Four paths from birational geometry to the elliptic genus

2025/05/25 by Andrzej Weber, Weber, Andrzej
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2505.19008

openalex publication_date 2025/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The article presents four reasons why the elliptic genus is the most general characteristic class that admits a generalization to singular spaces. We prove that the elliptic characteristic class (with an additional factor) is essentially the only characteristic class invariant under certain modifications, such as the Atiyah flop, Grassmannian flops, and modifications of Bott-Samelson resolutions.This result confirms and extends Totaro's result concerning the cobordism ring modulo classical flops. However, our approach is based on local calculus in equivariant cohomology.

Citations

• Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes
• Twisted motivic Chern class and stable envelopes
• A Schubert basis in equivariant elliptic cohomology
• Elliptic genera, real algebraic varieties and quasi-Jacobi forms
• A survey of characteristic classes of singular spaces
• Hirzebruch classes and motivic Chern classes for singular spaces
• Komplexe elliptische Geschlechter und S¹-aequivariante Kobordismustheorie (Complex elliptic genera and S¹-equivariant cobordism theory)
• McKay correspondence for elliptic genera
• Elliptic Genera of Singular Varieties
• Torification and Factorization of Birational Maps
• Stringy Hodge numbers of varieties with Gorenstein canonical singularities
• The moment map and equivariant cohomology

Related

• Singular elliptic genus of normal surfaces
• Algebraic elliptic cohomology theory and flops, I
• On orbifold elliptic genus
• The Witten genus and equivariant elliptic cohomology
• Elliptic genera, real algebraic varieties and quasi-Jacobi forms