Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2301.10525

Symplectomorphisms and spherical objects in the conifold smoothing

Ailsa Keating·Ivan Smith
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.SGmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ailsa Keating, Ivan Smith. 2026-04-14. Symplectomorphisms and spherical objects in the conifold smoothing. https://doi.org/10.1112/s0010437x24007425

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality

This paper relates the minimal Harder--Narasimhan slope to the surjectivity of real-linear Cauchy--Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

math.SG↗
arXiv2026

Welschinger invariants and the Conway polynomial

Welschinger showed that counts of connected holomorphic disks with Lagrangian boundary in symplectic 6-manifolds, meeting at least one boundary constraint, can be made invariant by correcting them with counts of disconnected disks weighted by certain "self-linking" numbers. We show his invariant is the lowest order term in an all-genus curve count where curves are weighted by the Conway polynomials of their boundaries. This in turn is a specialization of the skein-valued curve count, but can be defined without the 4-chain and vector field used in that setup.

math.SG↗
arXiv2026

KAM splittings and equidistributed periodic orbits for stable hypersurfaces

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

math.SG↗