The classification of isotrivially fibred surfaces with p (g) = q=2

An isotrivially fibred surface is a smooth projective surface endowed with a morphism onto a curve such that all the smooth fibres are isomorphic to each other. The first goal of this paper is to classify the isotrivially fibred surfaces with pg = q = 2 completing and extending a result of Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with pg = q = 2 and K2 = 4, 5 and a first example with K2 = 6.