On Morse-Bott degenerations with 𝐶𝑟𝑖𝑡(𝜋)≃ℂℙ^{𝕜} and Floer homology

Jerby, Yochay

doi:10.1090/S0002-9939-2014-12117-4

On Morse-Bott degenerations with $Crit(\pi ) \simeq \mathbb {C}P^k$ and Floer homology
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by Yochay Jerby PDF

Proc. Amer. Math. Soc. 142 (2014), 3731-3740 Request permission

Abstract:

Morse-Bott fibrations generalize the class of Lefschetz fibrations, allowing for non-isolated singularities. A Morse-Bott degeneration is a Morse-Bott fibration $\pi : X \rightarrow \mathbb {D}$ whose only singular value is $0 \in \mathbb {D}$. We show that Morse-Bott degenerations with $Crit(\pi )=\mathbb {C}P^k$ for $0<k$ admit restrictions which are unique to the non-isolated case. These restrictions are obtained as an application of methods of Floer homology for monotone Lagrangian submanifolds.

References

Michèle Audin, François Lalonde, and Leonid Polterovich, Symplectic rigidity: Lagrangian submanifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 271–321. MR 1274934, DOI 10.1007/978-3-0348-8508-9_{1}1
D. M. Austin and P. J. Braam, Morse-Bott theory and equivariant cohomology, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 123–183. MR 1362827
Paul Biran and Kai Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv. 76 (2001), no. 4, 712–753. MR 1881704, DOI 10.1007/s00014-001-8326-7
P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, preprint (2007). Can be found at +http://arxiv.org/pdf/0708.4221+.
Paul Biran and Octav Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), no. 5, 2881–2989. MR 2546618, DOI 10.2140/gt.2009.13.2881
C. H. Clemens Jr., Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities, Trans. Amer. Math. Soc. 136 (1969), 93–108. MR 233814, DOI 10.1090/S0002-9947-1969-0233814-9
S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), no. 2, 205–236. MR 1802722
Joel Fine and Dmitri Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geom. Topol. 14 (2010), no. 3, 1723–1763. MR 2679581, DOI 10.2140/gt.2010.14.1723
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. MR 316745, DOI 10.1215/kjm/1250523432
János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765–779. MR 1189503
John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
Yong-Geun Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993. MR 1223659, DOI 10.1002/cpa.3160460702
Yong-Geun Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 7 (1996), 305–346. MR 1389956, DOI 10.1155/S1073792896000219
Tim Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007), 759–828. MR 2302502, DOI 10.2140/gt.2007.11.759
Paul Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), no. 1, 103–149 (English, with English and French summaries). MR 1765826
Paul Seidel and Ivan Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006), no. 3, 453–514. MR 2254624, DOI 10.1215/S0012-7094-06-13432-4
Claire Voisin, Hodge theory and complex algebraic geometry. II, Reprint of the 2003 English edition, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2007. Translated from the French by Leila Schneps. MR 2449178
Katrin Wehrheim and Chris T. Woodward, Floer cohomology and geometric composition of Lagrangian correspondences, Adv. Math. 230 (2012), no. 1, 177–228. MR 2900542, DOI 10.1016/j.aim.2011.11.009