Inspired by that a problem, which I think about for quite a period of time while I am unable to give an answer, haunting me for a long time, I decide to use this place to put my reflections on some unsolved problems emerging from my own research. These problems are chosen to be of independent interests, regardless of how it relates to my original purposes. I invite people to think about them if you are interested, and feel free to talk with me about them if you like.

 

(1) Rationality of ball quotient surfaces associated with complex reflection groups

Along the way to investigate some problems around complex ball quotients, a seemingly quite natural question emerged unexpectedly on the way to those original destinations. That is, what is the variety type of ball quotients associated with complex reflection groups? Of course, one can ask the same question for any dimension and for any other discrete groups containing torsion elements. But for the moment, let us stick to dimension 2 and complex reflection groups, i.e., the ball quotient surfaces associated with a complex reflection group. All the known examples, such as Deligne-Mostow examples and some other such surfaces studied by Martin Deraux, point to that they are rational varieties (no counterexamples are found so far to my knowledge). I worked on this problem for a period of time but I did not succeed. I hope someone (counting myself) could prove (or disprove) it in a reasonably near future. I have to say it is beyond my ability to estimate the difficulty of this problem since I did not work it out, so please be cautious about it if you want to start to seriously think about it (but I suppose, with my best wishes, it should be of no harm to think about it just for fun). However, I do have some thoughts (maybe useless though) on it since I worked on it for some time anyway. So please feel free to discuss with me about it if you wish. (4 Oct. 2021)

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