Clearly, from my reputation record, there has been some vindictive downvoting in the last three days or so. MathOverfflow is a community of people, not saints, and things like this are expected to occur at one time or another. What can be done about it? The small loss of reputation points doesn't matter to me--I've deleted some of my own questions or answers, some with five points, or so, after I had published better on my website and/or had received no significant responses. But, I would like this childish behavior to be reprimanded, if possible, particularly since much younger researchers, or relatively new users, might be discouraged one way or another by it, and since I don't in principle agree with such gratuitous censorship.
Tom: after checking tools available to moderators, I see no evidence of systematic downvoting from any particular user. So at the moment I don't see evidence of downvoting that looks 'vindictive'. I realize that unexplained downvotes are an annoyance, and particularly if several happen during a short time frame they might look suspicious, but the system is so far not showing us that, and they could be from several different users who ('childishly' or not) might be reacting to a certain amount of noise they are hearing -- it's impossible to say.
As to what can be done: as quid suggested, alert the moderator team and tell them what you know or suspect, either by raising a flag or writing directly to email@example.com. Where you suspect revenge downvoting, you can mention time frames or suspected users; the moderators can then take a closer look. The Stack Exchange Community Team (firstname.lastname@example.org) can also have a look.
Targeted voting (either up or down), meaning voting not based on the merits or demerits of a post, is a site violation that moderators and SE Community Managers take seriously. You should always report such instances, as you suspect and as they arise.
There are other points of concern raised by your post. You mentioned your deleting questions. As I believe you know, the system design prohibits users from deleting questions that have either (1) an upvoted answer, (2) multiple answers (upvoted or not), or (3) an accepted answer; this is explained here: How does deleting work? What can cause a post to be deleted, and what does that actually mean? What are the criteria for deletion?.
Perhaps what you meant is that, where one of (1)-(3) applies, you have heavily edited a question to the point where it became virtually deleted, as happened earlier here: Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres This should not be done; in fact it's a site violation, as has been explained elsewhere. If there is a really good reason that a question (under those circumstances) needs to be deleted, this should be made clear to moderators at the site or system level through private message (via flag or email). On the other hand, if it is merely a situation where you are for some reason unhappy with the Community response to your question, then no, we moderators generally won't enable a question deletion or virtual deletion. Conflicts have to be resolved in a different way.
We are here to help if we can. In situations where an argument is going on in a comment thread (as at the aforementioned MO post), moderators seek to lower the noise level and bring the discussion back to the mathematics, which is what everyone should want. It is true that where there is sniping back and forth and flags are raised citing rudeness/offensiveness, I will go in and either delete comments if I judge they are adding nothing useful to the discussion, or if part of the comment is making a mathematical point, then I typically prune out non-constructive content but do what I can to leave the mathematical content intact. I try to be fair in doing so, but it's often difficult and unpleasant work, made more difficult when some users embed their mathematical points in rather blunt phrases (I can see this with some of your interlocutors). I also know full well that there are users who are going to protest loudly or cry 'censorship' every time I perform any edit at all in this capacity, which I do as part of my 'job' obligations but often with heavy heart.
To conclude, I am sad to see the threads after your posts degenerate into arguments and to see how unhappy certain community responses make you. I suggest that all participants get into a mindset of problem-solving and conflict resolution, de-escalate emotions, be considerate of others, bring a sense of humor and open-mindedness, and take a little break now and then to cool off if need be. Keep the focus on the math. Be the bigger man or woman and don't let yourself get rattled publicly. Contact the moderators privately when you need help.
Targeted voting as you describe it is not allowed. There is a script in place that undoes it in some cases. The precise workings of the script are not public, not to make it easy to work around it.
Moderators have some tools to track "strange" voting, too. But, they cannot see individual votes either.
One way to proceed is to wait (a day or so) to see if the script undoes the votes. However, this might not happen or not fully address the situation.
Another thing you can do is contact the moderators and explain the situation. You can do this:
flag the post in question, reason "other" and explain the situation.
write an email to the moderators, for details see Who are the MathOverflow moderators?
It is my understanding that they do react on such reports; as said the behavior is not allowed. However, a single or very few problematic votes might be impossible to trace.
Todd, read Schopenhauer to reflect on how too natural your sadness might be, but don't censor my opinions (biting or not) in response to others' opinions while retaining the initiating opinionated comments--the bias just looks too clubby and certainly provides a false record of thoughts for the remaining comments. Any venue in a fair system allows one to openly defend himself in debate (and not through a surrogate, particularly a possibly biased one at that appointed by the state). I do commiserate with you on the difficulties of being a moderator, though you did seek the position and were aware of the heavy, concomitant responsibilities.
I've frequently deleted my own questions and answers (and comments when no longer useful) in the past that have received no repsonses except perhaps upvotes or downvotes, especially when I've found a satisfactory answer myself and even more so if the answer is quite long and involved and would be better proferred in a note at my website/mini-arXiv to those who might be interested--notes that can be embellished and revised with new info without annoying the disinterested. Seems natural enough.
I reported two questions (Stasheff polytopes, Cavaliri's principle) to moderators that are quite old and have had no activity in months, but were downvoted within seconds of each other yesterday--a third very old question (an intriguing tapestry), the day before. Can't be a coincidence. Your 'apps' are apparently not adequate. I do not want to mention any suspected violators since such 'gossip' would also cast aspersions on the more or less innocent, which is unacceptable to me. Who may have cast the downvotes with whatever rationale is truly a mystery to me, but I'm sure it was motivated emotionally by the exchanges in the current situation due to the connection in time.
I initially replied to Ryan's first comment by directing him to take polemics to meta where people are stimulated by them. Then received the intended effect of Ryan's initial opinionated comment (bereft of additional mathematical content since the cited OEIS ref contains the correct relation to differential manifolds in the first comment there, so that seems only an entry point for his objective of politiking)--a downvote--and responded accordingly. As in "Why do the Bernoullis arise everywhere?", Ryan seems prone to denigrating posts according to his own personal tastes (which does seem to coincide with a coterie of similarly opinionated users, hopefully a minority, but a demonstrative one that tends to politik against questions, not suiting in format and/or content their tastes, with opinionated comments at best tangential to the math and veiled as constructive criticism). After an expected lively exchange, censorhip led me to delete any remaining comments, edited or not, to avoid a misrepresentation of my thoughts. I'd prefer to move the bulk of my question to my website where I can then allude to relevant ideas in OEIS entries without airing any dirty laundry or indeed non-productive question at MO, i.e., clean up in almost a literal sense for which I have received the disciplined and peer pressure badges, so such behavior is condoned by the exchange. The only obstacle is the one answer to my question, so I haven't deleted the question, but have deleted the text that unproductively addressed the comments and really is unnecessary.
Finally, I'd like to ask why there has NOT been more DOWNvoting on this meta question. People can still surprise me even after all these decades.