Mixed Hodge structures and <i>E</i>-polynomials

We provide a self-contained review of mixed Hodge structures (MHS) with a focus on the computation of E-polynomials for complex quasi-projective varieties. We discuss the discrepancy between MHS on diffeomorphic varieties, specifically for the cotangent bundle of an abelian variety. Furthermore, we compute the E-polynomials for S Ln(C) and PGLn(C) and explore the link between these invariants and point-counting over finite fields.