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We give a sharp lower bound for the selfintersection of a nef line bundle $L$on an irregular variety $X$ in terms of its continuous global sections and theAlbanese dimension of $X$, which we call the Generalized Clifford-Severiinequality. We also extend the result to nef vector bundles and give a slopeinequality for fibred irregular varieties. As a byproduct we obtain a lowerbound for the volume of irregular varieties; when $X$ is of maximal Albanesedimension the bound is $Vol(X) \geq 2 n! \chi\omega_X$ and it is sharp.

Generalized Clifford-Severi Inequality and the Volume of Irregular Varieties