Iterated residue, toric forms and Witten genus

TL;DR

The paper develops and applies the concept of iterated residue to generalized Bott manifolds, enabling explicit cohomological evaluations, and connects these to toric forms and the Witten genus. It yields new theta-function identities for Borisov–Gunnells toric forms, including twisted Rogers–Ramanujan-type formulas, and proves vanishing results under coprimality conditions for toric forms. It further derives Landweber–Stong-type vanishing results for the Witten genus of string complete intersections in two-staged generalized Bott manifolds, using double-residue reformulations and periodicity arguments. Collectively, the work demonstrates the power of iterated residues in producing concrete, modular-analytic consequences for toric and string-geometric invariants.

Abstract

We introduce the notion of {\em iterated residue} to study generalized Bott manifolds. When applying the iterated residues to compute the Borisov-Gunnells toric form and the Witten genus of certain toric varieties as well as complete intersections, we obtain interesting vanishing results and some theta function identities, one of which is a twisted version of a classical Rogers-Ramanujan type formula.

Theorems & Definitions (33)

• Remark 1
• Theorem 1.1: Theorem \ref{['main']}
• Remark 2
• Theorem 1.2: Theorem \ref{['main1']}
• Remark 3
• Proposition 1.1: Proposition \ref{['second']}
• Proposition 1.2: Proposition \ref{['Hir']}
• Theorem 1.3: Theorem \ref{['vanish toric']}
• Remark 4
• Theorem 1.4: Theorem \ref{['technique']}