Yhyjyk

I'm currently preparing an article for a very good journals in mathematics (AMS, Acta, Advances, etc.). I have never had a result worthy of these journals until now. I would like to know whether I can count on the editors to at least read my one-page introduction before making a decision. Or could they stop after the abstract? The reason I ask is currently my abstract is very short and gives zero background. If someone were to only read it, they could undervalue my paper.

No, there's no such guarantee. A top journal gets an awful lot of submissions, and you can expect them to be aggressive in rejecting papers which do not make a strong case for their importance, in the way that involves the least work for the editors.

If the editor, after reading the abstract, does not understand why the results would be highly interesting and significant, I would fully expect them to reject the paper without reading the introduction or any other part of the paper. (I can even imagine cases where they could make such a judgment by only reading the title.) As such, your abstract definitely should provide motivation for your results, such that a reader who is generally familiar with the area can see why they are of interest.

If your abstract just states the result ("we show that every snark is a boojum"), then it had better be such a well-known topic that every reader in the field would immediately know why it is of interest to know that every snark is a boojum. If that is not the case, then your abstract should briefly explain why people should care, perhaps relating it to previous work. ("Previous work of Smith showed that every reticulated snark is a boojum. In this paper, we show that the assumption of reticulation is unnecessary, thus resolving a conjecture of Jones.")

To my knowledge search engines for scientific publications like SCOPUS or ISI Web of Science don't even index any words of your manuscript apart from the abstract and the title (google scholar is indexing everything). Therefore, if the online submission system or journal guidelines for submission don't allow you to write a submission letter to further explain, why you think your manuscript is of high importance, then, as Nate pointed out, only title and abstract are read at all.

Therefore, writing a concise abstract and putting the right keywords into (-> search engines) it is crucial to get at all one foot into the review process.

On the other side I have read abstracts which consist of not much more then 1-2 lines/one sentence. The author is then mostly a famous professor who knows that he is known and his paper enters the review process. Still, I think when then a 3-4 pages manuscript is following for the sake of finding his manuscript via a search engine with some keywords, it would be nice to write a longer abstract.

Top journals are also more inclined to take a deeper look on a submission if the topic is trendy and more researchers care about it or work on it.

I also doubt that manuscripts are read in a linear way, from title to conclusion. With so much literature to choose and to read, the editor might overjump your introduction and look into the conclusion first. That's how I read published papers, I rather overfly the whole paper (conclusion paragraph, graphs,...) before deciding to read it completely from the beginning.