On the algebraicity of cycles via degenerations and refined unramified cohomology

The content of this thesis revolves around a central question in mathematics; the Hodge Conjecture. In particular we start out giving new counterexamples to the stronger, known to be false, Integral Hodge Conjecture. To do this, we develop Lefschetz theory type results for families where the special fibre is higher dimensional, called Morse-Bott degenerations. With these new results, it is possible to apply an approach by Shen to families defined by the variety of lines on Lefschetz pencils, obtaining the new counterexamples. Hereafter, we generalize the techniques used before, to provide new counterexamples to a variant of the Integral Hodge Conjecture, known as the Integral Tate Conjecture. More specifically, we generalise a specialisation argument of Colliot-Thélène and apply it to the refined unramified cohomology groups developed by Schreieder. The relation between the aforementioned groups and the Integral Tate Conjecture, given again by Schreieder, in turn proves the existence of these new counterexamples. We end this dissertation with an analysis of another variant of the Hodge Conjecture, called the Beilinson-Hodge Conjecture. In particular, we study the higher Chow groups and corresponding higher cycle class map of quasi-projective varieties over an algebraically closed field. In the main result of this final part, we show that the kernel and cokernel of this higher cycle class map are both related to the refined unramified cohomology groups in the form of a long exact sequence.