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The Brasselet-Schürmann-Yokura conjecture on $L$-classes of singular varieties

J. Fernández de Bobadilla, I. Pallarés

Abstract

In 2010, Brasselet, Schürmann and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky-MacPherson $L$-class $L_*(X)$ and the Hirzebruch homology class $T_{1*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions which we find of independent interest.

The Brasselet-Schürmann-Yokura conjecture on $L$-classes of singular varieties

Abstract

In 2010, Brasselet, Schürmann and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky-MacPherson -class and the Hirzebruch homology class for a compact complex algebraic variety that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions which we find of independent interest.

Paper Structure

This paper contains 17 sections, 13 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Characteristic classes, from manifolds to singular varieties
  3. Brasselet-Schürmann-Yokura's Hirzebruch classes of singular varieties
  4. Main results
  5. Perverse exact cubical hyperresolutions
  6. Independence of polarizations in the cobordism group
  7. Background
  8. An identity in the Grothendieck group of varieties
  9. Proof of Theorem \ref{['teo:equivalentRHM']}
  10. Rational homology manifolds
  11. Two spectral sequences of perverse sheaves
  12. Proof of Theorem \ref{['teo:equivalentRHM']}
  13. A computation in the cobordism group of self-dual complexes
  14. Polarizations at stalks of local systems
  15. Independence of the polarizing self-duality in the cobordism group
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $X$ is a compact complex algebraic variety that is a rational homology manifold, then

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6: BBD:1982, Saito:1989a, DeCM:2005
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 3.1
  • ...and 15 more