Morse–Bott split symplectic homology

Diogo, Luís; Lisi, Samuel

doi:10.1007/s11784-019-0714-y

Cited by 9 publications

(29 citation statements)

References 34 publications

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“…Proof. This result is similar to [19,Lemma 5.43]. We identify the simple Reeb orbits to which u converges at ±∞ with S 1 × S 1 .…”

Section: Pseudoholomorphic Spheres In σ and Meromorphic Sectionssupporting

confidence: 83%

“…and surjectivity and the statement about the kernel follow from [75, Proposition 2.2] together with [19,Lemma 5.20]. Case 2 is slightly different, since there is an additional puncture at P .…”

Section: From Floer Cylinders To Pseudoholomorphic Curvesmentioning

confidence: 90%

“…We will now define the gradings of the generators. For more details on these formulas, see [19,Section 3.1]. For a critical point p of f Y , and a multiplicity k, we define…”

Section: The Chain Complexmentioning

confidence: 99%

“…We say that a Floer cylinder is simple ifṽ : R × S 1 \ Γ → R × Y projects to a somewhere injective curve in Σ, or if it projects to a constant. Using the notation from [19], we write…”

Section: Morse-bott Split Symplectic Homologymentioning

confidence: 99%

“…Recall that Floer cascades are oriented using the following fiber sum convention (see also [19,Definition 7.5]).…”

Section: Orientationsmentioning

confidence: 99%

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Symplectic homology of complements of smooth divisors

Diogo

Lisi

2019

Journal of Topology

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If (X, ω) is a closed symplectic manifold, and Σ is a smooth symplectic submanifold Poincaré dual to a positive multiple of ω, then X \ Σ can be completed to a Liouville manifold (W, dλ).Under monotonicity assumptions on X and on Σ, we construct a chain complex whose homology computes the symplectic homology of W . We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to Σ and Gromov-Witten invariants of Σ.We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to Σ. Contents

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“…Proof. This result is similar to [19,Lemma 5.43]. We identify the simple Reeb orbits to which u converges at ±∞ with S 1 × S 1 .…”

Section: Pseudoholomorphic Spheres In σ and Meromorphic Sectionssupporting

confidence: 83%

Section: From Floer Cylinders To Pseudoholomorphic Curvesmentioning

confidence: 90%

“…We will now define the gradings of the generators. For more details on these formulas, see [19,Section 3.1]. For a critical point p of f Y , and a multiplicity k, we define…”

Section: The Chain Complexmentioning

confidence: 99%

Section: Morse-bott Split Symplectic Homologymentioning

confidence: 99%

“…Recall that Floer cascades are oriented using the following fiber sum convention (see also [19,Definition 7.5]).…”

Section: Orientationsmentioning

confidence: 99%

See 3 more Smart Citations

Symplectic homology of complements of smooth divisors

Diogo

Lisi

2019

Journal of Topology

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Sections and Unirulings of Families over $\mathbb{P}^{1}$

Pieloch

2024

Geom. Funct. Anal.

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We consider morphisms $\pi : X \to \mathbb{P}^{1}$ of smooth projective varieties over $\mathbb{C}$. We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π.To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

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Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

Bae,

Kang,

Kim

2024

Math. Ann.

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Let Y be a prequantization bundle over a closed spherically monotone symplectic manifold $$\Sigma $$ Σ . Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for Y in the following two settings. First, $$\Sigma $$ Σ is a symplectic hyperplane section of a closed symplectic manifold X satisfying a certain monotonicity condition; in this case, $$X {{\setminus }} \Sigma $$ X \ Σ is a Liouville filling of Y. Second, the minimal Chern number of $$\Sigma $$ Σ is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization $$\mathbb {R} \times Y$$ R × Y is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of $$X{\setminus }\Sigma $$ X \ Σ or $$\mathbb {R} \times Y$$ R × Y and the quantum homology of $$\Sigma $$ Σ . As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.

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Morse–Bott split symplectic homology

Cited by 9 publications

References 34 publications

Symplectic homology of complements of smooth divisors

Symplectic homology of complements of smooth divisors

Sections and Unirulings of Families over $\mathbb{P}^{1}$

Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

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