##
**Fukaya categories and Picard-Lefschetz theory.**
*(English)*
Zbl 1159.53001

Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-063-0/pbk). vi, 326 p. (2008).

The book under review is based on the author’s lectures to introduce the rather complicated Fukaya category through both algebra, geometry and topology. The subject of the Fukaya category is developed in Kontsevich’s homological mirror symmetry conjecture. The book contains basic subjects from basic definitions and concepts to basic results, and the research topics from the application of Fukaya category to the Lefschetz fibration and the Picard-Lefschetz theory. As the author explaines, many topics in the book are informal discussion. So it is really a book for non-specialists who would like to know the general ideas in this subject. Based on these special purposes of the authors’ aim, the review would be brief for non-specialist readers, and probably hard to say anything useful for experts in this research area (for that we would suggest the authors’ research articles).

Part I of the book is on the \(A_{\infty}\) category from the Fukaya’s original point of view [K. Fukaya, Morse theory, \(A_{\infty}\)-categories and Floer homologies. Proceedings of the GARC workshop on geometry and topology ’93 held at the Seoul National University, Seoul, Korea, July 1993. Seoul: Seoul National University, Lect. Notes Ser., Seoul. 18, 1–102 (1993; Zbl 0853.57030)]. In fact, for any \(Z-\)graded vector space A, a \(A_{\infty}\)-structure is a sequence of multilinear operations \(\{m_n: A^{\otimes n} \to A[2-n], n\geq 1\}\) satisfying

\[ \sum_{i+j=n+1}\sum_{0\leq j \leq i}\varepsilon (l, j)m_i(a_0, \cdots, a_{l-1}, m_j(a_l, \cdots, a_{l+j-1}, a_{l+j}, \cdots, a_n) = 0, \;\;\;m_1^2=0. \]

The generalization to \(A_{\infty}\)-categories was pioneered by Kontsevich. There are two problems in the Fukaya category: (i) morphisms can be defined only for transversal Lagrangian submanifolds (this is why one needs to define the \(A_{\infty}\)-pre-category), (ii) there are pseudo-holomorphic discs with boundary on a given Lagrangian. The author describes the homological algebra underlying Fukaya categories, and discusses at length identity morphisms.

Part II is the main thesis of the book to explain the definition of Fukaya’s \(A_{\infty}\)-category in a special case of exact symplectic manifolds. The general construction was in the fundamental work of Fukaya-Oh-Ohta-Ono. The author gives a wonderful exposition to lay the overview of Floer cohomology for exact Lagrangian submanifolds and the classcial methods pioneered by Gromov and Floer. He first defines an exact symplectic manifold with corners as an exact symplectic manifold such that its nonempty boundary is weakly \(J\)-convex (that is the condition to rule out \(J\)-holomorphic curves touches the boundary). The symplectic diffeomorphisms between these manifolds are exact conformally symplectic as given in Lemma 7.1. As the author states all proofs are given minimal sketches, many important claims in lemmas would be good exercises for people who are interesting in the field. The inhomogeneous pseudo-holomorphic maps possess certain transversality property for the definition of Floer cohomology.

The analytic setup for symplectic manifolds with nonempty boundary is quite challenging, as the usual exponential weighted Sobolev spaces won’t be able to guarantee the Fredholm property. The trace-classes or \(b\)-geometric analysis are suitable tools in this situation, but the book takes classical analytic methods for granted. One of the propositions shows that there are identity functors in the Fukaya \(A_{\infty}\)-pre-categroy in proposition 10.3, and the exact conformally symplectic diffeomorphisms gives rise to the quasi-isomorphic between the Fukaya \(A_{\infty}\)-pre-categories in Proposition 10.5. The author gives a very clear exposition on the determinant line bundles and spectral flows for the orientations and indices as well as the branes (which is equivalent to the Arg-functions used by Kontsevich and Soibelman) in section 11. The Fukaya category is given in section 12, although the signs and the Maslov indices in the multilinear operations \(m_n\) are quite hard to work out explicitly. The combinatorial Floer cohomology is also discussed in section 13, see Review of [M. Abouzaid, Adv. Math. 217, No. 3, 1192–1235 (2008; Zbl 1155.57029)]. The main difficulty in defining the equivariant Fukaya category is the equivariant transversality which is discussed in section 14c. Assumptions 14.3, 14.4 and 14.5 are purposely designed to have the nice equivariants property as expected. Whether one can achieve those assumptions by perturbations is another issue.

Part III of the book is to touch another rich topic in symplectic topology and geometry of Lefschetz fibrations. The fibre is an exact symplectic manifold with corners as described in Part II, and those vanishing cycles are the Lagrangian spheres for the objects of the Fukaya category, or for the extension of TQFT in this case. The section 17 gives a detailed description of those multilinear operations in the Lefschetz fibrations, the first \(m_n\) for \(n\leq 5\) are clearly explained in section 17f and 17g. The author explains the necessity of relative brane structures and the geometric twisting to have the equivalence of triangulated categories. The Fukaya category of a Lefschetz fibration is discussed in section 18. Other research topics on plurisubharmonic functions, Lefschetz pencils and Picard-Lefschetz data as well as Milnor fibres are very interesting to read. Over all this is an excellent book for non-specialists to grasp the Fukaya category and its application on the various symplectic manifolds.

Part I of the book is on the \(A_{\infty}\) category from the Fukaya’s original point of view [K. Fukaya, Morse theory, \(A_{\infty}\)-categories and Floer homologies. Proceedings of the GARC workshop on geometry and topology ’93 held at the Seoul National University, Seoul, Korea, July 1993. Seoul: Seoul National University, Lect. Notes Ser., Seoul. 18, 1–102 (1993; Zbl 0853.57030)]. In fact, for any \(Z-\)graded vector space A, a \(A_{\infty}\)-structure is a sequence of multilinear operations \(\{m_n: A^{\otimes n} \to A[2-n], n\geq 1\}\) satisfying

\[ \sum_{i+j=n+1}\sum_{0\leq j \leq i}\varepsilon (l, j)m_i(a_0, \cdots, a_{l-1}, m_j(a_l, \cdots, a_{l+j-1}, a_{l+j}, \cdots, a_n) = 0, \;\;\;m_1^2=0. \]

The generalization to \(A_{\infty}\)-categories was pioneered by Kontsevich. There are two problems in the Fukaya category: (i) morphisms can be defined only for transversal Lagrangian submanifolds (this is why one needs to define the \(A_{\infty}\)-pre-category), (ii) there are pseudo-holomorphic discs with boundary on a given Lagrangian. The author describes the homological algebra underlying Fukaya categories, and discusses at length identity morphisms.

Part II is the main thesis of the book to explain the definition of Fukaya’s \(A_{\infty}\)-category in a special case of exact symplectic manifolds. The general construction was in the fundamental work of Fukaya-Oh-Ohta-Ono. The author gives a wonderful exposition to lay the overview of Floer cohomology for exact Lagrangian submanifolds and the classcial methods pioneered by Gromov and Floer. He first defines an exact symplectic manifold with corners as an exact symplectic manifold such that its nonempty boundary is weakly \(J\)-convex (that is the condition to rule out \(J\)-holomorphic curves touches the boundary). The symplectic diffeomorphisms between these manifolds are exact conformally symplectic as given in Lemma 7.1. As the author states all proofs are given minimal sketches, many important claims in lemmas would be good exercises for people who are interesting in the field. The inhomogeneous pseudo-holomorphic maps possess certain transversality property for the definition of Floer cohomology.

The analytic setup for symplectic manifolds with nonempty boundary is quite challenging, as the usual exponential weighted Sobolev spaces won’t be able to guarantee the Fredholm property. The trace-classes or \(b\)-geometric analysis are suitable tools in this situation, but the book takes classical analytic methods for granted. One of the propositions shows that there are identity functors in the Fukaya \(A_{\infty}\)-pre-categroy in proposition 10.3, and the exact conformally symplectic diffeomorphisms gives rise to the quasi-isomorphic between the Fukaya \(A_{\infty}\)-pre-categories in Proposition 10.5. The author gives a very clear exposition on the determinant line bundles and spectral flows for the orientations and indices as well as the branes (which is equivalent to the Arg-functions used by Kontsevich and Soibelman) in section 11. The Fukaya category is given in section 12, although the signs and the Maslov indices in the multilinear operations \(m_n\) are quite hard to work out explicitly. The combinatorial Floer cohomology is also discussed in section 13, see Review of [M. Abouzaid, Adv. Math. 217, No. 3, 1192–1235 (2008; Zbl 1155.57029)]. The main difficulty in defining the equivariant Fukaya category is the equivariant transversality which is discussed in section 14c. Assumptions 14.3, 14.4 and 14.5 are purposely designed to have the nice equivariants property as expected. Whether one can achieve those assumptions by perturbations is another issue.

Part III of the book is to touch another rich topic in symplectic topology and geometry of Lefschetz fibrations. The fibre is an exact symplectic manifold with corners as described in Part II, and those vanishing cycles are the Lagrangian spheres for the objects of the Fukaya category, or for the extension of TQFT in this case. The section 17 gives a detailed description of those multilinear operations in the Lefschetz fibrations, the first \(m_n\) for \(n\leq 5\) are clearly explained in section 17f and 17g. The author explains the necessity of relative brane structures and the geometric twisting to have the equivalence of triangulated categories. The Fukaya category of a Lefschetz fibration is discussed in section 18. Other research topics on plurisubharmonic functions, Lefschetz pencils and Picard-Lefschetz data as well as Milnor fibres are very interesting to read. Over all this is an excellent book for non-specialists to grasp the Fukaya category and its application on the various symplectic manifolds.

Reviewer: Weiping Li (Stillwater)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53D40 | Symplectic aspects of Floer homology and cohomology |

32Q65 | Pseudoholomorphic curves |

53D12 | Lagrangian submanifolds; Maslov index |

16E45 | Differential graded algebras and applications (associative algebraic aspects) |

81T45 | Topological field theories in quantum mechanics |