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Hamiltonian Circle Actions On Symplectic Fano Manifolds

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Hamiltonian Circle Actions On Symplectic Fano Manifolds

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Author : Nicholas Lindsay
language : en
Publisher:
Release Date : 2018

Hamiltonian Circle Actions On Symplectic Fano Manifolds written by Nicholas Lindsay and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.

Hamiltonian Circle Actions On Symplectic Fano Manifolds

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Author : Nicholas James Lindsay
language : en
Publisher:
Release Date : 2018

Hamiltonian Circle Actions On Symplectic Fano Manifolds written by Nicholas James Lindsay and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.

In this thesis, we study symplectic manifolds satisfying certain co-homological conditions, in the presence of a Hamiltonian group action. In the case of 6-dimensional symplectic manifolds, the assumption of a Hamiltonian circle action allows us to use tools of 4-dimensional symplectic geometry, such as Seiberg-Witten theory, to prove restric-tions on their geometry and topology. The connection occurs primarily through symplectic reduction, and additionally when one considers col-lections of 4-dimensional symplectic submanifolds (i.e. fixed submani-folds, isotropy submanifolds or the symplectic fibre). The cohomological conditions of prime interest to us are the sym-plectic Fano condition and its relative version. The symplectic Fano condition is analogous to the Fano condition from algebraic geometry. In particular, symplectic manifolds satisfying this condition contain the class of smooth Fano varieties over the complex numbers. It is an open question to determine the extent to which these manifolds sat-isfy the boundedness of their algebraic counterparts in real dimensions 6 ≤ 2n ≤ 10. In this thesis, we give some positive progress towards a conjecture of Fine and Panov regarding the 6-dimensional case (see Conjecture 1.2). In particular, we prove that a symplectic Fano 6-manifold with a Hamiltonian circle action is simply connected and satisfies c1c2 = 24 (this result appeared in a joint work of the author and Panov [41]). In addition, we are able to show that there is at most one fixed surface of positive genus in such 6-manifolds, thus allowing us to bound the third Betti number in certain cases. Finally, we are able to show that these manifolds are symplectically birational to CP3, using the method of symplectic cuts.

Introduction To Symplectic Topology

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Author : Dusa McDuff
language : en
Publisher: Oxford University Press
Release Date : 2017-03-16

Introduction To Symplectic Topology written by Dusa McDuff and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-16 with Mathematics categories.

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

J Holomorphic Curves And Symplectic Topology

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Author : Dusa McDuff
language : en
Publisher: American Mathematical Society
Release Date : 2025-01-03

J Holomorphic Curves And Symplectic Topology written by Dusa McDuff and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-03 with Mathematics categories.

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 also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov–Witten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture. The main goal of this book is to establish the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four-dimensional manifolds. The second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.

Low Dimensional And Symplectic Topology

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Author : Michael Usher
language : en
Publisher: American Mathematical Soc.
Release Date : 2011

Low Dimensional And Symplectic Topology written by Michael Usher and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.

Every eight years since 1961, the University of Georgia has hosted a major international topology conference aimed at disseminating important recent results and bringing together researchers at different stages of their careers. This volume contains the proceedings of the 2009 conference, which includes survey and research articles concerning such areas as knot theory, contact and symplectic topology, 3-manifold theory, geometric group theory, and equivariant topology. Among other highlights of the volume, a survey article by Stefan Friedl and Stefano Vidussi provides an accessible treatment of their important proof of Taubes' conjecture on symplectic structures on the product of a 3-manifold and a circle, and an intriguing short article by Dennis Sullivan opens the door to the use of modern algebraic-topological techniques in the study of finite-dimensional models of famously difficult problems in fluid dynamics. Continuing what has become a tradition, this volume contains a report on a problem session held at the conference, discussing a variety of open problems in geometric topology.

Toric Topology

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Author : Megumi Harada
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Toric Topology written by Megumi Harada and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

Toric topology is the study of algebraic, differential, symplectic-geometric, combinatorial, and homotopy-theoretic aspects of a particular class of torus actions whose quotients are highly structured. The combinatorial properties of this quotient and the equivariant topology of the original manifold interact in a rich variety of ways, thus illuminating subtle aspects of both the combinatorics and the equivariant topology. Many of the motivations and guiding principles of the fieldare provided by (though not limited to) the theory of toric varieties in algebraic geometry as well as that of symplectic toric manifolds in symplectic geometry.This volume is the proceedings of the International Conference on Toric Topology held in Osaka in May-June 2006. It contains about 25 research and survey articles written by conference speakers, covering many different aspects of, and approaches to, torus actions, such as those mentioned above. Some of the manuscripts are survey articles, intended to give a broad overview of an aspect of the subject; all manuscripts consciously aim to be accessible to a broad reading audience of students andresearchers interested in the interaction of the subjects involved. We hope that this volume serves as an enticing invitation to this emerging field.

Abstracts Of Papers Presented To The American Mathematical Society

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Author : American Mathematical Society
language : en
Publisher:
Release Date : 1997

Abstracts Of Papers Presented To The American Mathematical Society written by American Mathematical Society and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.