Seminar on Mixed Hodge Structures (Fall 2021)

The topic (for now) is mixed Hodge structures, following Deligne's original papers [D2] and [D3]. I'm down to do other related things too, if anybody wants to.

Schedule

10/28
Kevin Chang
Intro to mixed Hodge structures; Hodge theory for smooth varieties
I will start by reviewing Hodge theory for smooth projective varieties and the relevant homological algebra. Following [D2], I will introduce mixed Hodge structures. I will finish by presenting the first part of the proof of the existence of mixed Hodge structures on arbitrary smooth varieties.
notes
11/04
Kevin Chang
Hodge theory for smooth varieties II
I will construct mixed Hodge structures for smooth varieties. I will then provide some applications and examples.
notes
11/11
Caleb Ji
Cohomological descent
Cohomological descent uses the simplicial theory of hypercoverings to formulate a derived category version of descent. In this talk, I will begin by introducing the preliminary notions of coskeleta and hypercoverings, which form a generalization of Cech theory. Using de Jong's alterations, one can construct a proper and regular hypercovering of singular varieties. Then I will define cohomological descent and describe some of its properties and applications. Finally I will state and explain the main theorem due to Deligne that says that proper hypercoverings are universally of cohomological descent, which is an important input used in his Hodge III paper.
notes
11/18
Kevin Chang
Applications and examples of mixed Hodge structures; Hodge theory for all varieties
In the first part of this talk, I will give some applications and examples of mixed Hodge structures on smooth varieties. In the second part of this talk, I will introduce the formalism of mixed Hodge complexes, which give a systematic way of producing mixed Hodge structures. I will then explain how to get a mixed Hodge complex from a simplicial resolution of an arbitrary variety.
notes the Grothendieck fact about the image of the cohomology of a smooth compactification
12/02
Kevin Chang
Hodge theory for all varieties II
I will finish the proof of the existence of mixed Hodge structures on all varieties, modulo some homological algebra.