Softcover ISBN: | 978-2-85629-972-2 |
Product Code: | AST/440 |
List Price: | $81.00 |
AMS Member Price: | $64.80 |
Softcover ISBN: | 978-2-85629-972-2 |
Product Code: | AST/440 |
List Price: | $81.00 |
AMS Member Price: | $64.80 |
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Book DetailsAstérisqueVolume: 440; 2023; 274 ppMSC: Primary 18; 35; 53
The aim of this paper is to apply the microlocal theory of sheaves of Kashiwara-Schapira to the symplectic geometry of cotangent bundles, following ideas of Nadler-Zaslow and Tamarkin.
The author recalls the main notions and results of the microlocal theory of sheaves, in particular the microsupport of sheaves. The microsupport of a sheaf\(F\) on a manifold \(M\) is a closed conic subset of the cotangent bundle \(T^{*}M\) which indicates in which directions we can modify a given open subset of \(M\) without modifying the cohomology of \(F\) on this subset. An important theorem of Kashiwara-Schapira says that the microsupport is coisotropic, and recent works of Nadler-Zaslow and Tamarkin study in the other direction the sheaves which have for microsupport a given Lagrangian submanifold \(A\), obtaining information on \(A\) in this way.
Nadler and Zaslow made the link with the Fukaya category, but Tamarkin only made use of the microlocal sheaf theory. The author moves in this direction and recovers several results of symplectic geometry with the help of sheaves. In particular, the author explains how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, and the existence of graph selectors. The author also proves a three cusps conjecture of Arnol'd about curves on the sphere. In the last sections, the author recovers more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and mathematicians interested in sheaves and symplectic geometry.
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The aim of this paper is to apply the microlocal theory of sheaves of Kashiwara-Schapira to the symplectic geometry of cotangent bundles, following ideas of Nadler-Zaslow and Tamarkin.
The author recalls the main notions and results of the microlocal theory of sheaves, in particular the microsupport of sheaves. The microsupport of a sheaf\(F\) on a manifold \(M\) is a closed conic subset of the cotangent bundle \(T^{*}M\) which indicates in which directions we can modify a given open subset of \(M\) without modifying the cohomology of \(F\) on this subset. An important theorem of Kashiwara-Schapira says that the microsupport is coisotropic, and recent works of Nadler-Zaslow and Tamarkin study in the other direction the sheaves which have for microsupport a given Lagrangian submanifold \(A\), obtaining information on \(A\) in this way.
Nadler and Zaslow made the link with the Fukaya category, but Tamarkin only made use of the microlocal sheaf theory. The author moves in this direction and recovers several results of symplectic geometry with the help of sheaves. In particular, the author explains how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, and the existence of graph selectors. The author also proves a three cusps conjecture of Arnol'd about curves on the sphere. In the last sections, the author recovers more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and mathematicians interested in sheaves and symplectic geometry.