We compute the integral Chow rings of $\overline {\mathcal {M}}_{1,n}$ for $n=3,4$. The alternative compactifications introduced by Smyth – and studied further by Lekili and Polishchuk – present each of these stacks as a sequence of weighted blow-ups and blow-downs from a weighted projective space. We compute all the integral Chow rings by repeated application of the blow-up formula.

We extend the notion of the J-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the J-invariant.

We show that the derived categories of perverse Nori motives investigate and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations. As a result, we obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We also prove that if a motivic t-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally, we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives.

Let S and T be smooth projective varieties over an algebraically closed field k. Suppose that S is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of k, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational and motivic functor. This generalizes Kahn’s result on the torsion order of S. We also exhibit an example of S over $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.

We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the J-invariant in terms of the ordinary (Chow) J-invariant.

A real variety whose real locus achieves the Smith–Thom equality is called maximal. This paper introduces new constructions of maximal real varieties, by using moduli spaces of geometric objects. We establish the maximality of the following real varieties:

• – moduli spaces of stable vector bundles of coprime rank and degree over a maximal real curve (recovering Brugallé–Schaffhauser’s theorem with a short new proof), which extends to moduli spaces of parabolic vector bundles;

• – moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal real curve, providing maximal hyper-Kähler manifolds in every even dimension;

• – if a real variety has maximal Hilbert square, then the variety and its Hilbert cube are maximal, which happens for all maximal real cubic 3-folds, but never for maximal real cubic 4-folds;

• – punctual Hilbert schemes on a maximal real surface with vanishing first $\mathbb {F}_2$-Betti number and connected real locus, such as $\mathbb {R}$-rational maximal real surfaces and some generalized Dolgachev surfaces;

• – moduli spaces of stable sheaves on an $\mathbb {R}$-rational maximal Poisson surface (e.g. the real projective plane).

We compute the integral Chow ring of the moduli stack of smooth elliptic curves with n marked points for $3\leq n\leq 10$.

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

We construct a collection of families of higher Chow cycles of type $(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.

We prove that the cohomology rings of the moduli space $M_{d,\chi }$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $\chi $-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,\chi }$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.

We generalize Bloch’s map on torsion cycles from algebraically closed fields to arbitrary fields. While Bloch’s map over algebraically closed fields is injective for zero-cycles and for cycles of codimension at most two, we show that the generalization to arbitrary fields is only injective for cycles of codimension at most two but, in general, not for zero-cycles. Our result implies that Jannsen’s cycle class map in integral $\ell $-adic continuous étale cohomology is, in general, not injective on torsion zero-cycles over finitely generated fields. This answers a question of Scavia and Suzuki.

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$, by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different.

For any odd integer $d$, we give a presentation for the integral Chow ring of the stack $\mathcal {M}_{0}(\mathbb {P}^r, d)$, as a quotient of the polynomial ring $\mathbb {Z}[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of $d$ and $r$, and a conjecture for a minimal set of generators.

For a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case $w=w^{sph}$ (the spherical weight structure on $SH$), we deduce the following converse to the stable Hurewicz theorem: $H^{sing}_{i}(M)=\{0\}$ for all $i<0$ if and only if $M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.

The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$, this is the case if there exists a distinguished triangle $LM\to M\to RM$, where $RM$ is an n-connected spectrum and $LM$ is an $m-1$-skeleton (of M) in the sense of Margolis’s definition; this happens whenever $H^{sing}_i(M)=\{0\}$ for $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$; those ‘send n-w-skeleta into $m-1$-w-skeleta’.

The goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on the one hand, how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the local action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the global action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud’s approach to rigid analytic geometry.

We determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$-bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$-algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$. From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$-algebra. Finally, when $k = \mathbb {C}$, the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.