Math 245A: Topics in algebraic geometry

Stanford University, Autumn 2026
Instructor: Wanchun (Rosie) Shen (wanchun@stanford.edu)
Office hour: TBD

Topic: Hodge theory and algebraic K-theory

In this course, we will study algebraic K-theory and Hodge theory, with an emphasis on singular algebraic varieties and on recent interactions between the two subjects. A central theme will be how Hodge-theoretic invariants of singularities can be used to study the homotopy invariance and vanishing of the algebraic K-groups.

Lectures

• Lecture 1: Course overview; preview: higher singularities and K-theory

Course plan

Subject to change.

• Part I. Algebraic K-theory
  • Grothendieck groups and K₀
  • Whitehead groups and K₁
  • Quillen’s higher K-groups
  • Localization and dévissage
  • Bass’ negative K-groups
  • Homotopy K-theory KH, K-regularity, and descent
• Part II. Hodge theory and singularities
  • Hodge decomposition and the Hodge filtration
  • de Rham cohomology
  • Deligne’s mixed Hodge theory
  • The Deligne–Du Bois complex
  • Du Bois and higher Du Bois singularities
  • Mixed Hodge modules and related notions of higher singularities
• Part III. From Hodge theory to algebraic K-theory
  • Hochschild and cyclic homology
  • The Hochschild–Kostant–Rosenberg theorem
  • Trace maps and Chern characters
  • Derived differential forms and cdh descent
  • Comparison between K, KH, and Hodge-theoretic invariants
  • Applications to K-regularity and singularities

References

• K-theory
  • Jonathan Rosenberg, Algebraic K-Theory and Its Applications
  • Charles Weibel, The K-book: An Introduction to Algebraic K-theory
  • Daniel Quillen, Higher Algebraic K-Theory I
• Hodge theory
  • Claire Voisin, Hodge Theory and Complex Algebraic Geometry I
  • Chris Peters and Joseph Steenbrink, Mixed Hodge Structures
  • Pierre Deligne, Théorie de Hodge II
  • Pierre Deligne, Théorie de Hodge III
  • Christian Schnell, An overview of Morihiko Saito’s theory of mixed Hodge modules