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Fundamental groups of rationally connected symplectic manifolds

Part of: Differential topology Symplectic geometry, contact geometry

Published online by Cambridge University Press:  27 August 2025

Alex Pieloch
Alex Pieloch*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA pieloch@mit.edu
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Abstract

We show that the fundamental group of every enumeratively rationally connected closed symplectic manifold is finite. In other words, if a closed symplectic manifold has a non-zero Gromov–Witten invariant with two point insertions, then it has finite fundamental group. We also show that if the spherical homology class associated with such a non-zero Gromov–Witten invariant is holomorphically indecomposable, then the rational second homology of the symplectic manifold has rank one.

Keywords

enumeratively rationally connectedfundamental groupsGromov–Witten invariants

MSC classification

Primary: 57R17: Symplectic and contact topology
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 6 , June 2025 , pp. 1458 - 1482
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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