On the slope conjecture of Barja and Stoppino for fibred surfaces

Let f : X ! B be a locally non-trivial relatively minimal fibration of genus g ≥ 2 with relative irregularity q f . It was conjectured by Barja and Stoppino that the slope λ _f ≥ ⁴ _g ⁽ _- ^{g -} _q ¹ _f ⁾. On the one hand, we show the lower bound λ _f > _g ⁴ _- ^(g _q ^- _f ¹ _/ ⁾ ₂ , and also prove the Barja-Stoppino conjecture when q _f is small with respect to g. On the other hand, we construct counterexamples violating the conjectured bound when g is odd and q _f = (g + 1)/2.