Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

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This paper is devoted to the irregular surfaces of general type having the smallest invariants, p_g = q = 1 . We consider the still unexplored case K^2 = 4 , classifying those whose Albanese morphism has general fibre of genus 2 and such that the direct image of the bicanonical sheaf under the Albanese morphism is a direct sum of line bundles. We find 8 unirational families, and we prove that all are irreducible components of the moduli space of minimal surfaces of general type. This is unexpected because the assumption on the direct image bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension strictly bigger than the expected one.