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 fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) >= 2n!chi(omega(X)) and it is sharp.

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