Mixed Hodge structures

The theory of Mixed Hodge Structures (M.H.S.) on the cohomology of an algebraic variety $X$ over complex numbers was found by Deligne in 1970. The case when $X$ is a Normal Crossing Divisor is fundamental. When the variety $X$ is embedded in a smooth ambient space we get the Mixed Hodge Structure using standard exact sequences in topology. This technique uses resolution of singularities one time for a complete variety and $2$ times for a quasi-projective one. As applications to the study of local cohomology we give the spectral sequence to the Mixed Hodge Structure on cohomology with support on a subspace $Y$.