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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 quasi-projective varieties?

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Also see… for further refs. etc. – Donu Arapura Aug 29 '11 at 2:51
I don't know if this is exactly what you are looking for but there are the two books by Claire Voisin on Hodge theory (Hodge theory and complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77). As far as I remember she discusses variations of Hodge structures. – DamienC Aug 29 '11 at 7:12

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 quasi-projective algebraic varieties (in Section 3). It is about 50 pages long, with the following (high-level) Table of Contents:

  1. Hodge structures
  2. Varieties with normal crossings
  3. Smooth quasi-projective varieties
  4. The theory of Deligne
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