Lefschetz fibrations and symplectic homology

By Mark McLean. 2007

97 citations recorded by the source corpus.

This paper is about the symplectic topology of Stein manifolds. If we have a symplectic manifold .V; !/, then we say it carries a Stein structure if there exists a complex structure J and an exhausting (ie proper and bounded from below) plurisubharmonic function W V ! R such that ! D dd , where d is defined by d.a/.X / WD da.JX /. The triple .V;J; / is called a Stein manifold. We say that a Stein manifold is of finite type if only has finitely many critical points, each of which is nondegenerate.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/839dc7e1960c990fce40aa1ac9af80c182582dfb

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Legendrian fronts for affine varieties (2016, background): “In short, any generic linear function on an affine variety has complex non– degenerate Morse singularities, and will therefore be a Lefschetz fibration up to isotopy [53].”
Altering symplectic manifolds by homologous recombination (2010, methodology): “More recently, Maydanskiy-Seidel [36] (based on earlier work of Maydanskiy [35]) constructed another nonstandard Liouville structure on the cotangent bundle of the n-sphere for any n ≥ 3, which is different from McLean’s because it does not contain any Lagrangian sphere.”
Contact geometry of the restricted three‐body problem (2010, background): “By Cieliebak’s [4] computation of the symplectic homology of a boundary-connected sum, see also McLean [14, Theorem 2.20] for an improvement concerning the ring structure, we know ) .”
Effect of Legendrian surgery (2009, background): “Their gradings are as follows: jaj D jcmaxj D jbmax 1 j D jb max 2 j 1; jemax j j D n 2; jcminj D jbmin 1 j D jb min 2 j D 1; jemin j j D 0: 7The first examples of exotic symplectic Weinstein structures on T Sn were constructed by M McLean in [35] and also by M Maydanskiy and P Seidel in [34].”
The diffeomorphism type of symplectic fillings (2016, methodology): “Stein manifolds of this kind exist by the work of Seidel–Smith [41] and McLean [31]: in all even dimensions 2n ≥ 8 there are infinitely many distinct finite type Stein manifolds diffeomorphic to R2n; they all have Stein handle decompositions involving critical handles.”
Rabinowitz Floer homology and symplectic homology (2009, methodology): “18, using the multiplicative structure in symplectic homology investigated by McLean [14].”
Topological quantum field theory structure on symplectic cohomology (2010, background): “For Liouville subdomainsW ⊂M , Viterbo [36] constructed a restriction map SH(M)→ SH(W ) and McLean [22] observed that it is a ring homomorphism.”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, background): “Symplectic homology has the structure of a ring with unit [55, 47, 52], the unit being the image of 1 ∈ H(W ) under the map H(W ) ≃ H2n(W,∂W ) → SHn(W ).”