Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2411.06012

Symplectic Hodge Theory on Lie Algebroids

Shane Rankin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We explore the natural analogues of the Brylinksi condition, Strong Lefschetz condition, and $d\delta$-lemma in Symplectic Geometry originally explored by Brylinksi, Mathieu, Yan, and Guillemin in the Symplectic Lie Algebroid case. The equivalence of the three conditions is re-established as a purely algebraic statement along with a primitive notion of the $d\delta$-lemma established by Tseng, Yau, and Ho. We then apply this algebraic theory to the desired geometric setting to show that these analogues hold, and finally provide a few classes of examples.

Explore related subjects

math.SGmath.DG

Keep this discovery

BibTeXRIS

Shane Rankin. 2024-11-08. Symplectic Hodge Theory on Lie Algebroids. https://arxiv.org/abs/2411.06012

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

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗