70 research outputs found
Fukaya categories of symmetric products and bordered Heegaard-Floer homology
The main goal of this paper is to discuss a symplectic interpretation of
Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of
Fukaya categories of symmetric products and Lagrangian correspondences. More
specifically, we give a description of the algebra A(F) which appears in the
work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer
homology for product Lagrangians in the symmetric product, and outline how
bordered Heegaard-Floer homology itself can conjecturally be understood in this
language.Comment: 54 pages, 11 figures; v3: minor revisions, to appear in J Gokova
Geometry Topolog
Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers
The first part of this paper is a review of the Strominger-Yau-Zaslow
conjecture in various settings. In particular, we summarize how, given a pair
(X,D) consisting of a Kahler manifold and an anticanonical divisor, families of
special Lagrangian tori in X-D and weighted counts of holomorphic discs in X
can be used to build a Landau-Ginzburg model mirror to X. In the second part we
turn to more speculative considerations about Calabi-Yau manifolds with
holomorphic involutions and their quotients. Namely, given a hypersurface H
representing twice the anticanonical class in a Kahler manifold X, we attempt
to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double
cover of X branched along H; unfortunately, the implications for mirror
symmetry are far from clear.Comment: 27 pages, 1 figur
Monodromy invariants in symplectic topology
This text is a set of lecture notes for a series of four talks given at
I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick
overview of symplectic topology and its main tools: symplectic manifolds,
almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants
and Floer homology. The second and third lectures focus on symplectic Lefschetz
pencils: existence (following Donaldson), monodromy, and applications to
symplectic topology, in particular the connection to Gromov-Witten invariants
of symplectic 4-manifolds (following Smith) and to Fukaya categories (following
Seidel). In the last lecture, we offer an alternative description of symplectic
4-manifolds by viewing them as branched covers of the complex projective plane;
the corresponding monodromy invariants and their potential applications are
discussed.Comment: 42 pages, notes of lectures given at IPAM, Los Angele
Symplectic maps to projective spaces and symplectic invariants
After reviewing recent results on symplectic Lefschetz pencils and symplectic
branched covers of CP^2, we describe a new construction of maps from symplectic
manifolds of any dimension to CP^2 and the associated monodromy invariants. We
also show that a dimensional induction process makes it possible to describe
any compact symplectic manifold by a series of words in braid groups and a word
in a symmetric group.Comment: 39 pages; to appear in Proc. 7th Gokova Geometry-Topology Conferenc
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