國立虎尾科技大學 |

Kuranishi Structures and Virtual Fundamental Chains

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Kuranishi Structures and Virtual Fundamental Chains/ by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono.
Author:	Fukaya, Kenji.
other author:	Oh, Yong-Geun.
Description:	XV, 638 p. 149 illus., 34 illus. in color.online resource. :
Contained By:	Springer Nature eBook
Subject:	Differential geometry. -
Online resource:	https://doi.org/10.1007/978-981-15-5562-6
ISBN:	9789811555626

Kuranishi Structures and Virtual Fundamental Chains
Fukaya, Kenji.

Kuranishi Structures and Virtual Fundamental Chains[electronic resource] /by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono. - 1st ed. 2020. - XV, 638 p. 149 illus., 34 illus. in color.online resource. - Springer Monographs in Mathematics,1439-7382. - Springer Monographs in Mathematics,.

1.Introduction -- 2.Notations and conventions -- 3.Kuranishi structure and good coordinate system -- 4.Fiber product of Kuranishi structures -- 5.Thickening of a Kuranishi structure -- 6.Multivalued perturbation -- 7.CF-perturbation and integration along the fiber (pushout) -- 8.Stokes' formula -- 9.From good coordinate system to Kuranishi structure and back with CF-perturbations -- 10.Composition formula of smooth correspondences -- 11.Construction of good coordinate system -- 12.Construction of CF-perturbations -- 13.Construction of multivalued perturbations -- 14.Zero and one dimensional cases via multivalued perturbation -- 15.Introduction to Part 2 -- 16.Linear K-system: Floer cohomology I: statement -- 17.Extension of Kuranishi structure and its perturbation from boundary to its neighborhood -- 18.Smoothing corners and composition of morphisms -- 19.Linear K-system: Floer cohomology II: proof -- 20.Linear K-system: Floer cohomology III: Morse case by multisection -- 21.Tree-like K-system: A1 structure I: statement -- 22.Tree-like K-system: A1 structure II: proof -- 23. Orbifold and orbibundle by local coordinate -- 24.Covering space of effective orbifold and K-space -- 25.Admissible Kuranishi structure -- 26.Stratified submersion to a manifold with corners -- 27.Local system and smooth correspondence in de Rham theory with twisted coefficients -- 28.Composition of KG and GG embeddings: Proof of Lemma 3.34 -- 29.Global quotient and orbifold. .

The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures. .

ISBN: 9789811555626

Standard No.: 10.1007/978-981-15-5562-6doiSubjects--Topical Terms:

882213
Differential geometry.

LC Class. No.: QA641-670

Dewey Class. No.: 516.36

Kuranishi Structures and Virtual Fundamental Chains
LDR:05054nam a22004095i 4500 001 1030540
003 DE-He213
005 20201016104702.0
007 cr nn 008mamaa
008 210318s2020 si | s |||| 0|eng d
020 $a 9789811555626 $9 978-981-15-5562-6
024 7 $a 10.1007/978-981-15-5562-6 $2 doi
035 $a 978-981-15-5562-6
050 4 $a QA641-670
072 7 $a PBMP $2 bicssc
072 7 $a MAT012030 $2 bisacsh
072 7 $a PBMP $2 thema
082 0 4 $a 516.36 $2 23
100 1 $a Fukaya, Kenji. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1327478
245 1 0 $a Kuranishi Structures and Virtual Fundamental Chains $h [electronic resource] / $c by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono.
250 $a 1st ed. 2020.
264 1 $a Singapore : $b Springer Singapore : $b Imprint: Springer, $c 2020.
300 $a XV, 638 p. 149 illus., 34 illus. in color. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a Springer Monographs in Mathematics, $x 1439-7382
505 0 $a 1.Introduction -- 2.Notations and conventions -- 3.Kuranishi structure and good coordinate system -- 4.Fiber product of Kuranishi structures -- 5.Thickening of a Kuranishi structure -- 6.Multivalued perturbation -- 7.CF-perturbation and integration along the fiber (pushout) -- 8.Stokes' formula -- 9.From good coordinate system to Kuranishi structure and back with CF-perturbations -- 10.Composition formula of smooth correspondences -- 11.Construction of good coordinate system -- 12.Construction of CF-perturbations -- 13.Construction of multivalued perturbations -- 14.Zero and one dimensional cases via multivalued perturbation -- 15.Introduction to Part 2 -- 16.Linear K-system: Floer cohomology I: statement -- 17.Extension of Kuranishi structure and its perturbation from boundary to its neighborhood -- 18.Smoothing corners and composition of morphisms -- 19.Linear K-system: Floer cohomology II: proof -- 20.Linear K-system: Floer cohomology III: Morse case by multisection -- 21.Tree-like K-system: A1 structure I: statement -- 22.Tree-like K-system: A1 structure II: proof -- 23. Orbifold and orbibundle by local coordinate -- 24.Covering space of effective orbifold and K-space -- 25.Admissible Kuranishi structure -- 26.Stratified submersion to a manifold with corners -- 27.Local system and smooth correspondence in de Rham theory with twisted coefficients -- 28.Composition of KG and GG embeddings: Proof of Lemma 3.34 -- 29.Global quotient and orbifold. .
520 $a The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures. Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved. Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures. .
650 0 $a Differential geometry. $3 882213
650 0 $a Hyperbolic geometry. $3 1264371
650 0 $a Polytopes. $3 884204
650 1 4 $a Differential Geometry. $3 671118
650 2 4 $a Hyperbolic Geometry. $3 1069789
700 1 $a Oh, Yong-Geun. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1294393
700 1 $a Ohta, Hiroshi. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1327479
700 1 $a Ono, Kaoru. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1327480
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9789811555619
776 0 8 $i Printed edition: $z 9789811555633
776 0 8 $i Printed edition: $z 9789811555640
830 0 $a Springer Monographs in Mathematics, $x 1439-7382 $3 1254272
856 4 0 $u https://doi.org/10.1007/978-981-15-5562-6
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)