How important is the own talent for research of your PhD supervisor? Currently I am in the process of finding a PhD. Some potential supervisors are more didactical than others, some are nicer and warmer than others, and some are more famous mathematicians than others. Concerning this last point, I guess it is very fair to say that some professors are simply more talented at research. My question for you is: how important is this? When looking for a supervisor, how much should I weigh how famous and/or talented at research he/she is?
Potential answers I could give myself:


*

*A supervisor talented at research will have very nice ideas and an inspiring way of thinking. From this I could learn a lot. So it is a good thing.

*When looking for a postdoc, a reference letter from a famous mathematician is valued more. So again it is a good thing (yes, sorry, this is something we unfortunately need to care about these days).

*The only thing that is important is his guidance. Any professor will be of much higher level than yourself, so what is important is how he manages to guide you through the research (this is a whole different thing than being able to do research yourself as well). So it doesn’t matter.
I am looking for people, preferably having finished their PhD or in the midst of it, that could give a personal opinion about this. Maybe with some anecdote?
P.S. I keep having Deligne and Grothendieck in mind. How much of Deligne’s development could be attributed to him being Grothendieck’s student?
P.P.S. Should I have asked this in academia stackexchange? I just feel this could differ A LOT among different research fields so it feels important to get answers from mathematicians.
 A: I have a PhD and have served as direct advisor for a few PhD students and on committees for several, though in physics and electrical engineering and computer science, not mathematics.  A few thoughts:
First, look deep into yourself and review your career so far and understand your strengths and weaknesses, and choose an advisor accordingly.  Some of my friends and colleagues in math are superb math problem SOLVERS, but not question POSERS.  If you're such a mathematician, choose an advisor who can help you identify great problems, and also teach you how to do so yourself (essential for a productive career).
Next, choose someone with whom you can work productively.  He or she need not be a close friend, or agree on politics or religion or sports teams or whatever, but when you spend an hour in his or her office, you should feel like you're making progress, that the advisor hears your challenges and helps you along.  Most good students doing world-class work hit stumbling blocks, and you need to know that your advisor can help you.  (He or she won't be travelling the world or is such a recluse or has so many students that it will take weeks to seem him or her, etc.)
Understand within yourself whether you know precisely the field and class of problems you'd like to address (topology, number theory, differential equations, ...) and choose accordingly.  But if you're still unsure, try to find an advisor who will support your exploration throughout a range of fields.
All other things being equal, a more famous, better connected advisor may help you land a job, but notice who is a rising star and will be making a name for him or herself in the three or four years it will take you to finish.  Being the first advisee of such a rising star can be a great help when you graduate, especially if potential employers want to move into the field pioneered by your advisor.  (My own boss was the first PhD student of just such a rising star at a major university, and their joint work has been cited many thousands of times.)
Importantly:  talk honestly to current PhD students and graduates of a candidate advisor, and see where they've found jobs, whether they would work again with that candidate advisor.
Most importantly:  talk to candidate advisors and ask about their advising methods.  You can get this person's views on previous students and compare these views to those from the students and former students with whom you've corresponded.
Good luck!
