[Paper Review] Generalized Witten Genus and Vanishing Theorems

Qingtao Chen, Fengling Han+1 more|arXiv (Cornell University)|Mar 11, 2010

Advanced Algebra and Geometry16 references3 citations

TL;DR

This paper introduces a generalized Witten genus for spin$^c$ manifolds, taking values in level 1 modular forms with integral Fourier coefficients on string$^c$ manifolds. It establishes Landweber-Stong type vanishing theorems for both the generalized Witten genus and a mod 2 analogue on specific classes of manifolds, including generalized complete intersections in products of complex projective spaces.

ABSTRACT

We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin$^c$ manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalized Witten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.

Motivation & Objective

• To extend the Witten genus to spin$^c$ manifolds using modular forms with integral Fourier coefficients.
• To define a mod 2 analogue of the Witten genus for 8k+2 dimensional spin manifolds.
• To establish Landweber-Stong type vanishing theorems for the generalized Witten genus on string$^c$ manifolds.
• To prove vanishing theorems for the mod 2 Witten genus on generalized complete intersections in products of complex projective spaces.
• To unify and generalize existing vanishing results in the context of modular forms and spin geometry.

Proposed method

• Construct a generalized Witten genus using equivariant elliptic cohomology and modular forms of level 1.
• Define string$^c$ manifolds as a natural class of spin$^c$ manifolds where the generalized genus takes values in modular forms with integral Fourier expansions.
• Apply the Landweber exact functor theorem to derive vanishing theorems for the generalized Witten genus.
• Introduce a mod 2 version of the Witten genus via reduction modulo 2 of the elliptic genus on 8k+2 dimensional spin manifolds.
• Use bordism-theoretic techniques and characteristic classes to analyze the vanishing behavior of the genus on complete intersections.
• Leverage the structure of products of complex projective spaces to study generalized complete intersections and their genus invariants.

Experimental results

Research questions

• RQ1 Can the Witten genus be extended to spin$^c$ manifolds with values in modular forms with integral Fourier coefficients?
• RQ2 What conditions on spin$^c$ manifolds ensure that the generalized Witten genus takes values in level 1 modular forms?
• RQ3 Does a mod 2 analogue of the Witten genus exist for 8k+2 dimensional spin manifolds, and what are its properties?
• RQ4 Under what topological conditions does the generalized Witten genus vanish on string$^c$ manifolds?
• RQ5 Do vanishing theorems analogous to Landweber-Stong hold for the generalized Witten genus on generalized complete intersections in products of complex projective spaces?

Key findings

• The generalized Witten genus is constructed for string$^c$ manifolds and takes values in level 1 modular forms with integral Fourier coefficients.
• A mod 2 analogue of the Witten genus is defined for 8k+2 dimensional spin manifolds, extending the classical genus to characteristic 2.
• Landweber-Stong type vanishing theorems are proven for the generalized Witten genus on string$^c$ manifolds.
• Vanishing theorems are established for the mod 2 Witten genus on generalized complete intersections in products of complex projective spaces.
• The results generalize classical vanishing theorems in elliptic cohomology and provide new obstructions to the existence of certain geometric structures.
• The framework unifies and extends previous results on genus vanishing via modular forms and bordism theory.