Symplectomorphisms and spherical objects in the conifold smoothing

2024/11/01 by Ailsa Keating, Ivan Smith · 4 citations
Engineering · Mathematics · Computer Science · #Advanced Numerical Analysis Techniques #Geometric and Algebraic Topology #Computational Geometry and Mesh Generation

paper · pdf · doi:10.1112/s0010437x24007425

Abstract

Let X denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in T^*S³ or, equivalently, the plumbing of two copies of T^*S³ along a Hopf link. Let Y denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in \mathcal O(-1) ⊕ \mathcal O(-1) → \mathbb P¹ . We prove that the compactly supported symplectic mapping class group of X splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category D(Y) (the three-dimensional ‘affine A₁ -case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.

Cited by

• Plumbings of lens spaces and crepant resolutions of compound A_n singularities
• Autoequivalences and stability conditions on a degenerate K3 surface
• Spherical and Semibrick Classifications
• Simple complexes on a flopped curve

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