Symplectic resolutions, Lefschetz property and formality

TL;DR

This paper introduces a method for resolving symplectic orbifolds into smooth manifolds and investigates how properties like formality and the Lefschetz property behave under these resolutions, including new counterexamples to existing conjectures.

Contribution

It provides a novel resolution technique for symplectic orbifolds and constructs the first example of a simply connected 8-dimensional symplectic manifold with the Lefschetz property but lacking formality.

Findings

Resolved symplectic orbifolds into smooth manifolds.

Compared formality and Lefschetz property between orbifolds and resolutions.

Constructed a counterexample to a conjecture on formality.

Abstract

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.