Symplectomorphisms of some Weinstein 4-manifolds

Hacking, Paul; Keating, Ailsa

doi:10.2140/gt.2026.30.645

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Symplectomorphisms of some Weinstein $4$-manifolds

Paul Hacking and Ailsa Keating

Geometry & Topology 30 (2026) 645–699

DOI: 10.2140/gt.2026.30.645

Abstract

Let $M$ be a Weinstein four-manifold mirror to $Y ∖ D$ for $(Y, D)$ a log Calabi–Yau surface; this is typically the Milnor fibre of a smoothing of a cusp singularity. We introduce two families of symplectomorphisms of $M$ : Lagrangian translations, which we prove are mirror to tensors with line bundles; and nodal slide recombinations, which we prove are mirror to automorphisms of $(Y, D)$ . The proof uses a detailed compatibility between the homological and SYZ viewpoints on mirror symmetry. Together with spherical twists, these symplectomorphisms are expected to generate all autoequivalences of the wrapped Fukaya category of $M$ that are compactly supported in a categorical sense. A range of applications is given.

Keywords

homological mirror symmetry, symplectomorphisms, log Calabi–Yau surfaces, Weinstein handlebodies

Mathematical Subject Classification

Primary: 14J17, 53D05, 53D37

References

Publication

Received: 29 March 2024

Revised: 23 April 2025

Accepted: 2 June 2025

Published: 16 March 2026

Proposed: Mohammed Abouzaid

Seconded: Ulrike Tillmann, András I Stipsicz

Authors

Paul Hacking
Department of Mathematics and Statistics University of Massachusetts Amherst, MA United States

Ailsa Keating
Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Open Access made possible by participating institutions via Subscribe to Open.

Volume 30, issue 2 (2026)