Chow Motives Research Articles

Higher Tate traces of Chow motives

Abstract We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

Algebraic cycles on Gushel-Mukai varieties

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.

Isotropic and numerical equivalence for Chow groups and Morava K-theories

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Bbb {F}}_{p}$\\end{document}-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and ⊗ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{A}}^{1}$\\end{document}-stable homotopy category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.

Chow motives of genus one fibrations

Chow motives of genus one fibrations

Cubic fourfolds, Kuznetsov components, and Chow motives

We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier–Mukai equivalent are isomorphic as Frobenius algebra objects. As a corollary, there exists a Galois-equivariant isomorphism between their \ell -adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is equivalent to the derived category of a K3 surface.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Torsion Motives

Abstract In this paper, we study Chow motives whose identity map is killed by a natural number. Examples of such objects were constructed by Gorchinskiy and Orlov [ 10]. We introduce various invariants of torsion motives, in particular, the $p$-level. We show that this invariant bounds from below the dimension of the variety a torsion motive $M$ is a direct summand of and imposes restrictions on motivic and singular cohomology of $M$. We study in more details the $p$-torsion motives of surfaces, in particular, the Godeaux torsion motive. We show that such motives are in $1$-to-$1$ correspondence with certain Rost cycle submodules of free modules over $H^*_{et}$. This description is parallel to that of mod-$p$ reduced motives of curves.

On the motive of O'Grady's six dimensional hyper-K\"{a}hler varieties

We prove that the rational Chow motive of a six dimensional hyper-K\"{a}hler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an abelian surface $A$ belongs to the tensor category of motives generated by the motive of $A$. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-K\"{a}hler varieties of OG6-type.

Proper connective differential graded algebras and their geometric realizations

We show that every proper connective dg algebra A admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. If A is moreover smooth, we compute the noncommutative Chow motive of A. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective dg algebras.

Critical varieties and motivic equivalence for algebras with involution

Motivic equivalence for algebraic groups was recently introduced by De Clercq [Compos. Math. 153 (2017), pp. 2195–2213], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F 2 \mathbb {F}_2 , this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.

Motivic Decompositions of Families With Tate Fibers: Smooth and Singular Cases

AbstractWe apply Wildeshaus’s theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger–Murre and Corti–Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky’s motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell $ invertible on $S$. This result, and some of Bondarko’s ideas, lead us to a generalized formulation of Corti–Hanamura’s conjecture. Secondly we establish the existence of the motivic decomposition when $f:X \rightarrow S$ is a projective quadric bundle over a characteristic $0$ base, which is either sufficiently general or whose discriminant locus is a normal crossing divisor. This provides a motivic lift of the Bernstein–Beilinson–Deligne decomposition in this setting.

Simplification of λ-ring expressions in the Grothendieck ring of Chow motives

The Grothendieck ring of Chow motives admits two natural opposite $\lambda$-ring structures, one of which is a special structure allowing the definition of Adams operations on the ring. In this work I present algorithms which allow an effective simplification of expressions that involve both $\lambda$-ring structures, as well as Adams operations. In particular, these algorithms allow the symbolic simplification of algebraic expressions in the sub-$\lambda$-ring of motives generated by a finite set of curves into polynomial expressions in a small set of motivic generators. As a consequence, the explicit computation of motives of some moduli spaces is performed, allowing the computational verification of some conjectural formulas for these spaces.

Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve

<abstract><p>We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $ 3 $ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.</p></abstract>

The generalized Franchetta conjecture for some hyper-Kähler varieties, II

We prove the generalized Franchetta conjecture for the locally complete family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr\`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.

Motives and the Pfaffian–Grassmannian equivalence

We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds, and verifications of Kimura's finite-dimensionality conjecture, Voevodsky's smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Motivic integration on the Hitchin fibration

We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently proven by Groechenig, Ziegler and the second author. To adapt their argument, which relies on p-adic integration, we use a version of motivic integration with values in rational Chow motives and the geometry of N\'eron models to evaluate such integrals on Hitchin fibers.

Supersingular O’Grady Varieties of Dimension Six

Abstract O’Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O’Grady’s construction to fields of positive characteristic $p\neq 2$, called OG6 varieties. Assuming $p\geq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin–Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.

ISOTROPIC MOTIVES

AbstractIn this article we introduce the local versions of the Voevodsky category of motives with$\mathbb{F} _p$-coefficients over a fieldk, parametrized by finitely generated extensions ofk. We introduce the so-calledflexiblefields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructedlocalmotivic categories are much simpler than theglobalone and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for$p=2$and study thelocal Chow motivic category. We introducelocal Chow groupsand conjecture that over flexible fields these should coincide withChow groups modulo numerical equivalence with$\mathbb{F} _p$-coefficients, which implies thatlocal Chow motivescoincide withnumerical Chow motives. We prove this conjecture in various cases.

Morphisms of Rational Motivic Homotopy Types

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.