##
**On symplectic uniruling of Hamiltonian fibrations.**
*(English)*
Zbl 1250.53079

Questions about the existence of rational curves in algebraic varieties are among the most fundamental in algebraic geometry. One natural such question is whether there is a rational curve through every point, a property called uniruledness. With the advent of Gromov’s theory of pseudoholomorphic curves an obvious extension of this question is whether in a given symplectic manifold there is a persistent pseudoholomorphic curve through every point, in the sense that there is a non-vanishing Gromov-Witten invariant counting genus zero curves with a point constraint (and possibly other constraints) – this notion is called symplectic uniruledness. Early work on this problem yielded close parallels with algebraic geometry: the work of Ruan and Kollar [Y. Ruan, Turk. J. Math. 23, No. 1, 161–231 (1999; Zbl 0967.53055)] showed that if a smooth complex projective variety is uniruled in the traditional sense then it is symplectically uniruled. Other classes of symplectic manifold have since been shown to be symplectically uniruled, for instance any symplectic manifold admitting a Hamiltonian circle action [D. McDuff, Duke Math. J. 146, No. 3, 449–507 (2009; Zbl 1183.53080)] or Hamiltonian fibrations over symplectic base manifolds whose fibres are uniruled [T.-J. Li and Y. Ruan, “Uniruled symplectic divisors”, arXiv:0711.4254].

The paper under review extends this list of results to include total spaces of Hamiltonian fibrations over a symplectically uniruled base manifold.

Theorem 1.2. Let \((F,\omega)\overset{\iota}{\hookrightarrow}(P,\omega_{P,\kappa})\overset{\pi}{\to}{(B,\omega_B)}\) be a cohomologically split Hamiltonian fibration. Assume that \((F,\omega)\) is semipositive relative to \(P\) and that \((B,\omega_B)\) is symplectically uniruled for some class \(\sigma_B\in H_2(B;\mathbb{Z})\) admitting only simple decompositions. Then \((P,\omega_{P,\kappa})\) is also symplectically uniruled.

A Hamiltonian fibration is a fibre bundle \(F\to P\to B\) with symplectic fibres \(F\) whose structure group is the Hamiltonian group of \(F\). These can be characterised as fibre bundles whose total space admits a closed 2-form which pulls back to the symplectic form on each fibre. To this 2-form we can add a large multiple \(\kappa\pi^*\omega_B\) of the pullback of the symplectic form \(\omega_B\) on \(B\) and we obtain a symplectic form \(\omega_{P,\kappa}\) on \(P\).

The paper assumes that \(B\) is symplectically uniruled and that the rational cohomology of \(P\) is (as a ring) the tensor product of the rational cohomologies of \(B\) and of \(F\) (this cohomological splitting has long been conjectured [F. Lalonde and D. McDuff, Topology 42, No. 2, 309–347 (2003; Zbl 1032.53077); errata ibid. 44, No. 6, 1301–1303 (2005)] to hold in general). Moreover the paper assumes two technical conditions: firstly that the uniruling of the base consists of simple stable pseudoholomorphic curves (“admitting only simple decompositions”); secondly that the fibre is semipositive relative to the total space. This semipositivity assumption states that for all classes \(A\in H_2(F,\mathbb{Z})\) in the image of the Hurewicz homomorphism having positive symplectic area: \[ c_1^v(\iota(A))\geq 3-n_P\Rightarrow c_1^v(\iota(A))\geq 0. \] Here \(c_1^v\) is the vertical first Chern class and \(\iota(A)\) is the image of \(A\) in \(H_2(P;\mathbb{Z})\). This assumption is necessary for achieving transversality for a class of fibred almost complex structures.

The main tool used to prove Theorem 1.2 is a product formula from [C. Hyvrier, J. Symplectic Geom. 10, No. 2, 247–324 (2012; Zbl 1273.53071)] which expresses the Gromov-Witten invariants of \(P\) in terms of the Gromov-Witten invariants of \(B\) and the Gromov-Witten invariants of a sub-fibration obtained by restricting \(P\to B\) to a holomorphic curve \(C\subset B\). The latter are shown to be nonzero in the case required for the proof of Theorem 1.2 by exhibiting a nontrivial Seidel element in the quantum cohomology of the fibre using a lemma of McDuff (Lemma 2.5).

The paper also explains generalisations to \(k\)-point Gromov-Witten invariants (mimicking rational connectedness in algebraic geometry) and also how the Weinstein conjecture for separating contact-type hypersurfaces follows from uniruledness of the ambient manifold, so the results of this paper can be seen as extending results of [H. Hofer and C. Viterbo, Commun. Pure Appl. Math. 45, No. 5, 583–622 (1992; Zbl 0773.58021); G. Lu, Math. Res. Lett. 7, No. 4, 383–387 (2000; Zbl 0983.53062)].

The paper under review extends this list of results to include total spaces of Hamiltonian fibrations over a symplectically uniruled base manifold.

Theorem 1.2. Let \((F,\omega)\overset{\iota}{\hookrightarrow}(P,\omega_{P,\kappa})\overset{\pi}{\to}{(B,\omega_B)}\) be a cohomologically split Hamiltonian fibration. Assume that \((F,\omega)\) is semipositive relative to \(P\) and that \((B,\omega_B)\) is symplectically uniruled for some class \(\sigma_B\in H_2(B;\mathbb{Z})\) admitting only simple decompositions. Then \((P,\omega_{P,\kappa})\) is also symplectically uniruled.

A Hamiltonian fibration is a fibre bundle \(F\to P\to B\) with symplectic fibres \(F\) whose structure group is the Hamiltonian group of \(F\). These can be characterised as fibre bundles whose total space admits a closed 2-form which pulls back to the symplectic form on each fibre. To this 2-form we can add a large multiple \(\kappa\pi^*\omega_B\) of the pullback of the symplectic form \(\omega_B\) on \(B\) and we obtain a symplectic form \(\omega_{P,\kappa}\) on \(P\).

The paper assumes that \(B\) is symplectically uniruled and that the rational cohomology of \(P\) is (as a ring) the tensor product of the rational cohomologies of \(B\) and of \(F\) (this cohomological splitting has long been conjectured [F. Lalonde and D. McDuff, Topology 42, No. 2, 309–347 (2003; Zbl 1032.53077); errata ibid. 44, No. 6, 1301–1303 (2005)] to hold in general). Moreover the paper assumes two technical conditions: firstly that the uniruling of the base consists of simple stable pseudoholomorphic curves (“admitting only simple decompositions”); secondly that the fibre is semipositive relative to the total space. This semipositivity assumption states that for all classes \(A\in H_2(F,\mathbb{Z})\) in the image of the Hurewicz homomorphism having positive symplectic area: \[ c_1^v(\iota(A))\geq 3-n_P\Rightarrow c_1^v(\iota(A))\geq 0. \] Here \(c_1^v\) is the vertical first Chern class and \(\iota(A)\) is the image of \(A\) in \(H_2(P;\mathbb{Z})\). This assumption is necessary for achieving transversality for a class of fibred almost complex structures.

The main tool used to prove Theorem 1.2 is a product formula from [C. Hyvrier, J. Symplectic Geom. 10, No. 2, 247–324 (2012; Zbl 1273.53071)] which expresses the Gromov-Witten invariants of \(P\) in terms of the Gromov-Witten invariants of \(B\) and the Gromov-Witten invariants of a sub-fibration obtained by restricting \(P\to B\) to a holomorphic curve \(C\subset B\). The latter are shown to be nonzero in the case required for the proof of Theorem 1.2 by exhibiting a nontrivial Seidel element in the quantum cohomology of the fibre using a lemma of McDuff (Lemma 2.5).

The paper also explains generalisations to \(k\)-point Gromov-Witten invariants (mimicking rational connectedness in algebraic geometry) and also how the Weinstein conjecture for separating contact-type hypersurfaces follows from uniruledness of the ambient manifold, so the results of this paper can be seen as extending results of [H. Hofer and C. Viterbo, Commun. Pure Appl. Math. 45, No. 5, 583–622 (1992; Zbl 0773.58021); G. Lu, Math. Res. Lett. 7, No. 4, 383–387 (2000; Zbl 0983.53062)].

Reviewer: Jonathan D. Evans (Oxford)

### MSC:

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

55R10 | Fiber bundles in algebraic topology |

### Keywords:

Hamiltonian fibration; Gromov-Witten invariant; symplectic uniruledness; Weinstein conjecture### Citations:

Zbl 0967.53055; Zbl 1183.53080; Zbl 1032.53077; Zbl 0773.58021; Zbl 0983.53062; Zbl 1273.53071
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\textit{C. Hyvrier}, Algebr. Geom. Topol. 12, No. 2, 1145--1163 (2012; Zbl 1250.53079)

### References:

[1] | A Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956) 157 · Zbl 0073.37503 |

[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 · Zbl 0773.58021 |

[4] | J Hu, T J Li, Y Ruan, Birational cobordism invariance of uniruled symplectic manifolds, Invent. Math. 172 (2008) 231 · Zbl 1163.53055 |

[5] | C Hyvrier, A product formula for Gromov-Witten invariants, to appear in J. Symplectic Geom. · Zbl 1273.53071 |

[6] | J K\?dra, Restrictions on symplectic fibrations, Differential Geom. Appl. 21 (2004) 93 · Zbl 1052.57038 |

[7] | J Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. 32, Springer (1996) · Zbl 0877.14012 |

[8] | J Kollár, Low degree polynomial equations: arithmetic, geometry and topology, Progr. Math. 168, Birkhäuser (1998) 255 · Zbl 0970.14001 |

[9] | F Lalonde, D McDuff, Symplectic structures on fiber bundles, Topology 42 (2003) 309 · Zbl 1032.53077 |

[10] | F Lalonde, D McDuff, L Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369 · Zbl 0907.58004 |

[11] | J Li, G Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds (editor R J Stern), First Int. Press Lect. Ser. I, Int. Press (1998) 47 · Zbl 0978.53136 |

[12] | T Li, Y Ruan, Uniruled symplectic divisors · Zbl 1319.14057 |

[13] | G Liu, G Tian, Weinstein conjecture and GW-invariants, Commun. Contemp. Math. 2 (2000) 405 · Zbl 1008.53071 |

[14] | G Lu, The Weinstein conjecture in the uniruled manifolds, Math. Res. Lett. 7 (2000) 383 · Zbl 0983.53062 |

[15] | D McDuff, Hamiltonian \(S^1\)-manifolds are uniruled, Duke Math. J. 146 (2009) 449 · Zbl 1183.53080 |

[16] | D McDuff, D Salamon, Introduction to symplectic topology, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press (1998) · Zbl 0844.58029 |

[17] | D McDuff, D Salamon, \(J\)-holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publ. 52, Amer. Math. Soc. (2004) · Zbl 1064.53051 |

[18] | Y Ruan, Virtual neighborhoods and pseudo-holomorphic curvesokova Geometry-Topology Conference” (1999) 161 · Zbl 0967.53055 |

[19] | Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259 · Zbl 0860.58005 |

[20] | P Seidel, \(\pi_1\) of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046 · Zbl 0928.53042 |

[21] | C Viterbo, A proof of Weinstein’s conjecture in \(\mathbfR^{2n}\), Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 337 · Zbl 0631.58013 |

[22] | A Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential Equations 33 (1979) 353 · Zbl 0388.58020 |

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