Symplectic homology of complements of smooth divisors

Diogo, Luís; Lisi, Samuel

doi:10.1112/topo.12105

Cited by 34 publications

(65 citation statements)

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“…This is obtained by means of generic choice of the geometric data, as opposed to using abstract perturbations. This is an important preliminary step in the computation of this chain complex in [DL18]. Furthermore, in [DL18], we justify that the homology of this complex is indeed the symplectic homology of W .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 59%

“…We begin by summarizing some constructions from [DL18], specifically Proposition 2.1 ([DL18, Lemma 2.2]). Let pW , dλq be a Liouville domain with boundary Y " BW .…”

Section: Set-upmentioning

confidence: 99%

“…Notice that these Hamiltonians are Morse-Bott non-degenerate except at tlog 2uˆY . These orbits will not play a role because they can never arise as boundaries of relevant moduli spaces (see [DL18,Lemma 4.8]).…”

Section: Set-upmentioning

confidence: 99%

“…We will see that this latter degeneration can't occur in low dimensional moduli spaces, at least under our monotonicity assumptions. For energy reasons, these Floer cylinders with cascades will not break at constant Hamiltonian trajectories in p´8, log 2sˆY (see [DL18,Lemma 4.8]).…”

Section: Split Symplectic Homology Moduli Spacesmentioning

confidence: 99%

“…For such a curve, we may assume that H is identically 0, and thus this example includes J Y -holomorphic curves. This is of independent interest, and is useful in [DL18]. Part of this was sketched in [EGH00, Section 2.9.2].…”

Section: 22mentioning

confidence: 99%

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Morse–Bott split symplectic homology

Diogo

Lisi

2019

J. Fixed Point Theory Appl.

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We introduce a chain complex associated to a Liouville domain pW , dλq whose boundary Y admits a Boothby-Wang contact form (i.e. is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of W , in the spirit of Morse-Bott homology [Bou02,Fra04,BO09b]. The homology of this complex is the symplectic homology ofLet X be obtained from W by collapsing the boundary Y along Reeb orbits, giving a codimension 2 symplectic submanifold Σ. Under monotonicity assumptions on X and Σ, we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential. Contents arXiv:1804.08013v2 [math.SG] 21 Nov 2018Definition 2.6. An admissible almost complex structure J X on X is compatible with ω and its restriction to ϕpUq is the push-forward by ϕ of a bundle almost complex structure on N Σ.An almost complex structure J W on pW, dλq is admissible if J W " ψ˚J X for an admissible J X . In particular, such almost complex structures are cylindrical and Reeb-invariant on W zW .A compatible almost complex structure J Y on the symplectization RˆY is admissible if J Y is cylindrical and Reeb-invariant.In the following, we will identify W with XzΣ by means of the diffeomorphism ψ and identify the corresponding almost complex structures. By an abuse of notation, 6 LUÍS DIOGO AND SAMUEL T. LISI we will both write π Σ : Y Ñ Σ to denote the quotient map that collapses the Reeb fibres, and π Σ : RˆY Ñ Σ to denote the composition of this projection with the projection to Y .Definition 2.7. Denote the space of almost complex structures in Σ that are compatible with ω Σ by J Σ .Let J Y denote the space of admissible almost complex structures on RˆY . Then, the projection dπ Σ induces a diffeomorphism between J Y and J Σ .Let J W denote the space of admissible almost complex structures on W . By Proposition 2.3, for any J W P J W , we obtain an almost complex structure J Σ .Denote this map by P : J W Ñ J Σ . This map is surjective and open by Proposition 2.3. For any given J Σ P J Σ , P´1pJ Σ q consist of almost complex structures on W that differ in W , or equivalently, can be identified (using ψ) with almost complex structures on X that differ in V " XzϕpUq. The chain complexWe will now describe the chain complex for the split symplectic homology associated to W .

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 59%

“…We begin by summarizing some constructions from [DL18], specifically Proposition 2.1 ([DL18, Lemma 2.2]). Let pW , dλq be a Liouville domain with boundary Y " BW .…”

Section: Set-upmentioning

confidence: 99%

Section: Set-upmentioning

confidence: 99%

Section: Split Symplectic Homology Moduli Spacesmentioning

confidence: 99%

Section: 22mentioning

confidence: 99%

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Morse–Bott split symplectic homology

Diogo

Lisi

2019

J. Fixed Point Theory Appl.

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Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel

2017

Algebra, Geometry, and Physics in the 21st Century

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Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and establish a relation with enumerative geometry.

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