Reading course for Ph.D. students at IMPA.
Contents: The purpose of the course is to learn some of the basics of Mixed Hodge theory, for computing the Mixed Hodge Structure of affine hypersurfaces admiting quasi-smooth compactifications inside weighted projective spaces or in other complete simplicial toric varieties.
References:
Mixed Hodge Structures, Chris A. M. Peters and Joseph H. M. Steenbrink.
Toric Varieties, David Cox, John Little and Hal Schenk.
Notes: My lecture notes.
Lecture 1: Reminding derived functors and spectral sequences.
Lecture 2: Direct image functor and Leray spectral sequence.
Lecture 3: Filtrations on complexes of sheaves and the weight filtration on smooth varieties.
Lecture 4: Mixed Hodge structures and pole order filtration.
Lecture 5: Steenbrink basis on smooth affine hypersurfaces and orbifolds.
Lecture 6: Steenbrink basis for quasi-homogeneous varieties and introduction to toric varieties.
Lecture 7: Toric varieties defined by a fan.
Lecture 8: Orbits, divisors and homogeneous coordinates.
Lecture 9: Mixed Hodge structures on toric orbifolds.