Mixed Hodge Structures

Description: This is a short reading course for Ph.D. students at IMPA. 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:

1.  Mixed Hodge Structures, Chris A. M. Peters and Joseph H. M. Steenbrink.

2.  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.