Algebraic & Geometric Topology

On symplectic uniruling of Hamiltonian fibrations

Clément Hyvrier

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Abstract

Under certain conditions of technical order, we show that closed connected Hamiltonian fibrations over symplectically uniruled manifolds are also symplectically uniruled. As a consequence, we partially extend to nontrivial Hamiltonian fibrations a result of Lu [Math. Res. Lett. 7 (2000) 383–387], stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type. The proof of our result is based on the product formula for Gromov–Witten invariants of Hamiltonian fibrations derived by the author in [arXiv 0904.1492].

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1145-1163.

Dates
Received: 6 April 2011
Revised: 17 February 2012
Accepted: 28 February 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715384

Digital Object Identifier
doi:10.2140/agt.2012.12.1145

Mathematical Reviews number (MathSciNet)
MR2928908

Zentralblatt MATH identifier
1250.53079

Subjects
Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 57R17: Symplectic and contact topology
Secondary: 55R10: Fiber bundles

Keywords
Hamiltonian fibration Gromov–Witten invariant symplectic uniruledness Weinstein Conjecture

Citation

Hyvrier, Clément. On symplectic uniruling of Hamiltonian fibrations. Algebr. Geom. Topol. 12 (2012), no. 2, 1145--1163. doi:10.2140/agt.2012.12.1145. https://projecteuclid.org/euclid.agt/1513715384


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References

  • A Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956) 157–202
  • B Chen, A-M Li, Symplectic virtual localization of Gromov–Witten invariants
  • H Hofer, C Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992) 583–622
  • J Hu, T-J Li, Y Ruan, Birational cobordism invariance of uniruled symplectic manifolds, Invent. Math. 172 (2008) 231–275
  • C Hyvrier, A product formula for Gromov–Witten invariants, to appear in J. Symplectic Geom.
  • J Kędra, Restrictions on symplectic fibrations, Differential Geom. Appl. 21 (2004) 93–112 With an appendix by the author and K Ono
  • J Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. 32, Springer, Berlin (1996)
  • J Kollár, Low degree polynomial equations: arithmetic, geometry and topology, from: “European Congress of Mathematics, Vol. I (Budapest, 1996)”, Progr. Math. 168, Birkhäuser, Basel (1998) 255–288
  • F Lalonde, D McDuff, Symplectic structures on fiber bundles, Topology 42 (2003) 309–347
  • F Lalonde, D McDuff, L Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369–385
  • J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: “Topics in symplectic $4$-manifolds (Irvine, CA, 1996)”, (R J Stern, editor), First Int. Press Lect. Ser. I, Int. Press, Cambridge, MA (1998) 47–83
  • T Li, Y Ruan, Uniruled symplectic divisors
  • G Liu, G Tian, Weinstein conjecture and GW–invariants, Commun. Contemp. Math. 2 (2000) 405–459
  • G Lu, The Weinstein conjecture in the uniruled manifolds, Math. Res. Lett. 7 (2000) 383–387
  • D McDuff, Hamiltonian $S^1$–manifolds are uniruled, Duke Math. J. 146 (2009) 449–507
  • D McDuff, D Salamon, Introduction to symplectic topology, second edition, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, New York (1998)
  • D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publ. 52, Amer. Math. Soc. (2004)
  • Y Ruan, Virtual neighborhoods and pseudo-holomorphic curves, from: “Proceedings of 6th Gökova Geometry-Topology Conference”, 23 (1999) 161–231
  • Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259–367
  • P Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046–1095
  • C Viterbo, A proof of Weinstein's conjecture in ${\bf R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 337–356
  • A Weinstein, On the hypotheses of Rabinowitz' periodic orbit theorems, J. Differential Equations 33 (1979) 353–358