Basic notions in cotangent Schubert calculus

Lecture given at the Introductory Workshop: Combinatorial Algebraic Geometry, ICERM, Brown University, February 1-5, 2021

Abstract:
A key notion of Schubert Calculus is the cohomology ring of a homogeneous space, together with a distinguished basis: the collection of Schubert classes. There is a one-parameter deformation of the notion “Schubert class’’, called Chern-Schwartz-MacPherson (CSM-) class (a.k.a. characteristic cycle class, or cohomological stable envelope). In this lecture/workshop we will define the CSM class, and illustrate many of its properties through examples.

Slides:

• Introduction and H^*_T(Gr)
• Grassmannian
• Schubert cells
• Schubert Classes Gr(2,4)
• CSM Classes
• R Matrix property of CSM classes
• MacPherson Property of CSM classes
• Why “Cotangent” Schubert calculus
• Recursion
• Formulas
• K Theoretic Version: motivic Chern class, convex polygons
• Enumerative geometry

Problems for the problem session.