Symplectomorphisms and spherical objects in the conifold smoothing

TL;DR

This paper studies symplectomorphisms and spherical objects in the conifold smoothing and resolution, revealing the infinite complexity of the symplectic mapping class group and classifying key objects in derived categories.

Contribution

It proves the symplectic mapping class group of the conifold smoothing has an infinite rank free subgroup and classifies spherical objects in the derived category of the resolution.

Findings

The symplectic mapping class group is infinitely generated.

The classification of spherical objects in the derived category of the conifold resolution.

The results rely on mirror symmetry techniques.

Abstract

Let $X$ denote the `conifold smoothing', the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$, or equivalently the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the `conifold resolution', by which we mean the complement of a smooth divisor in $\mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathbb{P}^1$. 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 $A_1$-case'). Our results build on work of Chan-Pomerleano-Ueda and Toda, and both theorems make essential use of working on the `other side' of the mirror.