It depends on the context. "The distant genealogy" indicates previous context which identifies a specific genealogy as "the distant genealogy". In this case "a more modern genealogy" would be correct, as it introduces a genealogy which contrasts with the distant one. However, if there is no previous context, then there should be zero article for "distant genealogy" and "more modern genealogy". The question illustrates the difficulties of articles, and the need to understand the meaning of the sentences accurately, and in context. Good question.

I would leave the articles out all together i.e. just say ...' I'm not talking about distant genealogy' .... and .... ' I am talking about more modern genealogy' .....

I am talking about A more modern genealogy. Hope this helps! :-)

For reference: The - specific - I am reading the paragraph you posted. A/An - not specific - I am reading a paragraph. No article - usually for plural, but has other rules - I am reading paragraphs. So as Diego pointed out, in that case you use "A"