Hodge theory & perverse sheaves (Spring 2027)

In broad strokes, the main question of the Hodge theory is "What is so special about algebraic varieties and algebraic maps between them from the standpoint of topology?" The study of this question was initiated by Lefschetz and Hodge and led to a surprisingly rich structure. A concise way to package these structures is the celebrated Decomposition Theorem of Beilinson-Bernstein-Deligne-Gabber, which describes an intricate relation between the cohomology groups of the source and the target of a proper map between algebraic varieties. 

The theory of perverse sheaves turns out to be a natural framework to study these structures. One of the key reasons for their appearance is that when working with algebraic varieties and maps between them we frequently unavoidably encounter singular algebraic varieties (e.g. as special fibers). This leads to a desire of a uniform treatment of singular and smooth varieties. Perverse sheaves lead to the definition of intersection homology, a homology theory of possibly singular algebraic varieties introduced by Goresky and MacPherson, which enjoys nice properties expected from smooth manifolds, such as Poincare duality. The perverse sheaves themselves are also of independent interest due to a wide range of applications in number theory, algebraic geometry, representation theory and mathematical physics.  

The main goal of the class is to give an introduction to the theory of perverse sheaves and, in particularly, the Decomposition Theorem. The focus is to see how these concepts arise naturally from geometry and topology and explore their concrete manifestations and applications. There are several drastically different approaches to the topic, and I will follow the one developed by Mark Andrea de Cataldo and Luca Migliorini, which, in my opinion, is the most geometric, illustrative, closest to the classic approaches and requires the least prerequisites. 

After that, depending on the remaining time and desire of the audience, we can either discuss some of the more advanced applications of the developed techniques or look into alternative approaches to the theory. 

The class will be meeting TuTh 3:30-4:45pm in PMA, room TBD. All the materials and announcements related to the class will be posted on this page. Office hours for the course are available by appointment. Don't hesitate to contact me at andrei.ionov@austin.utexas.edu if you have any questions or concerns.  

References: The main reference is "The Decomposition Theorem and the topology of algebraic maps" by Mark Andrea de Cataldo and Luca Migliorini (available here: https://arxiv.org/abs/0712.0349, both v1 and v2 are relevant and quite different). I will also use the shorter survey (https://arxiv.org/pdf/1506.03642) of the first named author for the exercises. An alternative approach, which I might also discuss to some extent if time permits, is described in the notes https://perso.pages.math.cnrs.fr/users/claude.sabbah/MHMProject/mhm.html (also plenty of good exercises here). Additional references will be provided as required.

Prerequisites: solid knowledge of topology at a graduate level (at least up until Poincare duality) and basic knowledge of category theory is assumed; some knowledge of algebraic and complex geometry will be very helpful but not strictly required. If you are new to some of these it might be helpful to attend the class concurrently with David Ben-Zvi's class.