On proper splinters in positive characteristic

While the splinter property is a local property for Noetherian schemes in characteristic zero, Bhatt observed that it imposes strong conditions on the global geometry of proper schemes in positive characteristic. We show that if a proper scheme over a field of positive characteristic is a splinter, then its Nori fundamental group scheme is trivial and its Kodaira dimension is negative. In another direction, Bhatt also showed that any splinter in positive characteristic is a derived splinter. We ask whether the splinter property is a derived invariant for projective varieties in positive characteristic and give a positive answer for normal Gorenstein projective varieties with big anticanonical divisor. We also show that global F-regularity is a derived invariant for normal Gorenstein projective varieties in positive characteristic.