This is the view that I'm familiar with about the decline of the Geonim:
The Mesopotamian Gaon reached the height of its prestige in the 10th century- with Saadia Gaon (d. 945), but the political split in the Islamic world at this time created difficulties of communication between the western communities in both Moslem and Christian countries. Through the growth of rabbinical centers in [...the west], the academies of Mesopotamia became less consulted and the Gaonate declined. Aryeh Grabois The Encyclopedia of Medieval Civilization p. 340
The political split he's referring to is the emergence and dominance of the first Shia Caliphates around the Middle East and in Africa.
I thought that the Fatamid Caliphate was particularly responsible for this "cutting off" because it almost bisected the Near East, and it was the center of Shi'ism. I also read that the Fatamid Caliphs sided with certain Jews to oppose Exilarch. However, the Exilarch is not synonymous with the Gaon, and they were usually at odds with each other.
Medieval Jewish tradition ascribes the creation of this position [Nagid] to the Fatimids' desire to remove the influence of the *exilarch on Egyptian Jewry. This view has been accepted by modern scholars. S.D. *Goitein, however, holds that the office of the Nagid developed independent of the aspirations and the policies of the Fatimids. Apparently the first of the negidim was Paltiel of Oria. https://www.jewishvirtuallibrary.org/fatimids
I'm beginning to question all of this now. I've recently learned that the Fatamids, while Shia, were forced to be tolerant because Egypt had a Sunni majority, and also numerous Jews and Christians. Many people moved to Egypt in the 1000's and then converted to Judaism (JVL.com "Fatamids").
Documents found in Cairo portray Talmudic Jews and Karaites living together more peacefully then one may think from reading traditional rabbinical sources that were hostile to them.
All of these things mentioned above lead me to believe that Talmudic Jews were not challenged in the Fatamid Caliphate. That leaves other Shias, the Buyids, Hamdanids, and the Qarmatians, as possible adversaries.
There were Gaon during the Buyid period. Sheria Gaon (d. 1006), Hai Gaon (d. 1038)
What I'm looking for is specific theories regarding the shift of Talmudic scholarship to the Mediterranean and Europe. I'd like to know what specifically cut them off, or if there is another detailed explanation for their movement out of Mesopotamia.
I have some other ideas, but can't substantiate them. These are that the progress of Jews in the west made the Geon obsolete, or that the stability of Egypt made Babylonia obsolete. I'd also be interested in additional or alternative complications to Jewish communications with the east, such as the recession of Islam in the Mediterranean, the rise of Venice or Pisa, or the Byzantines.