The integral Hodge conjecture for two-dimensional Calabi–Yau categories

Abstract

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.

Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

“…The authors use noncommutative motives to construct isogenies between intermediate jacobians. In [Per20], the author obtains similar results on intermediate jacobians, using only cohomological methods. Our technique is inspired by that of [BMMS12]; we construct an isomorphism between moduli spaces and use the Abel-Jacobi map to argue that this is sufficient to conclude.…”

Algebraic cycles on Gushel-Mukai varieties

“…Hence the cycle class map in (ii) is injective. The last assertion follows from the integral Hodge conjecture for GM fourfolds, which is proven by Perry in [37].…”

Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties

“…Let and let be a smooth projective variety over of dimension . Recall the definition of the degree Voisin group of (see [Voi16, Per22]): We say that satisfies the integral Hodge conjecture for -cycles up to factor if is annihilated by (in other words, if for every ).…”

“…Definition 5.1. Let d, k, n ∈ Z 1 and let X be a smooth projective variety over C of dimension d. Recall the definition of the degree 2d − 2k Voisin group of X (see [Voi16,Per22]):…”

“…can we bound the order of Z 2g−2 (A) in terms of g and n? For a smooth complex projective d-dimensional variety X, Z 2d−2 (X) is called the degree 2d − 2 Voisin group of X (see [Per22]), is a stably birational invariant [Voi16, Lemma 2.20], and related to the unramified cohomology groups by Colliot-Thélène and Voisin [CTV12] and Schreieder [Sch21]. We prove that if n…”