A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

We study a family of surfaces of general type with $$p_g=q=2$$ p g = q = 2 and $$K^2=7$$ K 2 = 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $$\mathcal {M}$$ M in the moduli space of surfaces of general type. In particular we prove that $$\mathcal {M}$$ M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map $$j^{dr}_n$$ j n dr on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $${\mathscr {U}}$$ U . Several algorithms forming part of the computation of finite level versions $$j^{dr}_n$$ j n dr of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $${\mathscr {U}}$$ U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $${\mathscr {U}}$$ U over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.

A Pair of Rigid Surfaces with pg= q = 2 and K2= 8 Whose Universal Cover is Not the Bidisk

We construct two complex-conjugated rigid minimal surfaces with $p_g\!=q=2$ and $K^2\!=8$ whose universal cover is not biholomorphic to the bidisk $\mathbb{H} \times \mathbb{H}$. We show that these are the unique surfaces with these invariants and Albanese map of degree 2, apart from the family of product-quotient surfaces given in [33]. This completes the classification of surfaces with $p_g=q=2, K^2=8$, and Albanese map of degree 2.

A higher-dimensional generalization of Lichtenbaum duality in terms of the Albanese map

In this article, we present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron–Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer–Manin pairing due to Saito–Sato and on Gabber–de Jong’s comparison result of cohomological and Azumaya–Brauer groups. We will also mention the local–global problem for the Albanese cokernel; the abelian group on the ‘local side’ turns out to be a finite group.