On symplectic uniruling of Hamiltonian fibrations

Hyvrier, Clément¹

¹Mathematics Department, Uppsala Universitet, Box 480, SE-75106 Uppsala, Sweden

Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 1145-1163

Résumé

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].

Détail

DOI : 10.2140/agt.2012.12.1145

Classification : 53D45, 57R17, 55R10
Keywords: Hamiltonian fibration, Gromov–Witten invariant, symplectic uniruledness, Weinstein Conjecture

Bibliographie

[1] A Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956) 157

[2] B Chen, A M Li, Symplectic virtual localization of Gromov–Witten invariants

[3] H Hofer, C Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992) 583

[4] J Hu, T J Li, Y Ruan, Birational cobordism invariance of uniruled symplectic manifolds, Invent. Math. 172 (2008) 231

[5] C Hyvrier, A product formula for Gromov–Witten invariants, to appear in J. Symplectic Geom.

[6] J Kȩdra, Restrictions on symplectic fibrations, Differential Geom. Appl. 21 (2004) 93

[7] J Kollár, Rational curves on algebraic varieties, 32, Springer (1996)

[8] J Kollár, Low degree polynomial equations: arithmetic, geometry and topology, from: "European Congress of Mathematics, Vol. I (Budapest, 1996)", Progr. Math. 168, Birkhäuser (1998) 255

[9] F Lalonde, D Mcduff, Symplectic structures on fiber bundles, Topology 42 (2003) 309

[10] F Lalonde, D Mcduff, L Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369

[11] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: "Topics in symplectic 4-manifolds (Irvine, CA, 1996)" (editor R J Stern), First Int. Press Lect. Ser. I, Int. Press (1998) 47

[12] T Li, Y Ruan, Uniruled symplectic divisors

[13] G Liu, G Tian, Weinstein conjecture and GW–invariants, Commun. Contemp. Math. 2 (2000) 405

[14] G Lu, The Weinstein conjecture in the uniruled manifolds, Math. Res. Lett. 7 (2000) 383

[15] D Mcduff, Hamiltonian S1–manifolds are uniruled, Duke Math. J. 146 (2009) 449

[16] D Mcduff, D Salamon, Introduction to symplectic topology, , The Clarendon Press, Oxford Univ. Press (1998)

[17] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004)

[18] Y Ruan, Virtual neighborhoods and pseudo-holomorphic curves, from: "Proceedings of 6th Gökova Geometry-Topology Conference" (1999) 161

[19] Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259

[20] P Seidel, π1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046

[21] C Viterbo, A proof of Weinstein’s conjecture in R2n, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 337

[22] A Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979) 353