Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Uniqueness Theorem for Gysin Coherent Characteristic Classes of Singular Spaces

Markus Banagl, Dominik Wrazidlo

Abstract

We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us e.g. to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky-MacPherson L-class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the L-class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincaré duality over the rationals.

The Uniqueness Theorem for Gysin Coherent Characteristic Classes of Singular Spaces

Abstract

We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of a recursive nature terminating on genera of explicit characteristic subvarieties that we construct. It enables us e.g. to apply intersection theory of Schubert varieties to obtain a uniqueness result for such characteristic classes in the homology of an ambient Grassmannian. Our framework applies in particular to the Goresky-MacPherson L-class by virtue of the Gysin restriction formula obtained by the first author in previous work. We illustrate our approach for a systematic computation of the L-class in terms of normally nonsingular expansions in examples of singular Schubert varieties that do not satisfy Poincaré duality over the rationals.
Paper Structure (20 sections, 34 theorems, 147 equations, 5 figures)

This paper contains 20 sections, 34 theorems, 147 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Whitney Transversality
  3. Generic Transversality
  4. Preliminaries on Schubert Varieties
  5. Partitions
  6. Flags
  7. Schubert Varieties
  8. The Homology of Schubert Varieties
  9. The Singular Set of a Schubert Variety
  10. Intersection Theory of Schubert Cycles
  11. Transversality of Flags
  12. Segre Products in Ambient Grassmannians
  13. Gysin Coherent Characteristic Classes
  14. Normally Nonsingular Expansion
  15. Normally Nonsingular Integration
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $c\ell$ and $\widetilde{c\ell}$ be Gysin coherent characteristic classes. If $c\ell^{\ast} = \widetilde{c\ell}^{\ast}$ and $|c\ell_{\ast}| = |\widetilde{c\ell}_{\ast}|$ for the associated genera, then we have $c\ell_{\ast}(i) = \widetilde{c\ell}_{\ast}(i)$ for all inclusions $i \colon X \rightar

Figures (5)

  • Figure 1: Relation between the Young diagrams $D_{c'}$, $D_{c"}$, and $D_{c' \sqcup c"}$.
  • Figure 2: Definition of the partition $a" \in \mathcal{P}(m", k")$ in terms of $a' \in \mathcal{P}(m', k')$. The Young diagram $D_{a"}$ is obtained by rotating the shown diagram $\widetilde{D}_{a"}$ by $180$ degrees.
  • Figure 3: (a) Definition of the Young diagram $D_{c}$. (b) Definition of the Young diagram $D_{\widetilde{c}}$.
  • Figure 4: (a) Definition of the Young diagram $D_{a}$ in terms of the Young diagram $D_{a"}$. (b) Definition of the Young diagram $D_{b}$ in terms of the Young diagram $D_{b'}$.
  • Figure 5: Induction scheme for the proof of \ref{['main result on Gysin coherent characteristic classes with respect to x']}. The outer induction runs over the dimension $g$ of the ambient Grassmannian $G$, starting with $g = 0$, while the inner induction runs for every fixed embedding $i' \colon X' \rightarrow G' = G_{k'}(\mathbb{C}^{m'+k'})$ in $\mathcal{X}$ with $\operatorname{dim}G' = g$ and $d' := \operatorname{dim}X'$ over the formal codimension $l := d' - |b'|$ of partitions $b' \in \mathcal{P}(m', k')$ in $X'$, starting with $l = 0$.

Theorems & Definitions (66)

  • Theorem 1.1: Uniqueness Theorem
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4: Base Change
  • proof
  • Proposition 2.5
  • proof
  • ...and 56 more