Lagrangian intersection Floer theory: anomaly and obstruction. Part I
ABSTRACT The studies on symplectic geometry have changed drastically after the method of pseudo-holomorphic curves was developed by Gromov and the elliptic Morse theory was invented by Floer. Many problems in symplectic geometry were solved by the techniques of pseudo-holomorphic curves, and the concept of symplectic topology gradually began to take shape. This progress was accompanied by parallel developments in physics most notably in closed string theory. Fukuya introduced the structure of an A ∞ -category in symplectic geometry whose objects are Lagrangian submanifolds and whose morphisms are the Floer cohomologies. Based on this algebraic framework, Kontsevich proposed the celebrated homological mirror symmetry between the derived category of coherent sheaves and the Fukaya category of Lagrangian submanifolds. Enhanced by the later development in open string theory of D-branes, this homological mirror symmetry has been a source of many new insights and progresses in both algebraic geometry and symplectic geometry as well as in physics. This book is the first volume of a two-volume monograph on the general Lagrangian Floer theory. The authors try to explain how the obstruction to and anomaly in the construction of Floer cohomology arise. They provide a precise formulation of the obstructions and then carry out detailed algebraic and analytic study of deformation theory of Floer cohomology. It turns out that even a description of such an obstruction requires new homological algebra of filtered A ∞ algebras. In addition, the authors provide various immediate applications of the Floer cohomology to problems in symplectic topology. This volume consists of six chapters. After an introduction, acknowledgments, and conventions in Chapter 1, Chapter 2 is mostly devoted to a review of Floer’s original version of Floer cohomology. Then, the authors give the definitions of various kinds of Maslov indices and of the stable map moduli space of pseudo-holomorphic discs on which the whole constructions in the book are based on. Next, they explain the basic idea of Floer’s construction of Lagrangian intersection Floer cohomology, and review the Floer cohomology of monotone Lagrangian submanifolds. Chapter 3 gives the definition of filtered A ∞ algebra and their deformation theory. Notions of obstruction cycles and of bounding cochains are introduced first in algebraic context. Then, a filtered A ∞ algebra is associated to each Lagrangian submanifold L over a countably generated complex on L. Next, the algebraic arguments explaining how to proceed from the filtered A ∞ algebra to the definitions of deformed Floer cohomology and of the moduli space ℳ(L) that parameterizes the Floer cohomologies are given. A deformed Floer cohomology on a single Lagrangian submanifold L is viewed as a version of Bott-Morse type Floer cohomology. To each pair of Lagrangian submanifolds a filtered A ∞ bimodule is associated. There are two other important issues discussed in Chapter 3. The first one is the issue of the unit of filtered A ∞ algebra and the relationship of the unit with the moduli space ℳ weak (L) of weak bounding cochains. The other issue is the deformation of the filtered A ∞ algebra by the cohomology classes from the ambient symplectic manifold. Chapter 4 is devoted to studying the homotopy equivalence of filtered A ∞ algebras. The first four sections are of purely algebraic nature. The notion of two filtered A ∞ homomorphisms to be homotopic to each other is introduced and basic properties about homotopy between filtered A ∞ homomorphisms are established. This leads to the definition of homotopy equivalent A ∞ algebras and a notion that a filtered A ∞ homomorphism is a weak homotopy equivalence. In fact, a filtered A ∞ homomorphism is a homotopy equivalence if and only if it is a weak homotopy equivalence. In the last section the geometric situation is considered and it is shown that the homotopy type of the filtered A ∞ algebra associated to a relatively spin Lagrangian submanifold L is independent of various choices involved. Chapter 5 deals with the case when there are two Lagrangian submanifolds or filtered A ∞ bimodules. The authors discuss relations of various Novikov rings which appear in the theory of Floer cohomologies, study homotopy between filtered A ∞ bimodule homomorphisms, and define homotopy equivalence of A ∞ bimodules. Finally, they construct homotopy equivalence between filtered A ∞ bimodules associated to a relatively spin pair of Lagrangian submanifolds. The purpose of Chapter 6 is to construct a spectral sequence from a filtered differential A 0,nov module and to prove its convergence.
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