scholarly journals A Symplectic Dynamics Proof of the Degree–Genus Formula

2021 ◽  
Author(s):  
Peter Albers ◽  
Hansjörg Geiges ◽  
Kai Zehmisch
Keyword(s):  
Field Theory ◽  
Hopf Fibration ◽  
Contact Form ◽  
Projective Curves ◽  
Symplectic Field Theory ◽  
Genus Formula ◽  
Symplectic Dynamics ◽  
Standard Contact ◽  
Complex Projective

AbstractWe classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

Quarto-quartic birational maps of ℙ3(ℂ)

2017 ◽  
Vol 28 (05) ◽  
pp. 1750037
Author(s):  
Julie Déserti ◽  
Frédéric Han
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Birational Maps ◽  
Dimension Formula ◽  
Trigonal Curves ◽  
Irreducible Components ◽  
Genus Formula ◽  
Complex Projective

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension [Formula: see text] from trigonal curves of degree [Formula: see text] and genus [Formula: see text]. Moreover, we show that the variety of [Formula: see text]-birational maps of [Formula: see text] has at least four irreducible components and describe three of them.

scholarly journals Non-commutative geometry and symplectic field theory

2007 ◽  
Vol 361 (6) ◽  
pp. 464-471 ◽  
Author(s):  
R.G.G. Amorim ◽  
M.C.B. Fernandes ◽  
F.C. Khanna ◽  
A.E. Santana ◽  
J.D.M. Vianna
Keyword(s):  
Field Theory ◽  
Symplectic Field Theory

scholarly journals Introduction to Symplectic Field Theory

2000 ◽  
pp. 560-673 ◽  
Author(s):  
Y. Eliashberg ◽  
A. Glvental ◽  
H. Hofer
Keyword(s):  
Field Theory ◽  
Symplectic Field Theory

scholarly journals Local contact homology and applications

2015 ◽  
Vol 07 (02) ◽  
pp. 167-238 ◽  
Author(s):  
Umberto L. Hryniewicz ◽  
Leonardo Macarini
Keyword(s):  
Field Theory ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Local Version ◽  
Contact Homology ◽  
Local Contact ◽  
Symplectic Field Theory ◽  
Uniformly Bounded

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll–Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits. Most of the results of this paper remain conjectural until the foundational issues of Symplectic Field Theory are resolved.

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