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Dualization invariance and a new complex elliptic genus

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Published/Copyright: October 30, 2012
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 692

Abstract.

We define a new elliptic genus ψ on the complex bordism ring. With coefficients in ℤ[1/2], we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences ℙ(E)-ℙ(E*) of projective bundles and their duals onto a polynomial ring on four generators in degrees 2, 4, 6 and 8. As an alternative geometric description of ψ, we prove that it is the universal genus which is multiplicative in projective bundles over Calabi–Yau 3-folds. With the help of the q-expansion of modular forms we will see that for a complex manifold M, the value ψ(M) is a holomorphic Euler characteristic. We also compare ψ with Krichever–Höhn's complex elliptic genus and see that their only common specializations are Ochanine's elliptic genus and the χy-genus.

First and foremost, I would like to thank D. Kotschick for his continued support and encouragement, and in particular for raising the question of determining the ideal ℐU. Moreover, I am deeply grateful to him for giving me ample helpful advice and corrections during the development of this paper, respectively my Master Thesis. Next, I would like to thank B. Totaro for raising the question which motivated Theorem 5.1, as well as C. McTague for providing me with some useful advice. Finally, thanks to the referees and to P. Landweber for helpful suggestions.

Received: 2011-10-3
Revised: 2012-7-16
Published Online: 2012-10-30
Published in Print: 2014-7-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Sur un problème de compatibilité local-global modulo p pour GL2
  3. Dualization invariance and a new complex elliptic genus
  4. The rigidity theorems for Lagrangian self-shrinkers
  5. On intermediate conformal nets
  6. On the blow-up of four-dimensional Ricci flow singularities
  7. Remarks on Hamilton's Compactness Theorem for Ricci flow
  8. Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
  9. Localizing the Elliott conjecture at strongly self-absorbing C*-algebras, II
https://doi.org/10.1515/crelle-2012-0085

Articles in the same Issue

  1. Frontmatter
  2. Sur un problème de compatibilité local-global modulo p pour GL2
  3. Dualization invariance and a new complex elliptic genus
  4. The rigidity theorems for Lagrangian self-shrinkers
  5. On intermediate conformal nets
  6. On the blow-up of four-dimensional Ricci flow singularities
  7. Remarks on Hamilton's Compactness Theorem for Ricci flow
  8. Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
  9. Localizing the Elliott conjecture at strongly self-absorbing C*-algebras, II
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