}$ to be weakly birational (i.e., that its $n+1$th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees $>n$ for some $nge 0$) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures $w_{Smooth}^{s}$ (where $s=(s_{j})$ are non-decreasing sequences parameterizing our aisles) that vastly generalize the Chow weight structures $w_{Chow}$ defined earlier. Using general abstract nonsense we also construct the corresponding adjacent $t-$structures $t_{Smooth}^{s}$ and prove that they give the birationality filtrations on $Ht^{eff}_{hom}$. Moreover, some of these weight structures induce weight structures on the corresponding $n-$birational motivic categories (these are the localizations by the levels of the slice filtrations). Our results also yield some new unramified cohomology calculations.
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
A plethora of spaces in Functional Analysis (Braun-Meise-Taylor and Carleman ultradifferentiable and ultraholomorphic classes; Orlicz, Besov, Lipschitz, Lebesque spaces, to cite the main ones) are defined by means of a weighted structure, obtained from a weight function or sequence subject to standard conditions entailing desirable properties (algebraic closure, stability under operators, interpolation, etc.) for the corresponding spaces. The aim of this paper is to stress or reveal the true nature of these diverse conditions imposed on weights, appearing in a scattered and disconnected way in the literature: they turn out to fall into the framework of O-regular variation, and many of them are equivalent formulations of one and the same feature. Moreover, we study several indices of regularity/growth for both functions and sequences, which allow for the rephrasing of qualitative properties in terms of quantitative statements.
Let $X$ be a compact manifold, $G$ a Lie group, $P to X$ a principal $G$-bundle, and $mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_bullet$ we previously studied orientations on the real determinant line bundle over $mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X times S^1$. Thus, if $P to X$ and $Q to X times S^1$ are principal $G$-bundles with $Q|_{Xtimes{1}} cong P$, we relate spin structures on $(mathcal{B}_P,F_bullet)$ to orientations on $(mathcal{B}_Q,E_bullet)$ for a certain class of operators $F_bullet$ on $X$ and $E_bullet$ on $Xtimes S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.
We define the Chow $t$-structure on the $infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{cheartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, cheartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
