We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

We prove some results on the nilpotent orbit theorem for complex variations of Hodge structure.

We prove analogues of Schur’s lemma for endomorphisms of extensions in Tannakian categories. More precisely, let $\mathbf {T}$ be a neutral Tannakian category over a field of characteristic zero. Let E be an extension of A by B in $\mathbf {T}$. We consider conditions under which every endomorphism of E that stabilises B induces a scalar map on $A\oplus B$. We give a result in this direction in the general setting of arbitrary $\mathbf {T}$ and E, and then a stronger result when $\mathbf {T}$ is filtered and the associated graded objects to A and B satisfy some conditions. We also discuss the sharpness of the results.

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$. This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field $k$ which contains all $\ell$-power roots of unity.

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 $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus $\textrm {HL}(S) \subset S$, which is a countable union of ‘special’ algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.

Let $X/{\mathbb C}$ be a smooth projective variety. We consider two integral invariants, one of which is the level of the Hodge cohomology algebra $H^*(X,{\mathbb C})$ and the other involving the complexity of the higher Chow groups ${\mathrm {CH}}^*(X,m;{\mathbb Q})$ for $m\geq 0$. We conjecture that these two invariants are the same and accordingly provide some strong evidence in support of this.

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.

We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$, for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call ‘Schiffer variations.’ We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.

Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.

Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.

Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

We consider Calabi–Yau n-folds X arising from certain hyperplane arrangements. Using Fu–Vial’s theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of X generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin’s conjecture for X, and Voevodsky’s smash-nilpotence conjecture for odd-dimensional X.

We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.

We introduce a weak Lefschetz-type result on Chow groups of complete intersections. As an application, we can reproduce some of the results in [P]. The purpose of this paper is not to reproduce all of [P] but rather illustrate why the aforementioned weak Lefschetz result is an interesting idea worth exploiting in itself. We hope the reader agrees.

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.