On the topological index of irregular surfaces

Abstract

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.

On the canonical map of surfaces with q ⩾ 6

“…By 1-connectedness of D and (A + E)B = 2, we must have A i (D − A i ) = 1, for i = 1, 2, and so we conclude as above that A i is 1-connected and A 2 i 0, for i = 1, 2. Then from 0 = (A + E) 2 …”

“…This is easily done under the assumption that S has an irrational pencil of genus > 1 (cf. [2,20]), but the matter becomes very hard if one assumes that S has no such pencil. Examples of surfaces with these properties are known only for q = 3, 4.…”

“…, α 4 such that α 1 ∧ α 2 + α 3 ∧ α 4 = 0. In [2] it is shown that a minimal generalized Lagrangian surface whose canonical system has no fixed part has K 2 8χ, and in [20] the weaker inequality K 2 7χ − 1 has been proven for all surfaces with p g = 2q − 3.…”

On surfaces with pg = 2q – 3

“…In [1] Barja, Naranjo and Pirola prove the inequality K 2 ≥ 8χ under some technical assumptions on the base locus of the canonical system. Furthermore, they make a detailed study of the case q = 4 (and hence p g = 5), showing that the inequality K 2 ≥ 8χ holds in this case without any extra assumption.…”

“…In the general case we prove inequalities for the invariants of S which are weaker than the one in [1] but require no extra assumptions:…”