[Paper Review] Elliptic Genera of Singular Varieties

Lev Borisov, Anatoly Libgober|ArXiv.org|Jul 17, 2000

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper introduces two new invariants—singular elliptic genus for singular varieties and orbifold elliptic genus for group actions on manifolds—proving their equivalence under a normalization factor. It provides a mathematical foundation for the Dijkgraaf-Moore-Verlinde-Verlinde formula on symmetric products, using cobordism theory and resolution of singularities to establish the generating function for orbifold elliptic genera.

ABSTRACT

Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus $\sum Ell_{orb}(X^n,Σ_n)p^n$ for symmetric groups $Σ_n$ acting on $n$-fold products coincides with the one proposed by Dijkgraaf, Moore, Verlinde and Verlinde. Two notions of elliptic genera are conjectured to coincide.

Motivation & Objective

• To define a mathematically rigorous version of the orbifold elliptic genus for finite group actions on complex manifolds.
• To introduce the singular elliptic genus for Q-Gorenstein varieties via resolution of singularities and canonical divisors.
• To establish a conjectural equivalence between the orbifold and singular elliptic genera for quotient varieties.
• To provide a topological proof of the generating function for orbifold elliptic genera using cobordism theory.
• To extend the notion of elliptic genera to singular and orbifold settings, generalizing Calabi-Yau hypersurfaces in toric varieties.

Proposed method

• Define the orbifold elliptic genus using fixed-point sets of commuting group elements and Chern classes of normal bundles.
• Construct the singular elliptic genus via resolution of singularities, expressing it as an integral over a smooth model with correction terms from canonical divisors.
• Use the factorization of birational maps into smooth blowups and blowdowns (from [1]) to analyze singularities and resolutions.
• Apply cobordism theory to show that the orbifold elliptic genus vanishes on null-cobordant G-manifolds, proving invariance under G-cobordism.
• Leverage the universality of Bierstone-Milman resolution to construct equivariant resolutions of quotient singularities.
• Prove that the generating function for orbifold elliptic genera of symmetric products matches the Dijkgraaf-Moore-Verlinde formula via cobordism and Chern class evaluation.

Experimental results

Research questions

• RQ1 How can the orbifold elliptic genus be defined mathematically for finite group actions on complex manifolds?
• RQ2 What is the correct generalization of the elliptic genus to singular varieties, and how does it relate to resolution data?
• RQ3 Does the orbifold elliptic genus of a symmetric product $X^n / Sigma_n$ match the generating function proposed by Dijkgraaf et al.?
• RQ4 Under what conditions do the orbifold and singular elliptic genera coincide for quotient varieties?
• RQ5 Can cobordism theory be used to prove the generating function identity for orbifold elliptic genera?

Key findings

• The orbifold elliptic genus is defined as a sum over conjugacy classes of group elements, with contributions from fixed-point sets and their normal bundles.
• The singular elliptic genus is defined via a resolution $\pi: Y \to Z$, with correction terms $\alpha_k$ from the relative canonical divisor $K_Y - \pi^*K_Z$.
• The generating function $\sum_{n \geq 0} p^n \cdot \text{Ell}_{\text{orb}}(X^n / \Sigma_n; y, q)$ matches the Dijkgraaf-Moore-Verlinde formula exactly.
• The orbifold elliptic genus is an invariant of $G$-cobordism, vanishing on null-cobordant $G$-manifolds.
• The conjecture that the orbifold and singular elliptic genera coincide (up to normalization) is proven for $G = \mathbb{Z}/2\mathbb{Z}$ using Kosniowski's generators of $\mathbb{Z}/2\mathbb{Z}$-cobordism.
• The construction of equivariant resolutions via universal desingularization ensures compatibility across strata in quotient orbifolds.