Let f : S −→ B be a non locally trivial relatively minimal fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.

We study three methods that prove the positivity of a natural numerical invariant associated to 1-parameter families of polarized varieties. All these methods involve different stability conditions. In dimension 2 we prove that there is a natural connection between them, related to a yet another stability condition, the linear stability. Finally we make some speculations and prove new results in higher dimension.

We give a sharp lower bound for the self-intersection of a nef line bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dimension of X, which we call the Generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibred irregular varieties. As a byproduct we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) ≥ 2 n! χ(ωX) and it is sharp.

When f : S → B is a surjective morphism of a complex, smooth surface S onto a complex, smooth, genus b curve B, such that the fibre F of f has genus g, it is well known that f * ω S/B = E is a locally free sheaf of rank g and degree d = X O S − (b − 1)(g − 1) and that f is not an holomorphic fibre bundle if and only if d > 0. In this case the slope, λ(f ) =

We study the topological index of some irregular surfaces that we call generalized Lagrangian. We show that under certain hypotheses on the base locus of the Lagrangian system the topological index is non-negative. For the minimal surfaces of general type with q = 4 and p g = 5 we prove the same statement without any hypothesis.