Gushel--Mukai varieties: intermediate Jacobians

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds. Comment: 48 pages. Latest addition to our series of articles on the geometry of Gushel-Mukai varieties; v2: minor stylistic improvements, results unchanged; v3: minor improvements; v4: final version, published in EPIGA

TATE’S CONJECTURE AND THE TATE–SHAFAREVICH GROUP OVER GLOBAL FUNCTION FIELDS

Let ${\mathcal{X}}$ be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of ${\mathcal{X}}$ is finite if and only Tate’s conjecture for divisors on $X$ holds and the Tate–Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.

DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety$X$and an effective Cartier divisor$D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of$X$along$D$has a canonical decomposition in terms of the Chow group of 0-cycles$\text{CH}_{0}(X)$and the Chow group of 0-cycles with modulus$\text{CH}_{0}(X|D)$on$X$. When$X$is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that$\text{CH}_{0}(X|D)$is torsion-free and there is an injective cycle class map$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$if$X$is affine. For a smooth affine surface$X$, this is strengthened to show that$K_{0}(X,D)$is an extension of$\text{CH}_{1}(X|D)$by$\text{CH}_{0}(X|D)$.

On the structure of generalized Albanese varieties

Given a smooth projective surface X over an algebraically closed field k and a modulus (an effective divisor) m on X, one defines the idle class group Cm(X) of X with modulus m (see 1, chapter III, section 4). The corresponding generalized Albanese variety Gum and the generalized Albanese map um:X|m|Gum have the following universal mapping property (2): if :XG is a rational map into a commutative algebraic group which induces a homomorphism Cm(X)G(k) (1, chapter III, proposition 1), then factors uniquely through um.

Towards a problem in deformations of polarized algebraicK3 surfaces

A nonsingular algebraic surfaceVis called aK3surface if i) 1 =i.e. a canonical divisorKvonVis linearly equivalent to zero; and ii). When the characteristic is zero, condition ii) is equivalent to ii)′q= dimension of the Albanese variety ofV= 0, and always ii) implies ii)′ as in fact≥q ≥0, but in non-zero characteristic it can happen that>q([6], [15]). When ii)′ is true, the algebraic and linear equivalences of divisors coincide onV, because of the duality between Picard and Albanese varieties ofV, [8].