On the Tate and Mumford-Tate conjectures in codimension one for varieties with h^{2,0}=1

TL;DR

This paper proves the Tate and Mumford-Tate conjectures for certain algebraic varieties with specific Hodge numbers, advancing understanding in algebraic geometry and number theory.

Contribution

It establishes the conjectures for varieties with h^{2,0}=1 under mild moduli assumptions, including some algebraic surfaces with p_g=1.

Findings

Proves Tate conjecture for divisor classes in specified varieties.

Establishes Mumford-Tate conjecture for degree 2 cohomology.

Applies results to algebraic surfaces with p_g=1.

Abstract

We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.