On the monodromy and mixed Hodge structure on cohomology of the infinite cyclic covering of the complement to a plane algebraic curve

The semisimplicity is proved of the Alexander automorphism (the monodromy operator) on the cohomology $H^1(X_\infty)_{\ne 1}$ of the infinite cyclic covering of the complement to a plane non-reduced algebraic curve, and, in particular, the semisimplicity of $H^1(X_\infty)$ in the case of an irreducible curve. A natural mixed Hodge structure on $H^1(X_\infty)$ is introduced and the irregularity of cyclic coverings of $P^2$ is calculated in terms of the number of roots of the Alexander polynomial of the branch curve.