If there is a positive mitzva of living in Eretz Yisrael, why did the scholars of Pumbedisa, Sura, Nahardai, Mechoza, etc. remain in Bavel. If it was to learn Torah from the giants, then we can focus the question on the roshei yeshivos themselves.
There is also the halachic opinion of Rav Yehudah, quoted in several places in the Gemara (Berachos 24b, Shabbos 41a, Kesubos 110b-111a) that it is in fact forbidden to move from Bavel to Eretz Yisrael. He bases this on his understanding of Jer. 27:22 ("they shall be brought to Bavel, and there they shall remain until the day I am mindful of them"), plus some verses in Song of Songs (the famous "three oaths").
Conditions in Israel at the time were generally worse than they were in Bavel. The Romans empire (especially after it converted to Christianity) treated the Jews much worse than the Persian government did. In Bavel, the Jews were granted a certain amount of autonomy and freedom, which allowed them to study and keep the Torah without persecution.
I've heard that the first scholar to go to Bavel was Rav, sent by Ribbi Yehuda HanNasi. He opened up a yeshiva in Sura for the sake of the general population that resided there at the time, which was uneducated to a certain degree. He created his own talmidim. At little later, Shemuel broke off and started a yeshiva in Naharda'a (which I think may have been later relocated to Pumbedita). These yeshivot grew and produced more and more talmidei chachamim to the extent that the Torah-center of the world shifted to Bavel. At this point, students gathered from across the world to study in these great yeshivot.
So, in a sense, Torah in Bavel started as an outreach program for the Jewish population already living there, and eventually developed into a community too large to be abandoned.
By the way, Wikipedia has a ton about this stuff.
The answer is really very simple. They were exiled and not allowed to move to Israel. They were not just leaders of their Yeshivas, they were also members of the Babylonian government, known as the Exilarch.
However, the history of the Exilarch before the destruction of the second temple, is a bit more complicated and I'm not aware of any good reason given for why they kept the Exilarch position in Babylon when Ezra was returning the Jews to Israel.