The central objects in the book are Lagrangian submanifolds and their invariants, such as Floer homology and its multiplicative structures, which together constitute the Fukaya category. The relevant aspects of pseudo-holomorphic curve theory are covered in some detail, and there is also a self-contained account of the necessary homological algebra. Generally, the emphasis is on simplicity rather than generality. The last part discusses applications to Lefschetz fibrations, and contains many previously unpublished results. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry.
Author(s): Paul Seidel
Series: Zurich Lectures in Advanced Mathematics
Publisher: European Mathematical Society
Year: 2008
Language: English
Pages: 334
Preface......Page 5
Contents......Page 7
Introduction......Page 9
I A_infinity-categories......Page 15
Definitions......Page 16
Identity morphisms and equivalences......Page 30
Exact triangles......Page 40
Idempotents......Page 63
Twisting......Page 70
Z/2-actions......Page 95
II Fukaya categories......Page 103
A little symplectic geometry......Page 104
Classical Floer theory......Page 108
The Fukaya category (preliminary version)......Page 121
Some basic properties......Page 141
Indices and determinant lines......Page 157
The Fukaya category (complete version)......Page 182
Polygons on surfaces......Page 198
Symplectic involutions......Page 206
III Picard–Lefschetz theory......Page 219
First notions......Page 220
Vanishing cycles and matching cycles......Page 228
Pseudo-holomorphic sections......Page 243
The Fukaya category of a Lefschetz fibration......Page 273
Algebraic varieties......Page 299
(A_m) type Milnor fibres......Page 312
Bibliography......Page 323
Symbols......Page 331
Index......Page 333
