Is there any English textbook about Deligne's mixed Hodge structures? Can you tell me about a reference where they are introduced at least for smooth quasiprojective varieties?

Maybe you already know of the notes by Benoît Audoubert and Orsola Tommasi entitled "Mixed Hodge Structures," notes on Audoubert's seminars on Mixed Hodge structures at the University of Nijmegen in 2002? These notes certainly discuss quasiprojective algebraic varieties (in Section 3). It is about 50 pages long, with the following (highlevel) Table of Contents:


