发表于2024-12-22
Fukaya Categories and Picard-Lefschetz Theory 2024 pdf epub mobi 电子书
由于in general定义Lagrangian Floer theory存在obstruction,因此本书讨论了exact symplectic manifold (with corners) [;M;]中的closed exact Lagrangian。这样做的好处是运用Stokes定理可以看出没有disc bubbling,从而这些Lagrangian submanifold tautologically unobstruc...
评分由于in general定义Lagrangian Floer theory存在obstruction,因此本书讨论了exact symplectic manifold (with corners) [;M;]中的closed exact Lagrangian。这样做的好处是运用Stokes定理可以看出没有disc bubbling,从而这些Lagrangian submanifold tautologically unobstruc...
评分由于in general定义Lagrangian Floer theory存在obstruction,因此本书讨论了exact symplectic manifold (with corners) [;M;]中的closed exact Lagrangian。这样做的好处是运用Stokes定理可以看出没有disc bubbling,从而这些Lagrangian submanifold tautologically unobstruc...
评分由于in general定义Lagrangian Floer theory存在obstruction,因此本书讨论了exact symplectic manifold (with corners) [;M;]中的closed exact Lagrangian。这样做的好处是运用Stokes定理可以看出没有disc bubbling,从而这些Lagrangian submanifold tautologically unobstruc...
评分由于in general定义Lagrangian Floer theory存在obstruction,因此本书讨论了exact symplectic manifold (with corners) [;M;]中的closed exact Lagrangian。这样做的好处是运用Stokes定理可以看出没有disc bubbling,从而这些Lagrangian submanifold tautologically unobstruc...
图书标签: 辛几何 数学 mathematics SG Math MP
Preface
The subject of Fukaya categories has a reputation for being hard to approach. This
is due to the amount of background knowledge required (taken from homological
algebra, symplectic geometry, and geometric analysis), and equally to the rather
complicated nature of the basic definitions. The present book is intended as a resource
for graduate students and researchers whowould like to learn about Fukaya categories,
and possibly use them in their own work. I have tried to focus on a rather basic subset
of topics, and to describe these as precisely as I could, filling in gaps found in some of
the early references. This makes for a rather austere style (for that reason, a thorough
study of this book should probably be complemented by reading some of the papers
dealing with applications). A second aim was to give an account of some previously
unpublished results, which connect Fukaya categories to the theory of Lefschetz
fibrations. This becomes predominant in the last sections, where the text gradually
turns into a research monograph.
I have borrowed liberally from the work of many people, first and foremost among
them Fukaya, Kontsevich, and Donaldson. Fukaya’s foundational contribution, of
course, was to introduce A1-structures into symplectic geometry. On the algebraic
side, he pioneered the use of the A1-version of the Yoneda embedding, which we
adopt systematically. Besides that, several geometric ideas, such as the role of Pin
structures, and the construction of A1-homomorphisms in terms of parametrized
moduli spaces, are taken from the work of Fukaya, Oh, Ohta and Ono. Kontsevich
introduced derived categories of A1-categories, and is responsible for much of
their theory, in particular the intrinsic characterization of exact triangles. He also
conjectured the relation between Dehn twist and twist functors, which is one of our
main results. Finally, in joint work with Barannikov, he suggested a construction
of Fukaya categories for Lefschetz fibrations; we use a superficially different, but
presumably equivalent, definition. Donaldson’s influence is equally pervasive. Besides
his groundbreaking work on Lefschetz pencils, he introduced matching cycles,
and proposed them as the starting point for a combinatorial formula for Floer cohomology,
which is indeed partly realized here. Other mathematicians have also made
important contributions. For instance, parts of our presentation of Picard–Lefschetz
theory reflect Auroux’ point of view. A result of Smith, namely that the vanishing
cycles in a four-dimensional Lefschetz pencil necessarily fill out the fibre, was crucial
in suggesting that such cycles might “split-generate” the Fukaya category. Besides
that, work of Fukaya–Smith on cotangent bundles provided a good testing-ground
for some of the more adventurous ideas about Lefschetz fibrations. Our approach
to transversality issues is the result of several conversations with Lazzarini. Finally,
Abouzaid’s suggestions greatly improved the discussion of symplectic embeddings.
Fukaya Categories and Picard-Lefschetz Theory 2024 pdf epub mobi 电子书