I first looked at the BABIP stats at The Hardball Times sometime last week, and it came as no surprise that the White Sox had the lowest BABIP in baseball at about .237. They've gotten better in the last week, and currently sport a .243 BABIP, which is still 30 points lower than any other team in the AL. Unfortunately, this isn't just a case of bad luck. The Sox have hit an inordinate amount of easily fielded balls that are (almost) always going to be turned into outs. (Think Joe Crede pop-ups.)
What's even more troubling is that the Sox currently lead the AL in Defensive Efficiency, which is essentially a different way of expressing BABIP since it's the measure of balls in play converted into outs, at an unsustainable .734. So while the offense will likely improve to post a passable .275 BABIP by the end of the year, the defense will likely degrade to a DER a hair above .700. The Sox should remain a mediocre club, except instead of a poor offense and good pitching, they'll feature a mediocre offense to go along with average run prevention.
On a more micro level, 4 of the Sox starters currently feature what appears to be an unsustainably high number of balls in play converted into outs. Here are the graphs (taken from Fan Graphs) to illustrate how Sox starters appear to be working outside career norms.
Javier Vazquez
Vazquez historically allows a high BABIP, .308 career average, yet he currently clock in with a .264 mark.
Jon Garland
While Garland is the one Sox pitcher who has consistently demonstrated the ability to have some control on balls in play, his true talent lies somewhere in the .275 BABIP range, not the .222 he's currently allowing.
Mark Buehrle
Buehrle is an interesting case. His initial big league dominance was largely BABIP driven, but over the last 4 years he's been entirely average in this category. If he is able to suppress BABIP, Buehrle can be one of the top pitchers in baseball, but I find it highly unlikely that he'll be able to sustain his current rate of .229 or match his first full season mark of .248.
Jose Contreras
Contreras' ERA fluctuation from season to season can almost entirely be traced to BABIP. With his declining strikeout rate, even his current .267 BABIP, which he seems unlikely to maintain given the sheer number of balls being put in play, may not be enough to render him an above average starter.
Finally, one more graph without much discussion. I ran out of time. Forgive me. A graphical look at last season's infield defense.

Crede + Uribe = best in baseball at covering the hole.
Iguchi + Konerko = sending out invitations to hit it between them