Uma Thurman's Baby Daddy Forcing Her to Choose Movie or the Kid

Uma Thurman is stuck between a rock and a hard place ... if she shoots a movie she won't see her daughter. That's the predicament she's in, thanks to her baby daddy.

Uma's ex-fiance, Arpad Busson, just filed legal docs asking a judge to prohibit her from taking their 4-year-old daughter to Europe while she shoots a movie ... sources familiar with the situation tell TMZ.

We're told Uma has been blindsided by the latest conflict ... she's supposed to leave in 2 days.

Arpad wants the kid to stay put in NYC, so now a judge will decide ... possibly creating a real-life "Sophie's Choice" for Uma.

TMZ broke the story ... Uma and Arpad's nasty custody battle heads to court in a few months to hash out visitation rights.