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Hodge rigidity of Chern classes

Yuxiang Liu, Artan Sheshmani, Shing-Tung Yau

Abstract

In this paper, we study the homogeneous components of the Chern--Schwartz--MacPherson (CSM) classes of Schubert cells. We prove that, under suitable conditions, each such component is represented by an irreducible subvariety. In particular, our result extends Huh's result \cite{Huh} by relaxing the regularity assumption on log resolutions. As a consequence, the conclusion holds for all cominuscule Schubert cells of classical type and for a large family of exceptional cases. We also obtain analogous results for certain Schubert varieties in symplectic Grassmannians and flag varieties.

Hodge rigidity of Chern classes

Abstract

In this paper, we study the homogeneous components of the Chern--Schwartz--MacPherson (CSM) classes of Schubert cells. We prove that, under suitable conditions, each such component is represented by an irreducible subvariety. In particular, our result extends Huh's result \cite{Huh} by relaxing the regularity assumption on log resolutions. As a consequence, the conclusion holds for all cominuscule Schubert cells of classical type and for a large family of exceptional cases. We also obtain analogous results for certain Schubert varieties in symplectic Grassmannians and flag varieties.

Paper Structure

This paper contains 26 sections, 28 theorems, 89 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Cominuscule Schubert varieties
  3. Symplectic Grassmannians and partial flag varieties
  4. Preliminaries
  5. Schubert varieties
  6. Grassmannians
  7. Partial flag varieties
  8. Orthogonal Grassmannians
  9. Orthogonal partial flag varieties
  10. Symplectic Grassmannians
  11. Symplectic partial flag varieties
  12. General homogeneous varieties
  13. Chern-Schwartz-MacPherson classes
  14. Hodge rigidity of CSM classes
  15. Cominuscule Schubert varieties
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $X$ be a Schubert variety in a homogeneous variety $G/P$, and let $X^\circ$ be the corresponding Schubert cell. Assume that there exists a $B$-equivariant log-resolution $\pi:Y\rightarrow X$ such that $Y$ has only finitely many $B$-orbits. Then each homogeneous component of $c_{SM}(X^\circ)$ is

Figures (2)

  • Figure 1: Marked Dynkin diagrams
  • Figure 2: Marked Dynkin diagram for $E_7/P_7$

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 53 more