Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-24T05:09:11.243Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EXISTENCE OF A GLOBAL NEIGHBOURHOOD

Part of: Complex manifolds Projective and enumerative geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  21 July 2015

TOM COATES  and
HIROSHI IRITANI
TOM COATES
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom e-mail: t.coates@imperial.ac.uk
HIROSHI IRITANI
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan e-mail: iritani@math.kyoto-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that a complex manifold M is locally embedded into a higher-dimensional neighbourhood as a submanifold. We show that, if the local neighbourhood germs are compatible in a suitable sense, then they glue together to give a global neighbourhood of M. As an application, we prove a global version of Hertling–Manin's unfolding theorem for germs of TEP structures; this has applications in the study of quantum cohomology.

MSC classification

Primary: 32Q99: None of the above, but in this section
Secondary: 32Q40: Embedding theorems 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 717 - 726
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1. Dubrovin, B., Geometry of 2D topological field theories, in Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Mathematics, vol. 1620 (Springer, Berlin, 1996), 120–348.Google Scholar
2. Dubrovin, B., Geometry and analytic theory of Frobenius manifolds, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra vol. II (1998), 315–326.Google Scholar
3. Hertling, C., tt * geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.Google Scholar
4. Hertling, C. and Manin, Y., Unfoldings of meromorphic connections and a construction of Frobenius manifolds, in Frobenius manifolds, Aspects of Mathematics, vol. E36 (Vieweg, Wiesbaden, 2004), 113–144.Google Scholar
5. Joyce, D., D-manifolds and d-orbifolds: A theory of derived differential geometry, Available at http://people.maths.ox.ac.uk/joyce/dmbook.pdf, 2012.Google Scholar
6. Kashiwara, M., D-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217 (American Mathematical Society, Providence, RI, 2003). (Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics.)Google Scholar
7. Reichelt, T., A construction of Frobenius manifolds with logarithmic poles and applications, Comm. Math. Phys. 287 (3) (2009), 1145–1187.Google Scholar
8. Shiota, M., Some results on formal power series and differentiable functions, Publ. Res. Inst. Math. Sci. 12 (1) (1967/77), 49–53.Google Scholar
9. Whitney, H. and Bruhat, F., Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33 (1959), 132–160.Google Scholar