# Seidel's morphism of toric 4-manifolds

@article{Anjos2014SeidelsMO,
title={Seidel's morphism of toric 4-manifolds},
author={S{\'i}lvia Anjos and R'emi Leclercq},
journal={arXiv: Symplectic Geometry},
year={2014}
}
• Published 30 June 2014
• Mathematics
• arXiv: Symplectic Geometry
Following McDuff and Tolman's work on toric manifolds [McDT06], we focus on 4-dimensional NEF toric manifolds and we show that even though Seidel's elements consist of infinitely many contributions, they can be expressed by closed formulas. From these formulas, we then deduce the expression of the quantum homology ring of these manifolds as well as their Landau-Ginzburg superpotential. We also give explicit formulas for the Seidel elements in some non-NEF cases. These results are closely…
6 Citations

## Figures from this paper

### Loops in the fundamental group of Symp ( CP 2 # 5 CP 2 , ω ) which are not represented by circle actions ∗

We study generators of the fundamental group of the group of symplectomorphisms Symp(CP2#5CP2, ω) for some particular symplectic forms. It was observed by J. Kȩdra in [13] that there are many

### Non-injectivity of Seidel's morphism

• Mathematics
• 2016
The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, only one of which being monotone. As

### Noncontractible Hamiltonian loops in the kernel of Seidel’s representation

• Mathematics
• 2017
The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone.

### THE FUNDAMENTAL GROUP OF Symp(S × S2#4CP)

• Mathematics
• 2019
This report concerns the work done under the guidance of professor Sílvia Anjos for the 2019 edition of the Gulbenkian project Novos Talentos em Matemática. In a succint way, the starting point was

### Loops in the fundamental group of $\mathrm{Symp} (\mathbb C\mathbb P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2)$ which are not represented by circle actions

• Mathematics
• 2019
We study the generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp} (\mathbb C\mathbb P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2, \omega)$ for some particular

### Lagrangian Circle Actions

We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the

## References

SHOWING 1-10 OF 45 REFERENCES

### Equivariant Gromov - Witten Invariants

The objective of this paper is to describe some construction and applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e. intersection theory on spaces of (pseudo-)

### Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold .M;!/ which upon passing to homology yields ring

### Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

• Mathematics
• 2009
In a recent paper, Donaldson (J. Gökova Geom. Topol. 3:1–8, 2009) explains how to use an older construction of Joyce (Duke Math. J. 77:519–552, 1995) to obtain four dimensional local models for

### Floer homology of families I

Floer theory is a certain kind of generalization of Morse theory, of which there are now a number of different flavors. These give invariants of symplectomorphisms, 3‐manifolds, Legendrian knots, and

### Topology of symplectomorphism groups of rational ruled surfaces

• Mathematics
• 1999
Let M be either S 2 S 2 or the one point blow-up CP 2 # CP 2 of CP 2 . In both cases M carries a family of symplectic forms ! , where > 1 determines the cohomology class [! ]. This paper calculates

### J-Holomorphic Curves and Symplectic Topology

• Mathematics
• 2004
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was

### Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces

• Mathematics
• 2010
In this paper, we compute the open Gromov-Witten invariants for every compact toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef). Unlike the Fano case, this involves

### The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

• Mathematics
• 2012
By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples,

### Introduction to toric varieties

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio

### Topological properties of Hamiltonian circle actions

• Mathematics
• 2004
This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,ω). Our main tool is the Seidel