The symplectic topology of some rational homology balls

Abstract

We study the symplectic topology of some finite algebraic quotients of the A n Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A n Milnor fibre coming from homological mirror symmetry. On the other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the A n Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.

Twin Lagrangian fibrations in mirror symmetry

“…This recovers the following result due to Lekili-Maydanskiy. Theorem 1.4 (Lekili-Maydanskiy [30]). There exist Floer theoretical essential tori T p,q ⊂ B p,q , and HF * (T p,q , T p,q ) ∼ = H * (T 2 , K) (15) additively.…”

“…This section contains two simple applications of twin Lagrangian fibrations studied in the last section, which are inspired respectively by Section 4.4 of the paper of Smith [45] and the work of Lekili-Maydanskiy [30]. To get a more explicit picture, we restrict ourselves here to the case of symplectic 4-manifolds.…”

“…As a geometric application of the structure of twin Lagrangian fibrations, we consider here the rational homology balls B p,q studied in [30]. These are Stein surfaces defined as the quotients of the A p−1 Milnor fiber X ∨ p−1 by certain finite group actions.…”

An-type surface singularity and nondisplaceable Lagrangian tori

“…Here we provide the computation of their potential functions with bulk deformation by using toric degeneration and admissible tuples in [7]. Also see [22] to count Maslov index 2 holomorphic disks by using Lefschetz fibrations and in [1], [20] from the perspective of mirror symmetry.…”

Koszul duality patterns in Floer theory

“…This is clear from the fact that the latter category have objects with infinite-dimensional endomorphisms (over K) but every object in the former has finite-dimensional endomorphisms. More strikingly, the monotone Lagrangian tori studied in [48] give objects in D π W(X Γ ) for Γ = A n with finite-dimensional endomorphisms and yet these do not belong to the category D π F(X Γ ). Though, one has to collapse the grading to Z 2 in order to admits these objects in F(X Γ ).…”