Existence of a motivic t-structure on Voevodsky's category of geometric motives and its relation to Grothendieck's standard conjectures

Voevodsky's triangulated category $DM_{gm}(k)$ of geometric motives with rational coefficients is expected to carry a motivic t-structure: a non-degenerate t-structure, compatible with the tensor product, under which the Betti and $\ell$ -adic realization functors are t-exact. Its heart would be the still-conjectural abelian category of mixed motives over $k$ . Beilinson (2010) proved that such a t-structure implies Grothendieck's standard conjectures (all of them in characteristic zero), and Bondarko (2015) proved the matching converse: the standard conjectures over algebraically closed fields imply a nice motivic t-structure, transversal to the Chow weight structure, over any reasonable base scheme; and over a field the existence of such a nice $t_\ell$ is equivalent to Beilinson's earlier conjectures A, B, C. The existence of the t-structure remains open in every natural generality; only small subcategories (1-motives, Artin–Tate motives, Artin motives) are settled unconditionally.

Background

For a field $k$ , Voevodsky's category $DM_{gm}(k)$ of geometric motives with $\mathbf{Q}$ -coefficients is a triangulated category (an additive category equipped with a shift functor and a class of distinguished triangles, abstracting the structure of a derived category) built from smooth algebraic varieties over $k$ by means of finite correspondences, i.e. formal sums of multi-valued algebraic maps; it is a rigid tensor category (a tensor category in which every object has a dual) receiving every reasonable Weil cohomology theory through realization functors; for $k$ of characteristic zero, each embedding $k\hookrightarrow\mathbf{C}$ yields a Betti realization, and for every prime $\ell$ invertible in $k$ there is an $\ell$ -adic étale realization into the derived category of finite-dimensional $\mathbf{Q}_\ell$ -vector spaces. A t-structure on a triangulated category is a pair of full subcategories of 'cohomologically bounded' objects making cohomology functors meaningful and producing an abelian heart, the way the standard t-structure on a derived category recovers the underlying abelian category. The category of geometric motives being triangulated, the expected home for an abelian category of mixed motives - through which the cohomology of an arbitrary variety would factor with its weight filtration - is the heart of a suitable t-structure on $DM_{gm}(k)$ . Beilinson (arXiv:1006.1116, Def. 1.2) made this precise: a t-structure is motivic if it is non-degenerate (its cohomology functor is conservative, i.e. detects isomorphisms), compatible with the tensor product, and compatible with a realization functor in the sense that tensoring and the realization are t-exact. Grothendieck's standard conjectures, formulated in the 1960s, are assertions of algebraicity for certain cohomological operators on the cohomology of smooth projective varieties: the Künneth type conjecture asserts that the Künneth projectors (the projectors onto the graded pieces of cohomology) are induced by algebraic correspondences, and the Lefschetz type conjecture asserts that an inverse to the Lefschetz operator of an ample class is induced by an algebraic correspondence; in characteristic zero the Lefschetz type conjecture is known to imply all the others, including the identity of homological and numerical equivalence of cycles. Beilinson proved that a motivic t-structure yields, for every smooth projective variety, a Künneth decomposition of its motive and hard Lefschetz isomorphisms in the heart, hence the standard conjectures of Künneth and Lefschetz type over any $k$ , and hence all standard conjectures in characteristic zero; he also characterized the t-structure intrinsically by a filtration of $DM_{gm}(k)$ by the dimension of supporting varieties (without mentioning realizations) and constructed on the heart a weight filtration with the expected vanishing of morphisms between irreducibles of different weights. A complementary structure is known unconditionally: Bondarko's theory of weight structures (axiomatizing stupid truncations of complexes rather than the canonical ones) produces the Chow weight structure $w_{Chow}$ on $DM_{gm}(k)$ , whose heart is the category of Chow motives (motives of smooth projective varieties with morphisms given by algebraic cycles modulo rational equivalence); the expected motivic t-structure should be transversal to $w_{Chow}$ - the two structures cross-cut so that the induced weight filtration on the motivic heart has semisimple graded pieces, mirroring Deligne's weight filtration for mixed Hodge structures. In earlier work (Irvine 1998, published 2002) Beilinson had formulated three conjectures A, B, C expressing strict compatibility of $\ell$ -adic étale (co)homology on the category of effective motives with such a t-structure. Bondarko (arXiv:1105.0420) proved that for a field $K$ the existence of a nice t-structure $t_\ell$ of this strictly compatible, transversal kind is equivalent to Beilinson's conjectures A–C, and that the standard conjectures over algebraically closed fields (including Murre's conjectures and homological = numerical equivalence) imply a nice motivic t-structure on the category of motives over any 'reasonable' base scheme, with the existence over a general base reduced to the case of algebraically closed fields (universal domains). Some small subcategories are understood unconditionally: Bondarko established the whole weight–t-structure picture for the derived category of 1-motives (Deligne's motives built from lattices, algebraic tori, and abelian varieties) over a smooth base and for Artin–Tate motives (extensions of Tate motives by Artin motives, the motives of zero-dimensional varieties) over number fields, and Ruimy constructed the ordinary motivic t-structure with t-exact $\ell$ -adic realization for integral Artin motives; on the motivic stable homotopy category $SH(k)$ Bachmann–Kong–Wang–Xu constructed unconditionally the 'Chow t-structure' with an explicitly computable heart, but that is a different category and does not provide a t-structure on $DM_{gm}(k)$ .

Problem Statement

Does there exist a motivic t-structure on Voevodsky's category $DM_{gm}(k)$ of geometric motives with $\mathbf{Q}$ -coefficients over a base field $k$ of characteristic zero - that is, a non-degenerate t-structure (conservative cohomology functor) compatible with the tensor product such that the Betti realization (for an embedding $k\hookrightarrow\mathbf{C}$ ) and, equivalently, the $\ell$ -adic realization for every prime $\ell$ invertible in $k$ are t-exact? Construct such a t-structure, equivalently establish Beilinson's conjectures A, B, C of 2002, and equivalently construct a nice t-structure $t_\ell$ strictly compatible with $\ell$ -adic étale (co)homology and transversal to the Chow weight structure on $DM_{gm}(k)$ , with the induced weight filtration on the heart having semisimple graded pieces; or prove that no such t-structure exists. By Beilinson's theorem a construction yields all Grothendieck standard conjectures in characteristic zero (and the Lefschetz and Künneth type ones over any field), and by Bondarko's theorem it then extends to a nice motivic t-structure over every reasonable characteristic-zero base scheme, with the converse implications equally available; the surviving open core is exactly the existence question over fields.

References

Alexander Beilinson - Remarks on Grothendieck's standard conjectures - arXiv:1006.1116 (2010) - https://arxiv.org/abs/1006.1116
Alexander Beilinson - Remarks on n-motives and correspondences at the generic point, in: Motives, polylogarithms and Hodge theory, Part I (Papers from the International Press conference, Irvine, CA, 1998), International Press Lecture Series vol. 3, International Press, Somerville, MA, 2002, pp. 35–46, ISBN 1-57146-090-X, zbMATH 1050.14015 - https://zbmath.org/?q=an%3A1050.14015
Mikhail V. Bondarko - Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do - arXiv:1105.0420; Compositio Mathematica 151 (2015) 917–956, doi:10.1112/S0010437X14007763 - https://arxiv.org/abs/1105.0420
M. V. Bondarko - Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) - arXiv:0704.4003; Journal of K-Theory 6 (2010) 387–504, doi:10.1017/is010012005jkt083 - https://arxiv.org/abs/0704.4003
Mikhail V. Bondarko - Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules (published as: Weight structures and "weights" on the hearts of t-structures) - arXiv:1011.3507; Homology, Homotopy and Applications 14 (2012) 239–261, doi:10.4310/hha.2012.v14.n1.a12 - https://arxiv.org/abs/1011.3507
Mikhail V. Bondarko - On t-structures adjacent and orthogonal to weight structures - arXiv:2403.07855 (2024, v3 2025) - https://arxiv.org/abs/2403.07855
Tom Bachmann, Hana Jia Kong, Guozhen Wang, Zhouli Xu - The Chow t-structure on the ∞-category of motivic spectra - arXiv:2012.02687 (2020, v3 2021) - https://arxiv.org/abs/2012.02687
Raphaël Ruimy - Integral Artin motives I: Smooth objects and the ordinary t-structure - arXiv:2211.02505 (2022, v4 2025) - https://arxiv.org/abs/2211.02505
Josiah Foster - The Lefschetz standard conjectures for 4d-dimensional Kummer-type hyper-Kaehler varieties - arXiv:2512.04114 (2025) - https://arxiv.org/abs/2512.04114