Narrow Results

This paper concerns hyperkähler fourfolds $X$ having a non-symplectic involution $\iota$. The Bloch–Beilinson conjectures predict the way $\iota$ should act on certain pieces of the Chow groups of $X$. The main result is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has consequences for the Chow ring of the quotient $X/\iota$.

For a smooth projective variety $X$, we consider when the diagonal ${\mathrm{\Delta }}_X$ is nef as a cycle on $X\times X$. In particular, we give a classification of complete intersections and smooth del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for spherical varieties.

We prove a number of basic vanishing results for modified diagonal classes. We also obtain some sharp results for modified diagonals of curves and abelian varieties, and we prove a conjecture of O’Grady about modified diagonals on double covers.

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

This paper is a work in progress on Bloch’s conjecture asserting the vanishing of the Pontryagin product of a $p$ codimensional cycle on an abelian variety by $p+1$ zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss in particular, the case of $3$-dimensional cycles.

We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space ${\mathbb{P}}^{n+1}$ of even dimension $n$, passing through the Fermat variety (with $d\ne 3,4,6$). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$. Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1,1,\dots ,1,2)$ attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether–Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL₂(𝔸_E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M_ét(π^∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M_ét(π^∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M_ét(π^∞) is spanned by algebraic cycles.